# Approximation of solutions to multidimensional parabolic equations by approximate approximations

*Authors*

- Lanzara, Flavia
- Maz'ya, Vladimir
- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 65D32 41A63 41A30 65-05

*Keywords*

- higher dimensions, parabolic equation, heat potential, separated representations, tensor product approximation

*Appeared in*

- Appl. Comput. Harmon. Anal., 41 (2016) pp. 749--767.

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# Accelerated rogue solitons triggered by background radiation

*Authors*

- Demircan, Ayhan
- Morgner, Uwe
- Amiranashvili, Shalva

ORCID: 0000-0002-8132-882X - Brée, Carsten
- Steinmeyer, Günter

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Re 42.65.Tg 42.81.Dp

*Keywords*

- Rogue waves, Pulse compression, Temporal solitons, Optical event horizons

*Appeared in*

- J. Opt., 18 (2016) pp. 1--37.

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# A boundary control problem for the pure Cahn--Hilliard equation with dynamic boundary conditions

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K55 35K50 82C26

*Keywords*

- Cahn--Hilliard equation, dynamic boundary conditions, phase separation, singular potentials, optimal control, optimality conditions

*Abstract*

A boundary control problem for the pure Cahn--Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first-order necessary conditions for optimality are proved.

*Appeared in*

- Adv. Nonlinear Anal., 4 (2015) pp. 311--325.

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# A reduced-order modeling for efficient design study of artificial valve in enlarged ventricular outflow tracts

*Authors*

- Caiazzo, Alfonso

ORCID: 0000-0002-7125-8645 - Guibert, Romain
- Vignon-Clementel, Irene E.

*2010 Mathematics Subject Classification*

- 62Z05 74L15 76Z05

*Keywords*

- Device design, percutaneous pulmonary valve replacement, proper orthogonal decomposition, finite element method, blood flow CFD, repaired Tetralogy of Fallot

*Abstract*

A computational approach is proposed for efficient design study of a reducer stent to be percutaneously implanted in enlarged right ventricular outflow tracts (RVOT). The need for such a device is driven by the absence of bovine or artificial valves which could be implanted in these RVOT to replace the absent or incompetent native valve, as is often the case over time after Tetralogy of Fallot repair. Hemodynamics are simulated in the stented RVOT via a reduce order model based on proper orthogonal decomposition (POD), while the artificial valve is modeled as a thin resistive surface. The reduced order model is obtained from the numerical solution on a reference device configuration, then varying the geometrical parameters (diameter) for design purposes. To validate the approach, forces exerted on the valve and on the reducer are monitored, varying with geometrical parameters, and compared with the results of full CFD simulations. Such an approach could also be useful for uncertainty quantification.

*Appeared in*

- Comput. Methods Biomech. Biomed. Engin., 19 (2016) pp. 1314--1318.

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# Analysis of algebraic flux correction schemes

*Authors*

- Barrenechea, Gabriel R.
- John, Volker

ORCID: 0000-0002-2711-4409 - Knobloch, Petr

*2010 Mathematics Subject Classification*

- 65N12 65N30

*Keywords*

- algebraic flux correction method, linear boundary value problem, well-posedness, discrete maximum principle, convergence analysis, convection-diffusion-reaction equations

*Abstract*

A family of algebraic flux correction schemes for linear boundary value problems in any space dimension is studied. These methods' main feature is that they limit the fluxes along each one of the edges of the triangulation, and we suppose that the limiters used are symmetric. For an abstract problem, the existence of a solution, existence and uniqueness of the solution of a linearized problem, and an a priori error estimate, are proved under rather general assumptions on the limiters. For a particular (but standard in practice) choice of the limiters, it is shown that a local discrete maximum principle holds. The theory developed for the abstract problem is applied to convection-diffusion-reaction equations, where in particular an error estimate is derived. Numerical studies show its sharpness.

*Appeared in*

- SIAM J. Numer. Anal., 54 (2016) pp. 2427--2451.

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# Analysis of a full space-time discretization of the Navier--Stokes equations by a local projection stabilization method

*Authors*

- Ahmed, Naveed

ORCID: 0000-0002-9322-0373 - Rebollo, Tomás Chacón
- John, Volker

ORCID: 0000-0002-2711-4409 - Rubino, Samuele

*2010 Mathematics Subject Classification*

- 65M12 65M60 76D05

*Keywords*

- evolutionary incompressible Navier--Stokes equations, high order term-by-term LPS scheme, finite element error analysis, high Reynolds number flows

*DOI*

*Abstract*

A finite element error analysis of a local projection stabilization (LPS) method for the time-dependent Navier--Stokes equations is presented. The focus is on the high-order term-by-term stabilization method that has one level, in the sense that it is defined on a single mesh, and in which the projection-stabilized structure of standard LPS methods is replaced by an interpolation-stabilized structure. The main contribution is on proving, theoretically and numerically, the optimal convergence order of the arising fully discrete scheme. In addition, the asymptotic energy balance is obtained for slightly smooth flows. Numerical studies support the analytical results and illustrate the potential of the method for the simulation of turbulent flows. Smooth unsteady flows are simulated with optimal order of accuracy.

*Appeared in*

- IMA J. Numer. Anal., 37 (2017), pp. 1437--1467, DOI 10.1093/imanum/drw048 .

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# Adaptive SDE based interpolation for random PDEs

*Authors*

- Anker, Felix
- Bayer, Christian

ORCID: 0000-0002-9116-0039 - Eigel, Martin
- Neumann, Johannes
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*Keywords*

- random PDE, stochastic differential equation, Feynman-Kac, interpolation, finite element, a posteriori error estimator, adaptive method, Euler Maruyama

*Abstract*

A numerical method for the fully adaptive sampling and interpolation of PDE with random data is presented. It is based on the idea that the solution of the PDE with stochastic data can be represented as conditional expectation of a functional of a corresponding stochastic differential equation (SDE). The physical domain is decomposed subject to a non-uniform grid and a classical Euler scheme is employed to approximately solve the SDE at grid vertices. Interpolation with a conforming finite element basis is employed to reconstruct a global solution of the problem. An a posteriori error estimator is introduced which provides a measure of the different error contributions. This facilitates the formulation of an adaptive algorithm to control the overall error by either reducing the stochastic error by locally evaluating more samples, or the approximation error by locally refining the underlying mesh. Numerical examples illustrate the performance of the presented novel method.

*Appeared in*

- Int. J. Uncertain. Quantif., 7 (2017), pp. 189--205; changed title: A fully adaptive interpolated stochastic sampling method for linear random PDEs

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# A random cloud model for the Wigner equation

*Authors*

- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 60J25 81Q05

*Keywords*

- Wigner equation, probabilistic representation, stochastic particle model, piecewise deterministic Markov process

*Abstract*

A probabilistic model for the Wigner equation is studied. The model is based on a particle system with the time evolution of a piecewise deterministic Markov process. Each particle is characterized by a real-valued weight, a position and a wave-vector. The particle position changes continuously, according to the velocity determined by the wave-vector. New particles are created randomly and added to the system. The main result is that appropriate functionals of the process satisfy a weak form of the Wigner equation.

*Appeared in*

- Kinet. Relat. Models, 9 (2016) pp. 217--235.

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# A random walk model for the Schrödinger equation

*Authors*

- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 35Q41 60J25 81Q05

*Keywords*

- Schrödinger equation, probabilistic representation, random walk model, piecewise deterministic Markov process

*Abstract*

A random walk model for the spatially discretized time-dependent Schrödinger equation is constructed. The model consists of a class of piecewise deterministic Markov processes. The states of the processes are characterized by a position and a complex-valued weight. Jumps occur both on the spatial grid and in the space of weights. Between the jumps, the weights change according to deterministic rules. The main result is that certain functionals of the processes satisfy the Schrödinger equation.

*Appeared in*

- Math. Comput. Simulation, 143 (2018), pp. 138--148, DOI 10.1016/j.matcom.2016.07.012 .

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# SDE based regression for random PDEs

*Authors*

- Anker, Felix
- Bayer, Christian

ORCID: 0000-0002-9116-0039 - Eigel, Martin
- Ladkau, Marcel
- Neumann, Johannes
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 35R60 47B80 60H35 65C20 65N12 65N22 65J10 65C05

*Keywords*

- partial differential equations with random coefficients, random PDE, uncertainty quantification, Feynman-Kac, stochastic differential equations, stochastic simulation, stochastic regression, Monte-Carlo, Euler-Maruyama

*Abstract*

A simulation based method for the numerical solution of PDE with random coefficients is presented. By the Feynman-Kac formula, the solution can be represented as conditional expectation of a functional of a corresponding stochastic differential equation driven by independent noise. A time discretization of the SDE for a set of points in the domain and a subsequent Monte Carlo regression lead to an approximation of the global solution of the random PDE. We provide an initial error and complexity analysis of the proposed method along with numerical examples illustrating its behaviour.

*Appeared in*

- SIAM J. Sci. Comput., 39 (2017) pp. A1168--A1200.

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# On the strongly damped wave equation with constraint

*Authors*

- Bonetti, Elena
- Rocca, Elisabetta
- Schimperna, Giulio
- Scala, Riccardo

*2010 Mathematics Subject Classification*

- 35L05 74D10 47H05 46A20

*Keywords*

- wave equation, strong damping, weak solution, maximal monotone operator, duality.

*Abstract*

A weak formulation for the so-called semilinear strongly damped wave equation with constraint is introduced and a corresponding notion of solution is de?ned. The main idea in this approach consists in the use of duality techniques in Sobolev-Bochner spaces, aimed at providing a suitable "relaxation" of the constraint term. A global in time existence result is proved under the natural condition that the initial data have finite "physical" energy.

*Appeared in*

- Comm. Partial Differential Equations, 42 (2017), pp. 1042-1064.

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# A density property for fractional weighted Sobolev spaces

*Authors*

- Dipierro, Serena
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 46E35 35A15

*Keywords*

- Weighted fractional Sobolev spaces, density properties

*Abstract*

Abstract. In this paper we show a density property for fractional weighted Sobolev spaces. That is, we prove that any function in a fractional weighted Sobolev space can be approximated by a smooth function with compact support. The additional difficulty in this nonlocal setting is caused by the fact that the weights are not necessarily translation invariant.

*Appeared in*

- Rend. Lincei Mat. Appl., 26 (2015) pp. 397--422.

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# Two convergence results for an alternation maximization procedure

*Authors*

- Andresen, Andreas
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62F10 62J12 62F25 62H12

*Keywords*

- profile maximum likelihood, local linear approximation, spread, local concentration, M-estimation, alternating procedure, EM algorithm

*Abstract*

Andresen and Spokoiny's (2013) ``critical dimension in semiparametric estimation`` provide a technique for the finite sample analysis of profile M-estimators. This paper uses very similar ideas to derive two convergence results for the alternating procedure to approximate the maximizer of random functionals such as the realized log likelihood in MLE estimation. We manage to show that the sequence attains the same deviation properties as shown for the profile M-estimator in Andresen and Spokoiny (2013), i.e. a finite sample Wilks and Fisher theorem. Further under slightly stronger smoothness constraints on the random functional we can show nearly linear convergence to the global maximizer if the starting point for the procedure is well chosen.

*Appeared in*

- J. Mach. Learn. Res., 17 (2016) pp. 1--53.

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# On microscopic origins of generalized gradient structures

*Authors*

- Liero, Matthias

ORCID: 0000-0002-0963-2915 - Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Peletier, Mark A.
- Renger, D. R. Michiel

ORCID: 0000-0003-3557-3485

*2010 Mathematics Subject Classification*

- 35K55 35Q82 49S05 49J40 49J45 60F10 60J25

*Keywords*

- Generalized gradient structure, gradient system, evolutionary Gamma-convergence, energy-dissipation principle, variational evolution, relative entropy, large-deviation principle

*Abstract*

Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures. A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials. A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary Gamma-limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of diffusion over a high barrier, which gives a reaction-diffusion system.

*Appeared in*

- Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), pp. 1--35, DOI 10.3934/dcdss.2017001 .

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# Brownian occupation measures, compactness and large deviations: Pair interaction

*Authors*

- Mukherjee, Chiranjib

*2010 Mathematics Subject Classification*

- 60J65 60J55 60F10

*Keywords*

- Compactness, large deviations, Gibbs measures on Brownian paths, Parabolic Anderson model

*Abstract*

Continuing with the study of compactness and large deviations initiated in citeMV14, we turn to the analysis of Gibbs measures defined on two independent Brownian paths in $R^d$ interacting through a mutual self-attraction. This is expressed by the Hamiltonian $intint_R^2d V(x-y) mu(d x)nu(d y)$ with two probability measures $mu$ and $nu$ representing the occupation measures of two independent Brownian motions. Due to the mixed product of two independent measures, the crucial shift-invariance requirement of citeMV14 is slightly lost. However, such a mixed product of measures inspires a compactification of the quotient space of orbits of product measures, which is structurally slightly different from the one introduced in citeMV14. The orbits of the product of independent occupation measures are embedded in such a compactfication and a strong large deviation principle for these objects enables us to prove the desired asymptotic localization properties of the joint behavior of two independent paths under the Gibbs transformation. As a second application, we study the spatially smoothened parabolic Anderson model in $R^d$ with white noise potential and provide a direct computation of the annealed Lyapunov exponents of the smoothened solutions when the smoothing parameter goes to $0$.

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# Efficient current injection into single quantum dots through oxide-confined pn-diodes

*Authors*

- Kantner, Markus
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Koprucki, Thomas

ORCID: 0000-0001-6235-9412 - Schulze, Jan-Hindrik
- Strittmatter, André
- Wünsche, Hans-Jürgen

*2010 Mathematics Subject Classification*

- 65N08, 65Z05

*2008 Physics and Astronomy Classification Scheme*

- 85.30.De, 85.60.Bt, 85.60.Jb

*Keywords*

- single-photon emitters, semiconductor device simulation, current confinement

*Abstract*

Current injection into single quantum dots embedded in vertical pn-diodes featuring oxide apertures is analyzed in the low-injection regime suitable for single-photon emitters. Experimental and theoretical evidence is found for a rapid lateral spreading of the carriers after passing the oxide aperture in the conventional pin-design. By an alternative design employing p-doping up to the oxide aperture the current spreading can be suppressed resulting in an enhanced current confinement and increased injection efficiencies, both, in the continuous wave and under pulsed excitation.

*Appeared in*

- IEEE Trans. Electron Devices, 63 (2016), pp. 2036--2042.

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# Dynamical systems with multiple, long delayed feedbacks: Multiscale analysis and spatio-temporal equivalence

*Authors*

- Yanchuk, Serhiy
- Giacomelli, Giovanni

*2010 Mathematics Subject Classification*

- 34K17 34K23 34K18

*2008 Physics and Astronomy Classification Scheme*

- 89.75.Kd, 02.30.Ks, 05.45.-a

*Keywords*

- Two delays, long delays, spatio-temporal representation, amplitude equations, Ginzburg-Landau equation, turbulence, spirals, defects

*Abstract*

Dynamical systems with multiple, hierarchically long delayed feedback are introduced and studied. Focusing on the phenomenological model of a Stuart-Landau oscillator with two feedbacks, we show the multiscale properties of its dynamics and demonstrate them by means of a space-time representation. For sufficiently long delays, we derive a normal form describing the system close to the destabilization. The space and temporal variables, which are involved in the space-time representation, correspond to suitable timescales of the original system. The physical meaning of the results, together with the interpretation of the description at different scales, is presented and discussed. In particular, it is shown how this representation uncovers hidden multiscale patterns such as spirals or spatiotemporal chaos. The effect of the delays size and the features of the transition between small to large delays is also analyzed. Finally, we comment on the application of the method and on its extension to an arbitrary, but finite, number of delayed feedback terms.

*Appeared in*

- Phys. Rev. E, 92 (2015) pp. 042903/1--042903/12.

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# Simultaneous statistical inference for epigenetic data

*Authors*

- Schildknecht, Konstantin
- Olek, Sven
- Dickhaus, Thorsten

*2010 Mathematics Subject Classification*

- 62J15 62G09 62P10

*Keywords*

- beta-value, closed test principle, family-wise error rate, methylation, multiple testing, multivariate statistics, permutation test, relative effect

*Abstract*

Epigenetic research leads to complex data structures. Since parametric model assumptions for the distribution of epigenetic data are hard to verify we introduce in the present work a nonparametric statistical framework for two-group comparisons. Furthermore, epigenetic analyses are often performed at various genetic loci simultaneously. Hence, in order to be able to draw valid conclusions for specific loci, an appropriate multiple testing correction is necessary. Finally, with technologies available for the simultaneous assessment of many interrelated biological parameters (such as gene arrays), statistical approaches also need to deal with a possibly unknown dependency structure in the data. Our statistical approach to the nonparametric comparison of two samples with independent multivariate observables is based on recently developed multivariate multiple permutation tests. We adapt their theory in order to cope with families of hypotheses regarding relative effects. Our results indicate that the multivariate multiple permutation test keeps the pre-assigned type I error level for the global null hypothesis. In combination with the closure principle, the family-wise error rate for the simultaneous test of the corresponding locus/parameter-specific null hypotheses can be controlled. In applications we demonstrate that group differences in epigenetic data can be detected reliably with our methodology.

*Appeared in*

- PLOS ONE, 10 (2015) pp. e0125587/1--e0125587/15.

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# Improving accuracy and temporal resolution of learning curve estimation for within- and across-session analysis

*Authors*

- Deliano, Matthias
- Tabelow, Karsten

ORCID: 0000-0003-1274-9951 - König, Reinhard
- Polzehl, Jörg

ORCID: 0000-0001-7471-2658

*2010 Mathematics Subject Classification*

- 62P10 62G05 62G15

*Abstract*

Estimation of learning curves is ubiquitously based on proportions of correct responses within moving trial windows. In this approach, it is tacitly assumed that learning performance is constant within the moving windows, which, however, is often not the case. In the present study we demonstrate that violations of this assumption lead to systematic errors in the analysis of learning curves, and we explored the dependency of these errors on window size, different statistical models, and learning phase. To reduce these errors for single subjects as well as on the population level, we propose adequate statistical methods for the estimation of learning curves and the construction of confidence intervals, trial by trial. Applied to data from a shuttle-box avoidance experiment with Mongolian gerbils, our approach revealed performance changes occurring at multiple temporal scales within and across training sessions which were otherwise obscured in the conventional analysis. The proper assessment of the behavioral dynamics of learning at a high temporal resolution clarified and extended current descriptions of the process of avoidance learning. It further disambiguated the interpretation of neurophysiological signal changes recorded during training in relation to learning.

*Appeared in*

- PLOS ONE, 11 (2016) pp. e0157355/1--e0157355/23, DOI 10.1371/journal.pone.0157355 .

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# Finite element methods for the incompressible Stokes equations with variable viscosity

*Authors*

- John, Volker

ORCID: 0000-0002-2711-4409 - Kaiser, Kristine
- Novo, Julia

*2010 Mathematics Subject Classification*

- 65N30

*Keywords*

- Incompressible Stokes equations, variable viscosity, finite element error analysis, dependency on the viscosity

*Abstract*

Finite element error estimates are derived for the incompressible Stokes equations with variable viscosity. The ratio of the supremum and the infimum of the viscosity appears in the error bounds. Numerical studies show that this ratio can be observed sometimes. However, often the numerical results show a weaker dependency on the viscosity.

*Appeared in*

- ZAMM Z. Angew. Math. Mech., 96 (2016) pp. 205--216.

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# Asymptotics and stability of a periodic solution to a singularly perturbed parabolic problem in case of a double root of the degenerate equation

*Authors*

- Butuzov, Valentin F.
- Nefedov, Nikolai N.
- Recke, Lutz
- Schneider, Klaus R.

*2010 Mathematics Subject Classification*

- 35B25 35B10 35K20 35K57

*Keywords*

- singularly perturbed reaction-diffusion equation, double root of the degenerate equation, initial boundary value problem, asymptotic expansion, asymptotically stable periodic solution, region of attraction

*Abstract*

For a singularly perturbed parabolic problem with Dirichlet conditions we prove the existence of a solution periodic in time and with boundary layers at both ends of the space interval in the case that the degenerate equation has a double root. We construct the corresponding asymptotic expansion in the small parameter. It turns out that the algorithm of the construction of the boundary layer functions and the behavior of the solution in the boundary layers essentially differ from that ones in case of a simple root. We also investigate the stability of this solution and the corresponding region of attraction.

*Appeared in*

- Autom. Control Comput Sci., 51:7 (2017) pp. 606--613.

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# Pricing under rough volatility

*Authors*

- Bayer, Christian

ORCID: 0000-0002-9116-0039 - Friz, Peter

ORCID: 0000-0003-2571-8388 - Gatheral, Jim

*2010 Mathematics Subject Classification*

- 91B25 62P05

*Keywords*

- Stochastic volatility, Fractional Brownian motion, Bergomi model

*Abstract*

From an analysis of the time series of volatility using recent high frequency data, Gatheral, Jaisson and Rosenbaum [SSRN 2509457, 2014] previously showed that log-volatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale. The resulting Rough Fractional Stochastic Volatility (RFSV) model is remarkably consistent with financial time series data. We now show how the RFSV model can be used to price claims on both the underlying and integrated volatility. We analyze in detail a simple case of this model, the rBergomi model. In particular, we find that the rBergomi model fits the SPX volatility markedly better than conventional Markovian stochastic volatility models, and with fewer parameters. Finally, we show that actual SPX variance swap curves seem to be consistent with model forecasts, with particular dramatic examples from the weekend of the collapse of Lehman Brothers and the Flash Crash.

*Appeared in*

- Quant. Finance, 16 (2016) pp. 887--904.

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# Transient pulse compression at a group velocity horizon

*Authors*

- Babushkin, Ihar
- Amiranashvili, Shalva

ORCID: 0000-0002-8132-882X - Brée, Carsten
- Morgner, Uwe
- Steinmeyer, Günter
- Demircan, Ayhan

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Re 42.65.Tg 42.81.Dp

*Keywords*

- Chirp, Pulse compression, Temporal solitons, Optical event horizons.

*Abstract*

Group-velocity matched cross-phase modulation between a fundamental soliton and a dispersive wave-packet has been previously suggested for optical switching applications similar to an optical transistor. Moreover, the nonlinear interaction in the resulting groupvelocity horizon can be exploited for adiabatic compression of the soliton down into the fewcycle regime. Here we show that both mechanisms can be combined. In such a transient compressor, parameters of the dispersive wave may then serve to actively control the soliton compression and adjust the pulse duration in the presence of disturbances. While a certain amount of control is already enabled by the delay between soliton and dispersive wave, the means of controlling the compression process are substantially enhanced by additionally manipulating the chirp of the dispersive wave. Moreover, controlling the chirp of the dispersive wave also enables correction for limitations of the compression scheme due to a self-frequency shift of the soliton or for uncompensated dispersion in the scheme. This substantially widens the practicality of the compression scheme and other applications of the highly efficient nonlinear interaction at the group-velocity horizon.

*Appeared in*

- IEEE Photon. J., 8 (2016) pp. 7803113/1--7803113/14, DOI 10.1109/JPHOT.2016.2570001 .

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# Improved dual meshes using Hodge-optimized triangulations for electromagnetic problems

*Authors*

- Schlundt, Rainer

ORCID: 0000-0002-4424-4301

*2010 Mathematics Subject Classification*

- 35Q61 65F10 65F15 65N22 65N50

*Keywords*

- Optimal triangulations, discrete Hodge star, Maxwell's equations, finite integration technique, microcell method, linear algebraic equations

*Abstract*

Hodge-optimized triangulations (HOT) can optimize the dual mesh alone or both the primal and dual meshes. They make them more self-centered while keeping the primal-dual orthogonality. The weights are optimized in order to improve one or more of the discrete Hodge stars. Using the example of Maxwell's equations we consider academic examples to demonstrate the generality of the approach.

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# Variational approaches and methods for dissipative material models with multiple scales

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 35Q74 47J35 49Sxx 74C15

*Keywords*

- Generalized gradient systems, energy-dissipation principle, evolutionary Gamma convergence, energeitc solutions, rate-independent systems, balanced-viscosity solutions, finite-strain elastoplasticity, laminate evolution

*Abstract*

In a first part we consider evolutionary systems given as generalized gradient systems and discuss various variational principles that can be used to construct solutions for a given system or to derive the limit dynamics for multiscale problems. These multiscale limits are formulated in the theory of evolutionary Gamma-convergence. On the one hand we consider the a family of viscous gradient system with quadratic dissipation potentials and a wiggly energy landscape that converge to a rate-independent system. On the other hand we show how the concept of Balanced-Viscosity solution arise as in the vanishing-viscosity limit. As applications we discuss, first, the evolution of laminate microstructures in finite-strain elastoplasticity and, second, a two-phase model for shape-memory materials, where H-measures are used to construct the mutual recovery sequences needed in the existence theory.

*Appeared in*

- A. Mielke, Chapter 5: Variational Approaches and Methods for Dissipative Material Models with Multiple Scales, in: Analysis and Computation of Microstructure in Finite Plasticity, S. Conti, K. Hackl, eds., vol. 78 of Lecture Notes in Applied and Computational Mechanics, Springer International Publishing, Heidelberg et al., 2015, pp. 125--155

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# p-Laplace thermistor modeling of electrothermal feedback in organic semiconductors

*Authors*

- Liero, Matthias

ORCID: 0000-0002-0963-2915 - Koprucki, Thomas

ORCID: 0000-0001-6235-9412 - Fischer, Axel
- Scholz, Reinhard
- Glitzky, Annegret

*2010 Mathematics Subject Classification*

- 35J92 65M08 35D30 35G60 35J57 35Q79 80M12 80A20

*Keywords*

- p-Laplace, stationary thermistor model, nonlinear coupled system, finite-volume approximation, existence and boundedness, self-heating, Arrhenius-like conductivity law, organic light-emitting diode

*Abstract*

In large-area Organic Light-Emitting Diodes (OLEDs) spatially inhomogeneous luminance at high power due to inhomogeneous current flow and electrothermal feedback can be observed. To describe these self-heating effects in organic semiconductors we present a stationary thermistor model based on the heat equation for the temperature coupled to a p-Laplace-type equation for the electrostatic potential with mixed boundary conditions. The p-Laplacian describes the non-Ohmic electrical behavior of the organic material. Moreover, an Arrhenius-like temperature dependency of the electrical conductivity is considered. We introduce a finite-volume scheme for the system and discuss its relation to recent network models for OLEDs. In two spatial dimensions we derive a priori estimates for the temperature and the electrostatic potential and prove the existence of a weak solution by Schauder's fixed point theorem.

*Appeared in*

- ZAMP Z. Angew. Math. Phys., 66 (2015) pp. 2957--2977.

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# Brownian occupation measures, compactness and large deviations

*Authors*

- Mukherjee, Chiranjib
- Varadhan, S. R. Srinivasa

*2010 Mathematics Subject Classification*

- 60J65 60J55 60F10

*Keywords*

- Brownian occupation measures, shift compactness, large deviations

*Abstract*

In proving large deviation estimates, the lower bound for open sets and upper bound for compact sets are essentially local estimates. On the other hand, the upper bound for closed sets is global and compactness of space or an exponential tightness estimate is needed to establish it. In dealing with the occupation measure $L_t(A)=frac1tint_0^t1_A(W_s) d s$ of the $d$ dimensional Brownian motion, which is not positive recurrent, there is no possibility of exponential tightness. The space of probability distributions $mathcal M_1(R^d)$ can be compactified by replacing the usual topology of weak c onvergence by the vague toplogy, where the space is treated as the dual of continuous functions with compact support. This is essentially the one point compactification of $R^d$ by adding a point at $infty$ that results in the compactification of $mathcal M_1(R^d)$ by allowing some mass to escape to the point at $infty$. If one were to use only test functions that are continuous and vanish at $infty$ then the compactification results in the space of sub-probability distributions $mathcal M_le 1(R^d)$ by ignoring the mass at $infty$. The main drawback of this compactification is that it ignores the underlying translation invariance. More explicitly, we may be interested in the space of equivalence classes of orbits $widetildemathcal M_1=widetildemathcal M_1(R^d)$ under the action of the translation group $R^d$ on $mathcal M_1(R^d)$. There are problems for which it is natural to compactify this space of orbits. We will provide such a compactification, prove a large deviation principle there and give an application to a relevant problem.

*Appeared in*

- Ann. Probab., 44 (2016), pp. 3934--3964.

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# Free energy, free entropy, and a gradient structure for thermoplasticity

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 35Q79 37D35 74C10 74F05 82B35

*Keywords*

- Generalized gradient systems, GENERIC, variational formulations, incremental minimization, entropy as driving functional, primal and dual, entropy-production potential, thermodynamic modeling, viscoplasticity

*Abstract*

In the modeling of solids the free energy, the energy, and the entropy play a central role. We show that the free entropy, which is defined as the negative of the free energy divided by the temperature, is similarly important. The derivatives of the free energy are suitable thermodynamical driving forces for reversible (i.e. Hamiltonian) parts of the dynamics, while for the dissipative parts the derivatives of the free entropy are the correct driving forces. This difference does not matter for isothermal cases nor for local materials, but it is relevant in the non-isothermal case if the densities also depend on gradients, as is the case in gradient thermoplasticity.

Using the total entropy as a driving functional, we develop gradient structures for quasistatic thermoplasticity, which again features the role of the free entropy. The big advantage of the gradient structure is the possibility of deriving time-incremental minimization procedures, where the entropy-production potential minus the total entropy is minimized with respect to the internal variables and the temperature.

We also highlight that the usage of an auxiliary temperature as an integrating factor in Yang/Stainier/Ortiz "A variational formulation of the coupled thermomechanical boundary-value problem for general dissipative solids" (*J. Mech. Physics Solids*, 54, 401-424, 2006) serves exactly the purpose to transform the reversible driving forces, obtained from the free energy, into the needed irreversible driving forces, which should have been derived from the free entropy. This reconfirms the fact that only the usage of the free entropy as driving functional for dissipative processes allows us to derive a proper variational formulation.

*Appeared in*

- Innovative Numerical Approaches for Multi-Field and Multi-Scale Problems. In Honor of Michael Ortiz's 60th Birthday., K. Weinberg, A. Pandolfi, eds., vol. 81 of Lecture Notes in Applied and Computational Mechanics, Springer International Publishing Switzerland, 2016, pp. 135--160

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# Sliding modes for a phase-field system

*Authors*

- Barbu, Viorel
- Colli, Pierluigi
- Gilardi, Gianni
- Marinoschi, Gabriela
- Rocca, Elisabetta

*2010 Mathematics Subject Classification*

- 34B15 82B26 34H05 93B52

*Keywords*

- phase field system, nonlinear boundary value problems, phase transition, sliding mode control, state-feedback control law

*Abstract*

In the present contribution the sliding mode control (SMC) problem for a phase-field model of Caginalp type is considered. First we prove the well-posedness and some regularity results for the phase-field type state systems modified by the state- feedback control laws. Then, we show that the chosen SMC laws force the system to reach within finite time the sliding manifold (that we chose in order that one of the physical variables or a combination of them remains constant in time). We study three different types of feedback control laws: the first one appears in the internal energy balance and forces a linear combination of the temperature and the phase to reach a given (space dependent) value, while the second and third ones are added in the phase relation and lead the phase onto a prescribed target $phi^*$. While the control law is non-local in space for the first two problems, it is local in the third one, i.e., its value at any point and any time just depends on the value of the state.

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# Option pricing in affine generalized Merton models

*Authors*

- Bayer, Christian
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 91G60 65C30

*Keywords*

- Affine processes, Merton jump-model, approximate affine characteristic function

*Abstract*

In this article we consider affine generalizations of the Merton jump diffusion model Merton (1976) and the respective pricing of European options. On the one hand, the Brownian motion part in the Merton model may be generalized to a log-Heston model, and on the other hand, the jump part may be generalized to an affine process with possibly state dependent jumps. While the characteristic function of the log-Heston component is known in closed form, the characteristic function of the second component may be unknown explicitly. For the latter component we propose an approximation procedure based on the method introduced in Belomestny, Kampen, Schoenmakers (2009). We conclude with some numerical examples.

*Appeared in*

- J.G.M. Schoenmakers, Ch. Bayer, Option pricing in affine generalized Merton models, J. Kallsen, A. Papapantoleon , eds., Springer Proceedings in Mathematics & Statistics, Springer International Publishing , Switzerland, 2016, pp. 219--239

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# Extremes of some Gaussian random interfaces

*Authors*

- Chiarini, Alberto
- Cipriani, Alessandra
- Hazra, Rajat Subhra

*2010 Mathematics Subject Classification*

- 60K35 60G15 60G70

*Keywords*

- Extreme value theory, Gaussian random interfaces, Stein-Chen method

*Abstract*

In this article we give a general criterion for some dependent Gaussian models to belong to maximal domain of attraction of Gumbel, following an application of the Stein-Chen method studied in citeAGG. We also show the convergence of the associated point process. As an application, we show the conditions are satisfied by some of the well-known supercritical Gaussian interface models, namely, membrane model, massive and massless discrete Gaussian free field, fractional Gaussian free field.

*Appeared in*

- J. Statist. Phys., 165 (2016) pp. 521--544.

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# Classification of coupled dynamical systems with multiple delays: Finding the minimal number of delays

*Authors*

- Lücken, Leonhard
- Pade, Jan Philipp
- Knauer, Kolja

*2010 Mathematics Subject Classification*

- 34K17 34K20 37L15, 37L05

*Keywords*

- multiple delays, networks with delay, coupled dynamical systems, delay differential equations, dynamical equivalence, semidynamical systems, cycle space

*Abstract*

In this article we study networks of coupled dynamical systems with time-delayed connections. If two such networks hold different delays on the connections it is in general possible that they exhibit different dynamical behavior as well. We prove that for particular sets of delays this is not the case. To this aim we introduce a componentwise timeshift transformation (CTT) which allows to classify systems which possess equivalent dynamics, though possibly different sets of connection delays. In particular, we show for a large class of semiflows (including the case of delay differential equations) that the stability of attractors is invariant under this transformation. Moreover we show that each equivalence class which is mediated by the CTT possesses a representative system in which the number of different delays is not larger than the cycle space dimension of the underlying graph. We conclude that the 'true' dimension of the corresponding parameter space of delays is in general smaller than it appears at first glance.

*Appeared in*

- SIAM J. Appl. Dyn. Syst., 14 (2015) pp. 286--304.

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# The effect on Fisher--KPP propagation in a cylinder with fast diffusion on the boundary

*Authors*

- Rossi, Luca
- Tellini, Andrea
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35K57 35B40 35K40 35B53

*Keywords*

- KPP equations, reaction-diffusion systems, different spatial dimensions, asymptotic speed of spreading

*Abstract*

In this paper we consider a reaction-diffusion equation of Fisher-KPP type inside an infinite cylindrical domain in $R^N+1$, coupled with a reaction-diffusion equation on the boundary of the domain, where potentially fast diffusion is allowed. We will study the existence of an asymptotic speed of propagation for solutions of the Cauchy problem associated with such system, as well as the dependence of this speed on the diffusivity at the boundary and the amplitude of the cylinder. When $N=1$ the domain reduces to a strip between two straight lines. This models the effect of two roads with fast diffusion on a strip-shaped field bounded by them.

*Appeared in*

- SIAM J. Math. Anal., 49 (2017), pp. 4595-4624.

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# Homogenization of Cahn--Hilliard-type equations via evolutionary Gamma-convergence

*Authors*

- Liero, Matthias

ORCID: 0000-0002-0963-2915 - Reichelt, Sina

*2010 Mathematics Subject Classification*

- 35B27 49J40 35K55 35K30 35B30 49J45

*Keywords*

- Evolutionary Γ-convergence, gradient systems, homogenization, Cahn-Hilliard equation, evolutionary variational inequality, energy-dissipation principle, two-scale convergence

*Abstract*

In this paper we discuss two approaches to evolutionary Γ-convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ Γ-convergence methods. In the first approach we consider families of uniformly convex energy functionals such that the limit passage of the time-dependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savaré 2010. The second approach uses the equivalent formulation of the gradient system via the energy-dissipation principle and follows the ideas of Sandier and Serfaty 2004. We apply both approaches to rigorously derive homogenization limits for Cahn-Hilliard-type equations. Using the method of weak and strong two-scale convergence via periodic unfolding, we show that the energy and dissipation functionals Γ-converge. In conclusion, we will give specific examples for the applicability of each of the two approaches.

*Appeared in*

- NoDEA Nonlinear Differential Equations Appl., (2018), published online on 29.01.2018, DOI 10.1007/s00030-018-0495-9 .

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# Hölder-estimates for non-autonomous parabolic problems with rough data

*Authors*

- Meinlschmidt, Hannes
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 35B65 35K10 35K15

*Keywords*

- Linear, second order parabolic equations, non-smooth data, Hölder continuity of the solution

*Abstract*

In this paper we establish Hölder estimates for solutions to non-autonomous parabolic equations on non-smooth domains which are complemented with mixed boundary conditions. The corresponding elliptic operators are of divergence type, the coefficient matrix of which depends only measurably on time. These results are in the tradition of the classical book of Ladyshenskaya et al., which also serves as the starting point for our investigations.

*Appeared in*

- Evol. Equ. Control Theory, 5 (2016), pp. 147--184.

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# Generalized gradient flow structure of internal energy driven phase field systems

*Authors*

- Bonetti, Elena
- Rocca, Elisabetta

*2010 Mathematics Subject Classification*

- 74N25 82B26 35A01 35A02

*Keywords*

- gradient flow, phase field systems, existence of weak solutions, uniqueness

*Abstract*

In this paper we introduce a general abstract formulation of a variational thermomechanical model, by means of a unified derivation via a generalization of the principle of virtual powers for all the variables of the system, including the thermal one. In particular, choosing as thermal variable the entropy of the system, and as driving functional the internal energy, we get a gradient flow structure (in a suitable abstract setting) for the whole nonlinear PDE system. We prove a global in time existence of (weak) solutions result for the Cauchy problem associated to the abstract PDE system as well as uniqueness in case of suitable smoothness assumptions on the functionals.

*Appeared in*

- ESAIM Control Optim. Calc. Var. 23 (2017) pp. 1201--1216, changed title: Unified gradient flow structure of phase field systems via a generalized principle of virtual powers.

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# Hölder estimates for parabolic operators on domains with rough boundary

*Authors*

- Disser, Karoline
- Rehberg, Joachim
- ter Elst, A. F. M.

*2010 Mathematics Subject Classification*

- 35K20 35B45 35B65 35B05

*Keywords*

- Parabolic initial boundary value problems, Hölder continuity

*Abstract*

In this paper we investigate linear parabolic, second-order boundary value problems with mixed boundary conditions on rough domains. Assuming only boundedness/ellipticity on the coefficient function and very mild conditions on the geometry of the domain -- including a very weak compatibility condition between the Dirichlet boundary part and its complement -- we prove Hölder continuity of the solution in space and time.

*Appeared in*

- Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), XVII (2017), pp. 65--79.

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# Vanishing viscosities and error estimate for a Cahn--Hilliard type phase field system related to tumor growth

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Rocca, Elisabetta
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35Q92 92C17 35K35 35K57 78M35 35B20 65N15 35R35

*Keywords*

- Tumor growth, Cahn--Hilliard system, reaction-diffusion equation, asymptotic analysis, error estimates

*Abstract*

In this paper we perform an asymptotic analysis for two different vanishing viscosity coefficients occurring in a phase field system of Cahn--Hilliard type that was recently introduced in order to approximate a tumor growth model. In particular, we extend some recent results obtained in [Colli-Gilardi-Hilhorst 2015], letting the two positive viscosity parameters tend to zero independently from each other and weakening the conditions on the initial data in such a way as to maintain the nonlinearities of the PDE system as general as possible. Finally, under proper growth conditions on the interaction potential, we prove an error estimate leading also to the uniqueness result for the limit system.

*Appeared in*

- Nonlinear Anal. Real World Appl., 26 (2015) pp. 93--108.

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# Anisotropic finite element mesh adaptation via higher dimensional embedding

*Authors*

- Dassi, Franco
- Si, Hang
- Perotto, Simona
- Streckenbach, Timo

*2010 Mathematics Subject Classification*

- 65M50

*Keywords*

- anisotropic meshes, mesh optimization, partial differential equations

*Abstract*

In this paper we provide a novel anisotropic mesh adaptation technique for adaptive finite element analysis. It is based on the concept of higher dimensional embedding, which was exploited in [1-4] to obtain an anisotropic curvature adapted mesh that fits a complex surface in ℝ^{3}. In the context of adaptive finite element simulation, the solution (which is an unknown function ƒ: Ω ⊂; ℝ^{d} → ℝ) is sought by iteratively modifying a finite element mesh according to a mesh sizing field described via a (discrete) metric tensor field that is typically obtained through an error estimator. We proposed to use a higher dimensional embedding, Φ_{f}(x) := (x_{1}, …, x_{d}, s f (x_{1}, …, x_{d}), s ∇ f (x_{1}, …, x_{d}))^{t}, instead of the mesh sizing field for the mesh adaption. This embedding contains both informations of the function ƒ itself and its gradient. An isotropic mesh in this embedded space will correspond to an anisotropic mesh in the actual space, where the mesh elements are stretched and aligned according to the features of the function ƒ. To better capture the anisotropy and gradation of the mesh, it is necessary to balance the contribution of the components in this embedding. We have properly adjusted Φ_{f}(**x**) for adaptive finite element analysis. To better understand and validate the proposed mesh adaptation strategy, we first provide a series of experimental tests for piecewise linear interpolation of known functions. We then applied this approach in an adaptive finite element solution of partial differential equations. Both tests are performed on two-dimensional domains in which adaptive triangular meshes are generated. We compared these results with the ones obtained by the software BAMG - a metric-based adaptive mesh generator. The errors measured in the L_{2} norm are comparable. Moreover, our meshes captured the anisotropy more accurately than the meshes of BAMG.

*Appeared in*

- Procedia Engineering, 124 (2015) pp. 265--277.

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# Graph properties for nonlocal minimal surfaces

*Authors*

- Dipierro, Serena
- Savin, Ovidiu
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 49Q05 35R11 53A10

*Keywords*

- Nonlocal minimal surfaces, graph properties, regularity theory

*Abstract*

In this paper we show that a nonlocal minimal surface which is a graph outside a cylinder is in fact a graph in the whole of the space. As a consequence, in dimension $3$, we show that the graph is smooth. The proofs rely on convolution techniques and appropriate integral estimates which show the pointwise validity of an Euler-Lagrange equation related to the nonlocal mean curvature.

*Appeared in*

- Calc. Var. Partial Differ. Equ., 55 (2016) pp. 86/1--86/25.

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# A temperature-dependent phase-field model for phase separation and damage

*Authors*

- Heinemann, Christian
- Kraus, Christiane
- Rocca, Elisabetta
- Rossi, Riccarda

*2010 Mathematics Subject Classification*

- 35D30 74G25 93C55 82B26 74A45

*Keywords*

- damage, phase separation, thermoviscoelasticity, global-in-time entropic weak solutions, existence, time discretization

*Abstract*

In this paper we study a model for phase separation and damage in thermoviscoelastic materials. The main novelty of the paper consists in the fact that, in contrast with previous works in the literature (cf., e.g., [C. Heinemann, C. Kraus: Existence results of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage. Adv. Math. Sci. Appl. 21 (2011), 321--359] and [C. Heinemann, C. Kraus: Existence results for diffuse interface models describing phase separation and damage. European J. Appl. Math. 24 (2013), 179--211]), we encompass in the model thermal processes, nonlinearly coupled with the damage, concentration and displacement evolutions. More in particular, we prove the existence of "entropic weak solutions", resorting to a solvability concept first introduced in [E. Feireisl: Mathematical theory of compressible, viscous, and heat conducting fluids. Comput. Math. Appl. 53 (2007), 461--490] in the framework of Fourier-Navier-Stokes systems and then recently employed in [E. Feireisl, H. Petzeltová, E. Rocca: Existence of solutions to a phase transition model with microscopic movements. Math. Methods Appl. Sci. 32 (2009), 1345--1369], [E. Rocca, R. Rossi: "Entropic" solutions to a thermodynamically consistent PDE system for phase transitions and damage. SIAM J. Math. Anal., 47 (2015), 2519--2586] for the study of PDE systems for phase transition and damage. Our global-in-time existence result is obtained by passing to the limit in a carefully devised time-discretization scheme.

*Appeared in*

- Arch. Ration. Mech. Anal., 225 (2017) pp. 177--247.

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# Optimal control of doubly nonlinear evolution equations governed by subdifferentials without uniqueness of solutions

*Authors*

- Farshbaf Shaker, Mohammad Hassan
- Yamazaki, Noriaki

*2010 Mathematics Subject Classification*

- 35K86 49J20 49J40

*Keywords*

- optimal control, doubly nonlinear evolution equations, subdifferentials, without uniqueness

*Abstract*

In this paper we study an optimal control problem for a doubly nonlinear evolution equation governed by time-dependent subdifferentials. We prove the existence of solutions to our equation. Also, we consider an optimal control problem without uniqueness of solutions to the state system. Then, we prove the existence of an optimal control which minimizes the nonlinear cost functional. Moreover, we apply our general result to some model problem.

*Appeared in*

- System Modeling and Optimization, 27th IFIP TC 7 Conference, CSMO 2015, Sophia Antipolis, France, June 29 - July 3, 2015, Revised Selected Papers, L. Bociu, J.-A. Désidéri, A. Habbal, eds., Springer International Publishing AG, Cham, pp. 261-271

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# On the evolution by fractional mean curvature

*Authors*

- Sáez, Mariel
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R11 35K93 53A10

*Keywords*

- nonlocal mean curvature, geometric motions, evolving surfaces

*Abstract*

In this paper we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions equations for fractional geometric quantities that yield preservation of certain quantities (such as positive fractional curvature) and smoothness of graphical evolutions.

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# Turbulence in the Ott--Antonsen equation for arrays of coupled phase oscillators

*Authors*

- Wolfrum, Matthias
- Gurevich, Svetlana
- Omel'chenko, Oleh

ORCID: 0000-0003-0526-1878

*2010 Mathematics Subject Classification*

- 34C15 37N20 37N25

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Xt, 89.75.Kd

*Keywords*

- coupled oscillators, Kuramoto model, twisted states, turbulence, Ott/Antonsen, Eckhaus scenario

*Abstract*

In this paper we study the transition to synchrony in an one-dimensional array of oscillators with non-local coupling. For its description in the continuum limit of a large number of phase oscillators, we use a corresponding Ott-Antonsen equation, which is an integro-differential equation for the evolution of the macroscopic profiles of the local mean field. Recently, it has been reported that in the spatially extended case at the synchronization threshold there appear partially coherent plane waves with different wave numbers, which are organized in the well-known Eckhaus scenario. In this paper, we show that for Kuramoto-Sakaguchi phase oscillators the phase lag parameter in the interaction function can induce a Benjamin-Feir type instability of the partially coherent plane waves. The emerging collective macroscopic chaos appears as an intermediate stage between complete incoherence and stable partially coherent plane waves. We give an analytic treatment of the Benjamin-Feir instability and its onset in a codimension-two bifurcation in the Ott-Antonsen equation as well as a numerical study of the transition from phase turbulence to amplitude turbulence inside the Benjamin-Feir unstable region.

*Appeared in*

- Nonlinearity, 29 (2016) pp. 257--270.

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# Uniform approximation of the CIR process via exact simulation at random times

*Authors*

- Milstein, Grigori N.
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 65C30 60H35

*Keywords*

- Cox-Ingersoll-Ross process, Sturm-Liouville problem, Bessel functions, confluent hypergeometric equation

*Abstract*

In this paper we uniformly approximate the trajectories of the Cox-Ingersoll-Ross (CIR) process. At a sequence of random times the approximate trajectories will be even exact. In between, the approximation will be uniformly close to the exact trajectory. From a conceptual point of view the proposed method gives a better quality of approximation in a path-wise sense than standard, or even exact simulation of the CIR dynamics at some deterministic time grid.

*Appeared in*

- Adv. Appl. Probab., 48 (2016) pp. 1095--1116.

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# Obstacle mean-field game problem

*Authors*

- Gomes, Diogo A.
- Patrizi, Stefania

*2010 Mathematics Subject Classification*

- 35J87 49L99

*Keywords*

- mean-field games, obstacle problem, penalization method

*Abstract*

In this paper, we introduce and study a first-order mean-field game obstacle problem. We examine the case of local dependence on the measure under assumptions that include both the logarithmic case and power-like nonlinearities. Since the obstacle operator is not differentiable, the equations for first-order mean field game problems have to be discussed carefully. Hence, we begin by considering a penalized problem. We prove this problem admits a unique solution satisfying uniform bounds. These bounds serve to pass to the limit in the penalized problem and to characterize the limiting equations. Finally, we prove uniqueness of solutions.

*Appeared in*

- Interfaces Free Bound., 17 (2015) pp. 55--68.

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# Splitting methods for SPDEs: From robustness to financial engineering, optimal control and nonlinear filtering

*Authors*

- Bayer, Christian

ORCID: 0000-0002-9116-0039 - Oberhauser, Harald

*2010 Mathematics Subject Classification*

- 60H15 60H35 65C30

*Keywords*

- Splitting methots, SPDEs, rough paths, Ninomiya-Victoit method

*Abstract*

In this survey chapter we give an overview of recent applications of the splitting method to stochastic (partial) differential equations, that is, differential equations that evolve under the influence of noise. We discuss weak and strong approximations schemes. The applications range from the management of risk, financial engineering, optimal control and nonlinear filtering to the viscosity theory of nonlinear SPDEs.

*Appeared in*

- Splitting Methods in Communication, Imaging, Science, and Engineering, R. Glowinski, S.J. Osher, W. Yin, eds., Scientific Computation, Springer International P ublishing Switzerland, 2017, pp. 499--539.

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# Thin-film models for viscoelastic liquid bi-layers

*Authors*

- Jachalski, Sebastian
- Münch, Andreas
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 76A20 76Mxx

*Keywords*

- fluid dynamics, viscoelasticty, thin-film models, two-phase flow, asymptotic methods, numerical solution

*Abstract*

In this work we consider a two-layer system of viscoelastic liquids of corotational Jeffreys' type dewetting from a Newtonian liquid substrates. We derive conditions that allow for the first time the asymptotically consistent reduction of the free boundary problem for the two-layer system to a system of coupled thin-film equations that incorporate the full nonlinear viscoelastic rheology. We show that these conditions are controlled by the order of magnitude of the viscosity ratio of the liquid layers and their thickness ratio. For pure Newtonian flow, these conditions lead to a thin-film model that couples a layer with a parabolic flow field to a layer described by elongational flow. For this system we establish asymptotic regimes that relate the viscosity ratio to a corresponding apparent slip. We then use numerical simulations to discuss the characteristic morphological and dynamical properties of viscoelastic films of corotational Jeffreys' type dewetting from a solid as well as liquid substrate.

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# Generalized Post--Widder inversion formula with application to statistics

*Authors*

- Belomestny, Denis
- Mai, Hilmar
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 62G07 65R32

*Keywords*

- Laplace transform, inversion formula, Post-Widder formula, variance-mean mixtures, density estimation

*Abstract*

In this work we derive an inversion formula for the Laplace transform of a density observed on a curve in the complex domain, which generalizes the well known Post-Widder formula. We establish convergence of our inversion method and derive the corresponding convergence rates for the case of a Laplace transform of a smooth density. As an application we consider the problem of statistical inference for variance-mean mixture models. We construct a nonparametric estimator for the mixing density based on the generalized Post-Widder formula, derive bounds for its root mean square error and give a brief numerical example.

*Appeared in*

- J. Math. Anal. Appl., 455 (2017) pp. 89--104.

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# A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media

*Authors*

- Farshbaf Shaker, Mohammad Hassan
- Heinemann, Christian

*2010 Mathematics Subject Classification*

- 35Q74 49J20 35A01 35A02 35D35 35M33 35M87 74A45 74D10 74F99 74H20 74H25 74P99

*Keywords*

- damage processes, phase field model, viscoelasticity, nonlinear parabolic inclusions, well-posedness, optimal control

*Abstract*

In this work we investigate a phase field model for damage processes in two-dimensional viscoelastic media with nonhomogeneous Neumann data describing external boundary forces. In the first part we establish global-in-time existence, uniqueness, a priori estimates and continuous dependence of strong solutions on the data. The main difficulty is caused by the irreversibility of the phase field variable which results in a constrained PDE system. In the last part we consider an optimal control problem where a cost functional penalizes maximal deviations from prescribed damage profiles. The goal is to minimize the cost functional with respect to exterior forces acting on the boundary which play the role of the control variable in the considered model. To this end, we prove existence of minimizers and study a family of "local'' approximations via adapted cost functionals.

*Appeared in*

- Math. Models Methods Appl. Sci., 25 (2015) pp. 2749--2793.

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# Emergence and combinatorial accumulation of jittering regimes in spiking oscillators with delayed feedback

*Authors*

- Klinshov, Vladimir
- Lücken, Leonhard
- Shchapin, Dmitry
- Nekorkin, Vladimir
- Yanchuk, Serhiy

*2010 Mathematics Subject Classification*

- 37G15 37N20 92B25

*2008 Physics and Astronomy Classification Scheme*

- 87.19.ll, 05.45.Xt, 87.19.lr, 89.75.Kd

*Keywords*

- Phase oscillator, delayed feedback, pulsatile feedback, jitter, degenerate bifurcation, PRC

*Abstract*

Interaction via pulses is common in many natural systems, especially neuronal. In this article we study one of the simplest possible systems with pulse interaction: a phase oscillator with delayed pulsatile feedback. When the oscillator reaches a specific state, it emits a pulse, which returns after propagating through a delay line. The impact of an incoming pulse is described by the oscillator's phase reset curve (PRC). In such a system we discover an unexpected phenomenon: for a sufficiently steep slope of the PRC, a periodic regular spiking solution bifurcates with several multipliers crossing the unit circle at the same parameter value. The number of such critical multipliers increases linearly with the delay and thus may be arbitrary large. This bifurcation is accompanied by the emergence of numerous ``jittering'' regimes with non-equal interspike intervals (ISIs). The number of the emergent solutions increases exponentially with the delay. We describe the combinatorial mechanism that underlies the emergence of such a variety of solutions. In particular, we show how each periodic solution consisting of different ISIs implies the appearance of multiple other solutions obtained by rearranging of these ISIs. We show that the theoretical results for phase oscillators accurately predict the behavior of an experimentally implemented electronic oscillator with pulsatile feedback.

*Appeared in*

- Phys. Rev. E, 92 pp. 042914/1--042914/15.

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# Memory and adaptive behaviour in population dynamics: Anti-predator behaviour as a case study

*Authors*

- Pimenov, Alexander
- Kelly, Thomas C.
- Korobeinikov, Andrei
- O'Callaghan, Michael J.
- Rachinskii, Dmitry

*2010 Mathematics Subject Classification*

- 92D25 34C55

*Keywords*

- Bi-stability, Preisach operator, hysteresis, adaptation, predator-prey model, refuge

*Abstract*

Memory enables to forecast future on the basis of experience, and thus, in some form, is principally important for the development of flexible adaptive behaviour by animal communities. To model memory, in this paper we use the concept of hysteresis, which mathematically is described by the Preisach operator. As case study, we consider anti-predator adaptation in the classic Lotka-Volterra predator-prey model. Despite its simplicity, the model allows to naturally incorporate essential features of an adaptive system and memory. Our analysis and simulations show that a system with memory can have a continuum of equilibrium states with non-trivial stability properties.

*Appeared in*

- J. Math. Biol., (2016) pp. 1--27.

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# Bounded-hop percolation

*Authors*

- Hirsch, Christian

*2010 Mathematics Subject Classification*

- 60K35 60D05

*Keywords*

- Ad hoc network, chemical distance, connection probability, continuum percolation

*Abstract*

Motivated by an application in wireless telecommunication networks, we consider a two-type continuum-percolation problem involving a homogeneous Poisson point process of users and a stationary and ergodic point process of base stations. Starting from a randomly chosen point of the Poisson point process,we investigate distribution of the minimum number of hops that are needed to reach some point of the second point process.In the supercritical regime of continuum percolation, we use the close relationship between Euclidean and chemical distance to identify the distributional limit of the rescaled minimum number of hops that are needed to connect a typical Poisson point to a point of the second point process as its intensity tends to infinity. In particular, we obtain an explicit expression for the asymptotic probability that a typical Poisson point connects to a point of the second point process in a given number of hops.

*Appeared in*

- Journal of Applied Probability 53 (2016), 833-845.

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# Modeling high resolution MRI: Statistical issues with low SNR

*Authors*

- Polzehl, Jörg

ORCID: 0000-0001-7471-2658 - Tabelow, Karsten

ORCID: 0000-0003-1274-9951

*2010 Mathematics Subject Classification*

- 62G05 62P10

*Abstract*

Noise is a common issue for all Magnetic Resonance Imaging (MRI) techniques and obviously leads to variability of the estimates in any model describing the data. A number of special MR sequences as well as increasing spatial resolution in MR experiments further diminish the signal-to-noise ratio (SNR). However, with low SNR the expected signal deviates from its theoretical value. Common modeling approaches therefore lead to a bias in estimated model parameters. Adjustments require an analysis of the data generating process and a characterization of the resulting distribution of the imaging data. We provide an adequate quasi-likelihood approach that employs these characteristics. We elaborate on the effects of typical data preprocessing and analyze the bias effects related to low SNR for the example of the diffusion tensor model in diffusion MRI. We then demonstrate that the problem is relevant even for data from the Human Connectome Project, one of the highest quality diffusion MRI data available so far.

*Appeared in*

- J. Amer. Statist. Assoc., 111 (2016), pp. 1480--1490, changed title: Low SNR in diffusion MRI models.

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# Multi-stability and polariton formation in microcavity polaritonic waveguides

*Authors*

- Slavcheva, Gabriela
- Gorbach, Andrey V.
- Pimenov, Alexander
- Vladimirov, Andrei G.
- Skryabin, Dmitry

*2008 Physics and Astronomy Classification Scheme*

- 05.45.-a 42.65.Pc 42.65.Tg

*Keywords*

- polaritons, solitons, microcavities, multi-stability, tilted wavguides

*Abstract*

Nonlinear polaritons in microcavity waveguides are demonstrated to exhibit multi-stable behaviour and rich dynamics, including filamentation and soliton formation. We find that the multi-stability originates from co-existense of different transverse modes of the po- laritonic waveguide. Modulational stability and conditions for multi-mode polariton solitons are studied. Soliton propagation in tilted, relative to the pump momentum, waveguides is demonstrated and a critical tilt angle for the soliton propagation is found.

*Appeared in*

- Opt. Lett., 40 (2015) pp. 1787--1790.

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# Multistable jittering in oscillators with pulsatile delayed feedback

*Authors*

- Klinshov, Vladimir
- Lücken, Leonhard
- Shchapin, Dmitry
- Nekorkin, Vladimir
- Yanchuk, Serhiy

*2010 Mathematics Subject Classification*

- 37G15 37N20 92B25

*2008 Physics and Astronomy Classification Scheme*

- 87.19.ll, 05.45.Xt, 87.19.lr, 89.75.Kd

*Keywords*

- phase oscillator, delayed feedback, pulsatile feedback, jitter, degenerate bifurcation, PRC

*Abstract*

Oscillatory systems with time-delayed pulsatile feedback appear in various applied and theoretical research areas, and received a growing interest in recent years. For such systems, we report a remarkable scenario of destabilization of a periodic regular spiking regime. At the bifurcation point numerous regimes with non-equal interspike intervals emerge. We show that the number of the emerging, so-called ``jittering'' regimes grows emphexponentially with the delay value. Although this appears as highly degenerate from a dynamical systems viewpoint, the ``multi-jitter'' bifurcation occurs robustly in a large class of systems. We observe it not only in a paradigmatic phase-reduced model, but also in a simulated Hodgkin-Huxley neuron model and in an experiment with an electronic circuit.

*Appeared in*

- Phys. Rev. Lett., 114 (2015) pp. 178103/1--178103/5.

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# Optimal L2 velocity error estimate for a modified pressure-robust Crouzeix--Raviart Stokes element

*Authors*

- Linke, Alexander

ORCID: 0000-0002-0165-2698 - Merdon, Christian
- Wollner, Winnifried

*2010 Mathematics Subject Classification*

- 76M10 76D07

*Keywords*

- incompressible Navier--Stokes equations, mixed finite elements, a priori error analysis, duality argument

*Abstract*

Recently, a novel approach for the robust discretization of the incompressible Stokes equations was proposed that slightly modifies the nonconforming Crouzeix--Raviart element such that its velocity error becomes pressure-independent. The modification results in an O(h) consistency error that allows straightforward proofs for the optimal convergence of the discrete energy norm of the velocity and of the L2 norm of the pressure. However, though the optimal convergence of the velocity in the L2 norm was observed numerically, it appeared to be nontrivial to prove. In this contribution, this gap is closed. Moreover, the dependence of the error estimates on the discrete inf-sup constant is traced in detail, which shows that classical error estimates are extremely pessimistic on domains with large aspect ratios. Numerical experiments in 2D and 3D illustrate the theoretical findings.

*Appeared in*

- IMA J. Numer. Anal., 37 (2017), pp. 354--374, DOI 10.1093/imanum/drw019 .

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# On a long range segregation model

*Authors*

- Caffarelli, Luis
- Patrizi, Stefania
- Quitalo, Veronica

*2010 Mathematics Subject Classification*

- 35J60 35R35 35B65 35Q92

*Keywords*

- regularity for viscosity solutions, segregation of populations

*Abstract*

Segregation phenomena occurs in many areas of mathematics and science: from equipartition problems in geometry, to social and biological processes (cells, bacteria, ants, mammals) to finance (sellers and buyers). There is a large body of literature studying segregation models where the interaction between species is punctual. There are many processes though, where the growth of a population at a point is inhibited by the populations in a full area surrounding that point. This work is a first attempt to study the properties of such a segregation process.

*Appeared in*

- J. Eur. Math. Soc., 19 (2017), pp. 3575-3628.

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# More specific signal detection in functional magnetic resonance imaging by false discovery rate control for hierarchically structured systems of hypotheses

*Authors*

- Schildknecht, Konstantin
- Tabelow, Karsten

ORCID: 0000-0003-1274-9951 - Dickhaus, Thorsten

*2010 Mathematics Subject Classification*

- 62J15 62F03 62P10

*Keywords*

- Functional magnetic resonance imaging, grouped hypotheses, multiple hypotheses testing, voxels

*Abstract*

Signal detection in functional magnetic resonance imaging (fMRI) inherently involves the problem of testing a large number of hypotheses. A popular strategy to address this multiplicity is the control of the false discovery rate (FDR). In this work we consider the case where prior knowledge is available to partition the set of all hypotheses into disjoint subsets or families, e. g., by a-priori knowledge on the functionality of certain regions of interest. If the proportion of true null hypotheses differs between families, this structural information can be used to %relax multiplicity and to increase statistical power.

We propose a two-stage multiple test procedure which first excludes those families from the analysis for which there is no strong evidence for containing true alternatives. We show control of the family-wise error rate at this first stage of testing. Then, at the second stage, we proceed to test the hypotheses within each non-excluded family and obtain asymptotic control of the FDR within each family in this second stage. Our main mathematical result is that this two-stage strategy implies asymptotic control of the FDR with respect to all hypotheses.

In simulations we demonstrate the increased power of this new procedure in comparison with established procedures in situations with highly unbalanced families. Finally, we apply the proposed method to simulated and to real fMRI data.

*Appeared in*

- PLOS ONE, 11 (2016) pp. e0149016/1--e0149016/21, DOI 10.1371/journal.pone.0149016 .

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# Noise enhanced coupling between two oscillators with long-term plasticity

*Authors*

- Lücken, Leonhard
- Popovych, Oleksandr V.
- Tass, Peter A.
- Yanchuk, Serhiy

*2010 Mathematics Subject Classification*

- 37N20 92B25 34F05

*2008 Physics and Astronomy Classification Scheme*

- 05.45Xt, 87.19La, 87.18.Tt, 87.19.lw

*Keywords*

- Coupled oscillators, plasticity, noise, multistability, STDP, phase oscillators, Hodkin-Huxley model

*Abstract*

Spike time-dependent plasticity is a fundamental adaptation mechanism of the nervous system. It induces structural changes of synaptic connectivity by regulation of coupling strengths between individual cells depending on their spiking behavior. As a biophysical process its functioning is constantly subjected to natural fluctuations. We study theoretically the influence of noise on a microscopic level by considering only two coupled neurons. Adopting a phase description for the neurons we derive a two-dimensional system which describes the averaged dynamics of the coupling strengths. We show that a multistability of several coupling configurations is possible, where some configurations are not found in systems without noise. Intriguingly, it is possible that a strong bidirectional coupling, which is not present in the noise-free situation, can be stabilized by the noise. This means that increased noise, which is normally expected to desynchronize the neurons, can be the reason for an antagonistic response of the system, which organizes itself into a state of stronger coupling and counteracts the impact of noise. This mechanism, as well as a high potential for multistability, is also demonstrated numerically for a coupled pair of Hodgkin-Huxley neurons.

*Appeared in*

- Phys. Rev. E,
**93**(2016), pp. 032210/1--032210/15.

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# Aging in the GREM-like trap model

*Authors*

- Gayrard, Veronique
- Gün, Onur

*2010 Mathematics Subject Classification*

- 82C44 60G55 60K37 60J27

*Keywords*

- Random walk, random environment, trap models, aging, spin glasses, aging

*Abstract*

The GREM-like trap model is a continuous time Markov jump process on the leaves of a finite volume *L*-level tree whose transition rates depend on a *trapping landscape* built on the vertices of the whole tree. We prove that the natural two-time correlation function of the dynamics ages in the infinite volume limit and identify the limiting function. Moreover, we take the limit *L*→ ∞ of the two-time correlation function of the infinite volume *L*-level tree. The aging behavior of the dynamics is characterized by a collection of clock processes, one for each level of the tree. We show that for any *L*, the joint law of the clock processes converges. Furthermore, any such limit can be expressed through Neveu's continuous state branching process. Hence, the latter contains all the information needed to describe aging in the GREM-like trap model both for finite and infinite levels.

*Appeared in*

- Markov Process. Related Fields, 22 (2016), pp. 165--202.

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# New insights on the interfacial tension of electrochemical interfaces and the Lippmann equation

*Authors*

- Dreyer, Wolfgang
- Guhlke, Clemens
- Landstorfer, Manuel
- Müller, Rüdiger

*2010 Mathematics Subject Classification*

- 78A57 35Q70 35C20

*2008 Physics and Astronomy Classification Scheme*

- 82.45.Mp 68.05.-n 61.25.Mv

*Keywords*

- Lippmann equation, Electrochemistry, liquid-liquid interface, asymptotic analysis

*Abstract*

The Lippmann equation is considered as universal relationship between interfacial tension, double layer charge, and cell potential. Based on the framework of continuum thermo-electrodynamics we provide some crucial new insights to this relation. In a previous work we have derived a general thermodynamic consistent model for electrochemical interfaces, which showed a remarkable agreement to single crystal experimental data. Here we apply the model to a curved liquid metal electrode. If the electrode radius is large compared to the Debye length, we apply asymptotic analysis methods and obtain the Lippmann equation. We give precise definitions of the involved quantities and show that the interfacial tension of the Lippmann equation is composed of the surface tension of our general model, and contributions arising from the adjacent space charge layers. This finding is confirmed by a comparison of our model to experimental data of several mercury-electrolyte interfaces. We obtain qualitative and quantitative agreement in the 2V potential range for various salt concentrations. We also discuss the validity of our asymptotic model when the electrode curvature radius is comparable to the Debye length.

*Appeared in*

- European J. Appl. Math., 29 (2018), pp. 708--753, DOI 10.1017/S0956792517000341 .

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# Grad-div stabilization for the evolutionary Oseen problem with inf-sup stable finite elements

*Authors*

- de Frutos, Javier
- Garc'ıa-Archilla, Bosco
- John, Volker

ORCID: 0000-0002-2711-4409 - Novo, Julia

*2010 Mathematics Subject Classification*

- 65M60

*Keywords*

- time-dependent Oseen equations, inf-sup stable pairs of finite element spaces, grad-div stabilization, backward Euler scheme, two-step backward differentiation scheme (BDF2), Crank--Nicolson scheme, uniform error estimates

*Abstract*

The approximation of the time-dependent Oseen problem using inf-sup stable mixed finite elements in a Galerkin method with grad-div stabilization is studied. The main goal is to prove that adding a grad-div stabilization term to the Galerkin approximation has a stabilizing effect for small viscosity. Both the continuous-in-time and the fully discrete case (backward Euler method, the two-step BDF, and Crank--Nicolson schemes) are analyzed. In fact, error bounds are obtained that do not depend on the inverse of the viscosity in the case where the solution is sufficiently smooth. The bounds for the divergence of the velocity as well as for the pressure are optimal. The analysis is based on the use of a specific Stokes projection. Numerical studies support the analytical results.

*Appeared in*

- J. Sci. Comput., 66 (2016) pp. 991--1024.

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# On the divergence constraint in mixed finite element methods for incompressible flows

*Authors*

- John, Volker

ORCID: 0000-0002-2711-4409 - Linke, Alexander

ORCID: 0000-0002-0165-2698 - Merdon, Christian
- Neilan, Michael
- Rebholz, Leo G.

*2010 Mathematics Subject Classification*

- 65N30 76M10

*Keywords*

- incompressible Navier--Stokes and Stokes equations, divergence-free properties, mixed finite elements, pressure-robust discretization

*Abstract*

The divergence constraint of the incompressible Navier--Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely. As a result, these methods introduce a pressure-dependent consistency error which can potentially pollute the computed velocity. These methods are not robust in the sense that a contribution from the right-hand side, which influences only the pressure in the continuous equations, impacts both velocity and pressure in the discrete equations. This paper reviews the theory and practical implications of relaxing the divergence constraint. Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, $bH(mathrmdiv)$-conforming finite elements, and mixed methods with an appropriate reconstruction of the test functions. Numerical examples illustrate both the potential effects of using non-robust discretizations and the improvements obtained by utilizing pressure-robust discretizations.

*Appeared in*

- SIAM Rev., 59 (2017) pp. 492--544, DOI 10.1137/15M1047696 .

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# On indecomposable polyhedra and the number of interior Steiner points

*Authors*

- Goerigk, Nadja
- Si, Hang

*2010 Mathematics Subject Classification*

- 65D18 68U05 65M50 65N50

*Keywords*

- Indecomposable polyhedra, Steiner points, tetrahedralization, Schönhardt polyhedron, Bagemihl polyhedron, Chazelle polyhedron

*Abstract*

The existence of 3d it indecomposable polyhedra, that is, the interior of every such polyhedron cannot be decomposed into a set of tetrahedra whose vertices are all of the given polyhedron, is well-known. While the geometry and combinatorial structure of such polyhedra are much less studied. In this article, we first investigate the geometry of some well-known examples, the so-called it Schönhardt polyhedron citeSchonhardt1928 and the Bagemihl's generalization of it citeBagemihl48-decomp-polyhedra, which will be called it Bagemihl polyhedra. We provide a construction of an interior point, so-called it Steiner point, which can be used to tetrahedralize the Schönhardt and the Bagemihl polyhedra. We then provide a construction of a larger class of three-dimensional indecomposable polyhedra which often appear in grid generation problems. We show that such polyhedra have the same combinatorial structure as the Schönhardt and Bagemihl polyhedra, but they may need more than one interior Steiner point to be tetrahedralized. Given such a polyhedron with $n ge 6$ vertices, we show that it can be tetrahedralized by adding at most $leftlceil fracn - 52rightrceil$ interior Steiner points. %, is sufficient to decompose it. We also show that this number is optimal in the worst case.

*Appeared in*

- Procedia Engineering, Volume 124, 2015, Pages 343--355

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# Optimal control of the sweeping process over polyhedral controlled sets

*Authors*

- Colombo, Giovanni
- Henrion, René
- Hoang, Nguyen D.
- Mordukhovich, Boris S.

*2010 Mathematics Subject Classification*

- 49J52 49J53 49K24 49M25 90C30

*Keywords*

- optimal control, sweeping process, moving controlled polyhedra, dissipative differential inclusions, discrete approximations, variational analysis, generalized differentiation

*Abstract*

The paper addresses a new class of optimal control problems governed by the dissipative and discontinuous differential inclusion of the sweeping/Moreau process while using controls to determine the best shape of moving convex polyhedra in order to optimize the given Bolza-type functional, which depends on control and state variables as well as their velocities. Besides the highly non-Lipschitzian nature of the unbounded differential inclusion of the controlled sweeping process, the optimal control problems under consideration contain intrinsic state constraints of the inequality and equality types. All of this creates serious challenges for deriving necessary optimality conditions. We develop here the method of discrete approximations and combine it with advanced tools of first-order and second-order variational analysis and generalized differentiation. This approach allows us to establish constructive necessary optimality conditions for local minimizers of the controlled sweeping process expressed entirely in terms of the problem data under fairly unrestrictive assumptions. As a by-product of the developed approach, we prove the strong W^{1,2}-convergence of optimal solutions of discrete approximations to a given local minimizer of the continuous-time system and derive necessary optimality conditions for the discrete counterparts. The established necessary optimality conditions for the sweeping process are illustrated by several examples.

*Appeared in*

- J. Differential Equations, 260 (2016) pp. 3397--3447.

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# Feasibility of nominations in stationary gas networks with random load

*Authors*

- Gotzes, Claudia
- Heitsch, Holger
- Henrion, René
- Schultz, Rüdiger

*2010 Mathematics Subject Classification*

- 90B15 90C15

*Keywords*

- stationary gas networks, random nominations, spheric-radial decomposition, Gaussian probability of feasible loads

*Abstract*

The paper considers the computation of the probability of feasible load constellations in a stationary gas network with uncertain demand. More precisely, a network with a single entry and several exits with uncertain loads is studied. Feasibility of a load constellation is understood in the sense of an existing flow meeting these loads along with given pressure bounds in the pipes. In a first step, feasibility of deterministic exit loads is characterized algebraically and these general conditions are specified to networks involving at most one cycle. This prerequisite is essential for determining probabilities in a stochastic setting when exit loads are assumed to follow some (joint) Gaussian distribution when modeling uncertain customer demand. The key of our approach is the application of the spheric-radial decomposition of Gaussian random vectors coupled with Quasi Monte-Carlo sampling. This approach requires an efficient algorithmic treatment of the mentioned algebraic relations moreover depending on a scalar parameter. Numerical results are illustrated for different network examples and demonstrate a clear superiority in terms of precision over simple generic Monte-Carlo sampling. They lead to fairly accurate probability values even for moderate sample size.

*Appeared in*

- Math. Methods Oper. Res., 84 (2016) pp. 427--457, changed title: On the quantification of nomination feasibility in stationary gas networks with random load.

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# On tetrahedralisations of reduced Chazelle polyhedra with interior Steiner points

*Authors*

- Si, Hang
- Goerigk, Nadja

*2010 Mathematics Subject Classification*

- 65D18 68U05 65M50, 65N50

*Keywords*

- indecomposable polyhedron, Chazelle polyhedron, Schönhardt polyhedron, Steiner points, tetrahedralisation, edge flip

*Abstract*

The polyhedron constructed by Chazelle, known as Chazelle polyhedron [4], is an important example in many partitioning problems. In this paper, we study the problem of tetrahedralising a Chazelle polyhedron without modifying its exterior boundary. It is motivated by a crucial step in 3d finite element mesh generation in which a set of arbitrary boundary constraints (edges or faces) need to be entirely preserved. We first reduce the volume of a Chazelle polyhedron by removing the regions that are tetrahedralisable. This leads to a 3d polyhedron which may not be tetrahedralisable unless extra points, so-called Steiner points, are added. We call it a reduced Chazelle polyhedron. We define a set of interior Steiner points that ensures the existence of a tetrahedralisation of the reduced Chazelle polyhedron. Our proof uses a natural correspondence that any sequence of edge flips converting one triangulation of a convex polygon into another gives a tetrahedralization of a 3d polyhedron which have the two triangulations as its boundary. Finally, we exhibit a larger family of reduced Chazelle polyhedra which includes the same combinatorial structure of the Schönhardt polyhedron. Our placement of interior Steiner points also applies to tetrahedralise polyhedra in this family.

*Appeared in*

- 25th International Meshing Roundtable, S. Canann, S. Owen, H. Si, eds., vol. 163 of Procedia Engineering, Elsevier, Amsterdam, 2016, pp. 33--45

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# Guaranteed error control for the pseudostress approximation of the Stokes equations

*Authors*

- Bringmann, Philipp
- Carstensen, Carsten
- Merdon, Christian

*2010 Mathematics Subject Classification*

- 65N30 65N15 76D07

*Keywords*

- nonconforming finite element method, Crouzeix-Raviart element, Stokes equations, pseudostress finite element method, adaptive finite element method, a posteriori error estimation

*Abstract*

The pseudostress approximation of the Stokes equations rewrites the stationary Stokes equations with pure (but possibly inhomogeneous) Dirichlet boundary conditions as another (equivalent) mixed scheme based on a stress in H(div) and the velocity in $L^2$. Any standard mixed finite element function space can be utilized for this mixed formulation, e.g. the Raviart-Thomas discretization which is related to the Crouzeix-Raviart nonconforming finite element scheme in the lowest-order case. The effective and guaranteed a posteriori error control for this nonconforming velocity-oriented discretization can be generalized to the error control of some piecewise quadratic velocity approximation that is related to the discrete pseudostress. The analysis allows for local inf-sup constants which can be chosen in a global partition to improve the estimation. Numerical examples provide strong evidence for an effective and guaranteed error control with very small overestimation factors even for domains with large anisotropy.

*Appeared in*

- Numer. Methods Partial Differential Equations, 32 (2016) pp. 1411--1432.

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# Adaptive stochastic Galerkin FEM with hierarchical tensor representations

*Authors*

- Eigel, Martin
- Pfeffer, Max
- Schneider, Reinhold

*2010 Mathematics Subject Classification*

- 35R60 47B80 60H35 65C20 65N12 65N22 65J10

*Keywords*

- partial differential equations with random coefficients, tensor representation, tensor train, uncertainty quantification, stochastic finite element methods, operator equations, adaptive methods, ALS, low-rank, reduced basis methods

*Abstract*

The solution of PDE with stochastic data commonly leads to very high-dimensional algebraic problems, e.g. when multiplicative noise is present. The Stochastic Galerkin FEM considered in this paper then suffers from the curse of dimensionality. This is directly related to the number of random variables required for an adequate representation of the random fields included in the PDE. With the presented new approach, we circumvent this major complexity obstacle by combining two highly efficient model reduction strategies, namely a modern low-rank tensor representation in the tensor train format of the problem and a refinement algorithm on the basis of a posteriori error estimates to adaptively adjust the different employed discretizations. The adaptive adjustment includes the refinement of the FE mesh based on a residual estimator, the problem-adapted stochastic discretization in anisotropic Legendre Wiener chaos and the successive increase of the tensor rank. Computable a posteriori error estimators are derived for all error terms emanating from the discretizations and the iterative solution with a preconditioned ALS scheme of the problem. Strikingly, it is possible to exploit the tensor structure of the problem to evaluate all error terms very efficiently. A set of benchmark problems illustrates the performance of the adaptive algorithm with higher-order FE. Moreover, the influence of the tensor rank on the approximation quality is investigated.

*Appeared in*

- Numer. Math., 136 (2017) pp. 765--803.

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# Stability of concentrated suspensions under Couette and Poiseuille flow

*Authors*

- Ahnert, Tobias
- Münch, Andreas
- Niethammer, Barbara
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 76E17 76E15 76T20

*2008 Physics and Astronomy Classification Scheme*

- 83.80.Hj

*Keywords*

- stability analysis, suspensions, yield stress, multiphase flow model

*Abstract*

The stability of two-dimensional Poiseuille flow and plane Couette flow for concentrated suspensions is investigated. Linear stability analysis of the two-phase flow model for both flow geometries shows the existence of a convectively driven instability with increasing growth rates of the unstable modes as the particle volume fraction of the suspension increases. In addition it is shown that there exists a bound for the particle phase viscosity below which the two-phase flow model may become ill-posed as the particle phase approaches its maximum packing fraction. The case of two-dimensional Poiseuille flow gives rise to base state solutions that exhibit a jammed and unyielded region, due to shear-induced migration, as the maximum packing fraction is approached. The stability characteristics of the resulting Bingham-type flow is investigated and connections to the stability problem for the related classical Bingham-flow problem are discussed.

*Appeared in*

- J. Engrg. Math., 111 (2018), pp. 51--77, DOI 10.1007/s10665-018-9954-x .

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# Inverse modeling of thin layer flow cells for detection of solubility, transport and reaction coefficients from experimental data

*Authors*

- Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434 - Linke, Alexander

ORCID: 0000-0002-0165-2698 - Merdon, Christian
- Neumann, Felix
- Streckenbach, Timo
- Baltruschat, Helmut
- Khodayari, Mehdi

*2010 Mathematics Subject Classification*

- 76D07 65N30 65N08

*2008 Physics and Astronomy Classification Scheme*

- 47.10.ad 47.11.Fg 47.11.Hj 82.20.Wt

*Keywords*

- Stokes equations, mixed finite elements, convection diffusion equation, finite volume method, electrochemical flow cell

*Abstract*

Thin layer flow cells are used in electrochemical research as experimental devices which allow to perform investigations of electrocatalytic surface reactions under controlled conditions using reasonably small electrolyte volumes. The paper introduces a general approach to simulate the complete cell using accurate numerical simulation of the coupled flow, transport and reaction processes in a flow cell. The approach is based on a mass conservative coupling of a divergence-free finite element method for fluid flow and a stable finite volume method for mass transport. It allows to perform stable and efficient forward simulations that comply with the physical bounds namely mass conservation and maximum principles for the involved species. In this context, several recent approaches to obtain divergence-free velocities from finite element simulations are discussed. In order to perform parameter identification, the forward simulation method is coupled to standard optimization tools. After an assessment of the inverse modeling approach using known realistic data, first results of the identification of solubility and transport data for O2 dissolved in organic electrolytes are presented. A plausibility study for a more complex situation with surface reactions concludes the paper and shows possible extensions of the scope of the presented numerical tools.

*Appeared in*

- Electrochimica Acta, 211 (2016) pp. 1--10.

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# Well-posedness of space and time dependent transport equations on a network

*Authors*

- Roggensack, Arne

*2010 Mathematics Subject Classification*

- 35R02 35F46

*Keywords*

- Transport equation, PDE on a network, coupled boundary value problem, renormalization property, existence and uniqueness

*Abstract*

This article is concerned with the study of weak solutions of a linear transport equation on a bounded domain with coupled boundary data for general non smooth space and time dependent velocity fields. The existence of solutions, its uniqueness and the continuous dependence of the solution on the initial and boundary data as well on the velocity is proven. The results are based on the renormalization property. At the end, the theory is shown to be applicable to the continuity equation on a network.

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# A rate-independent gradient system in damage coupled with plasticity via structured strains

*Authors*

- Thomas, Marita
- Bonetti, Elena
- Rocca, Elisabetta
- Rossi, Riccarda

*2010 Mathematics Subject Classification*

- 74C05 74E10 74R05 74R20 49S05 49J40 35K86

*Keywords*

- rate-independent systems, tensorial damage model, isotropic damage, plasticity, structured strain, energetic solutions, existence results

*Abstract*

This contribution deals with a class of models combining isotropic damage with plasticity. It has been inspired by a work by Freddi and Royer-Carfagni, including the case where the inelastic part of the strain only evolves in regions where the material is damaged. The evolution both of the damage and of the plastic variable is assumed to be rate-independent. Existence of solutions is established in the abstract energetic framework elaborated by Mielke and coworkers.

*Appeared in*

- Gradient flows: from theory to application, B. Düring, C.-B. Schönlieb, M.-TH. Wolfram, eds., vol. 54 of ESAIM Proceedings and Surveys, EDP Sciences, 2016, pp. 54--69

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# On spurious oscillations due to irrotational forces in the Navier--Stokes momentum balance

*Authors*

- Linke, Alexander

ORCID: 0000-0002-0165-2698 - Merdon, Christian

*2010 Mathematics Subject Classification*

- 76D05 76M10

*2008 Physics and Astronomy Classification Scheme*

- 47.11.Fg

*Keywords*

- incompressible Navier--Stokes equations, mixed finite element methods, benchmarks, spurious velocity oscillations

*Abstract*

This contribution studies the influence of the pressure on the velocity error in finite element discretisations of the Navier--Stokes equations. Three simple benchmark problems that are all close to real-world applications convey that the pressure can be comparably large and is not to be underestimated. For widely used finite element methods like the Taylor--Hood finite element method, such relatively large pressures can lead to spurious oscillations and arbitrarily large errors in the velocity, even if the exact velocity is in the ansatz space. Only mixed finite element methods, whose velocity error is pressure-independent, like the Scott--Vogelius finite element method can avoid this influence.

*Appeared in*

- J. Comput. Phys., 313 (2016) pp. 654--661.

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# Some remarks on a model for rate-independent damage in thermo-visco-elastodynamics

*Authors*

- Lazzaroni, Giuliano
- Rossi, Riccarda
- Thomas, Marita
- Toader, Rodica

*2010 Mathematics Subject Classification*

- 35Q74 74H20 74R05 74C05 74F05

*Keywords*

- Partial damage, rate-independent systems, elastodynamics, phase-field models, heat equation, energetic solutions

*DOI*

*Abstract*

This note deals with the analysis of a model for partial damage, where the rate-independent, unidirectional flow rule for the damage variable is coupled with the rate-dependent heat equation, and with the momentum balance featuring inertia and viscosity according to Kelvin-Voigt rheology. The results presented here combine the approach from [Roubiček M2AS'09, SIAM'10] with the methods from Lazzaroni/Rossi/Thomas/Toader [WIAS Preprint 2025]. The present analysis encompasses, differently from [Roubiček SIAM'10], the monotonicity in time of damage and the dependence of the viscous tensor on damage and temperature, and, unlike [WIAS Preprint 2025], a nonconstant heat capacity and a time-dependent Dirichlet loading.

*Appeared in*

- MURPHYS-HSFS-2014: 7th International Workshop on MUlti-Rate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and Slow-Fast Systems (HSFS), O. Klein, M. Dimian, P. Gurevich, D. Knees, D. Rachinskii, S. Tikhomirov, eds., vol. 727 of Journal of Physics: Conference Series, IOP Publishing, 2016, pp. 012009/1--012009/20.

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# Critical objective size and calmness modulus in linear programming

*Authors*

- Cánovas, Maria J.
- Henrion, René
- Parra, Juan
- Toledo, F. Javier

*2010 Mathematics Subject Classification*

- 90C31 49J53 49K40 90C05

*Keywords*

- variational analysis, calmness, linear programming

*Abstract*

This paper introduces the concept of critical objective size associated with a linear program in order to provide operative point-based formulas (only involving the nominal data, and not data in a neighborhood) for computing or estimating the calmness modulus of the optimal set (argmin) mapping under uniqueness of nominal optimal solution and perturbations of all coefficients. Our starting point is an upper bound on this modulus given in citeCHPTmp. In this paper we prove that this upper bound is attained if and only if the norm of the objective function coefficient vector is less than or equal to the critical objective size. This concept also allows us to obtain operative lower bounds on the calmness modulus. We analyze in detail an illustrative example in order to explore some strategies that can improve the referred upper and lower bounds.

*Appeared in*

- Set-Valued Var. Anal., 24 (2016) pp. 565--579.

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# On an application of Tikhonov's fixed point theorem to a nonlocal Cahn--Hilliard type system modeling phase separation

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K40 35K86 45K05 47H10 80A22

*Keywords*

- Cahn-Hilliard system, nonlocal energy, phase separation, singular potentials, initial-boundary value problem, Tikhonov's fixed point theorem

*Abstract*

This paper investigates a nonlocal version of a model for phase separation on an atomic lattice that was introduced by P. Podio-Guidugli in *Ric. Mat.* **55** (2006) 105-118. The model consists of an initial-boundary value problem for a nonlinearly coupled system of two partial differential equations governing the evolution of an order parameter ρ and the chemical potential μ. Singular contributions to the local free energy in the form of logarithmic or double-obstacle potentials are admitted. In contrast to the local model, which was studied by P. Podio-Guidugli and the present authors in a series of recent publications, in the nonlocal case the equation governing the evolution of the order parameter contains in place of the Laplacian a nonlocal expression that originates from nonlocal contributions to the free energy and accounts for possible long-range interactions between the atoms. It is shown that just as in the local case the model equations are well posed, where the technique of proving existence is entirely different: it is based on an application of Tikhonov's fixed point theorem in a rather unusual separable and reflexive Banach space.

*Appeared in*

- J. Differential Equations, 260 (2016), pp. 7940--7964.

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# Asymptotic analyses and error estimates for a Cahn--Hilliard type phase field system modelling tumor growth

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Rocca, Elisabetta
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35Q92 92C17 35K35 35K57 78M35 35B20 65N15 35R35

*Keywords*

- tumor growth, Cahn-Hilliard system, reaction-diffusion equation, asymptotic analysis, error estimates

*Abstract*

This paper is concerned with a phase field system of Cahn--Hilliard type that is related to a tumor growth model and consists of three equations in gianni terms of the variables order parameter, chemical potential and nutrient concentration. This system has been investigated in the recent papers citeCGH and citeCGRS gianni from the viewpoint of well-posedness, long time bhv and asymptotic convergence as two positive viscosity coefficients tend to zero at the same time. Here, we continue the analysis performed in citeCGRS by showing two independent sets of results as just one of the coefficents tends to zero, the other remaining fixed. We prove convergence results, uniqueness of solutions to the two resulting limit problems, and suitable error estimates

*Appeared in*

- Discrete Contin. Dyn. Syst., 10 (2017), pp. 37--54.

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# Robust optimal stopping

*Authors*

- Krätschmer, Volker
- Ladkau, Marcel
- Laeven, Roger J. A.
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266 - Stadje, Mitja

*2010 Mathematics Subject Classification*

- 49L20 60G40 91B06

*Keywords*

- Optimal stopping, model uncertainty, robustness, convex risk measures, ambiguity aversion, duality, BSDEs, Monte Carlo simulation, regression, relative entropy

*Abstract*

This paper studies the optimal stopping problem in the presence of model uncertainty (ambiguity). We develop a method to practically solve this problem in a general setting, allowing for general time-consistent ambiguity averse preferences and general payoff processes driven by jump-diffusions. Our method consists of three steps. First, we construct a suitable Doob martingale associated with the solution to the optimal stopping problem represented by the Snell envelope using backward stochastic calculus. Second, we employ this martingale to construct an approximated upper bound to the solution using duality. Third, we introduce backward-forward simulation to obtain a genuine upper bound to the solution, which converges to the true solution asymptotically. We analyze the asymptotic behavior and convergence properties of our method. We illustrate the generality and applicability of our method and the potentially significant impact of ambiguity to optimal stopping in a few examples.

*Appeared in*

- Math. Oper. Res., 43 (2018), pp. 1177--1209, DOI 10.1287/moor.2017.0899 .

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# On the grad-div stabilization for the steady Oseen and Navier--Stokes equations

*Authors*

- Ahmed, Naveed

ORCID: 0000-0002-9322-0373

*2010 Mathematics Subject Classification*

- 35Q30 76M10 65L60

*Keywords*

- incompressible Navier--Stokes equations, mixed fnite elements, grad-div stabilization, error estimates, stabilization parameter

*DOI*

*Abstract*

This paper studies the parameter choice in the grad-div stabilization applied to the generalized problems of Oseen type. Stabilization parameters based on minimizing the H¹ (Ω) error of the velocity are derived which do not depend on the viscosity parameter. For the proposed parameter choices, the H¹ (Ω) error of the velocity is derived that shows a direct dependence on the viscosity parameter. Differences and common features to the situation for the Stokes equations are discussed. Numerical studies are presented which confirm the theoretical results. Moreover, for the Navier- Stokes equations, numerical simulations were performed on a two-dimensional ow past a circular cylinder. It turns out, for the MINI element, that the best results can be obtained without grad-div stabilization.

*Appeared in*

- Calcolo, pp. , DOI 10.1007/s10092-016-0194-z .

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# A time domain sampling method for inverse acoustic scattering problems

*Authors*

- Guo, Yukun
- Hömberg, Dietmar
- Hu, Guanghui
- Li, Jingzhi
- Liu, Hongyu

*2010 Mathematics Subject Classification*

- 74J20 74J25 35R30

*Keywords*

- inverse acoustic scattering, sampling method, time domain, dynamic measurements, Kirschhoff migration

*Abstract*

This work concerns the inverse scattering problems of imaging unknown/inaccessible scatterers by transient acoustic near-field measurements. Based on the analysis of the migration method, we propose efficient and effective sampling schemes for imaging small and extended scatterers from knowledge of time-dependent scattered data due to incident impulsive point sources. Though the inverse scattering problems are known to be nonlinear and ill-posed, the proposed imaging algorithms are totally ``direct'' involving only integral calculations on the measurement surface. Theoretical justifications are presented and numerical experiments are conducted to demonstrate the effectiveness and robustness of our methods. In particular, the proposed static imaging functionals enhance the performance of the total focusing method (TFM) and the dynamic imaging functionals show analogous behavior to the time reversal inversion but without solving time-dependent wave equations.

*Appeared in*

- J. Comput. Phys., 314 (2016) pp. 647--660.

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# A new perspective on the electron transfer: Recovering the Butler--Volmer equation in non-equilibrium thermodynamics

*Authors*

- Dreyer, Wolfgang
- Guhlke, Clemens
- Müller, Rüdiger

*2010 Mathematics Subject Classification*

- 35Q60 35Q79 80A17

*2008 Physics and Astronomy Classification Scheme*

- 82.45.Gj 82.45.Fk

*Keywords*

- Butler-Volmer, thermodynamics, electrolyte, double-layer

*Abstract*

Understanding and correct mathematical description of electron transfer reaction is a central question in electrochemistry. Typically the electron transfer reactions are described by the Butler-Volmer equation which has its origin in kinetic theories. The Butler-Volmer equation relates interfacial reaction rates to bulk quantities like the electrostatic potential and electrolyte concentrations. Since in the classical form, the validity of the Butler-Volmer equation is limited to some simple electrochemical systems, many attempts have been made to generalize the Butler-Volmer equation. Based on non-equilibrium thermodynamics we have recently derived a reduced model for the electrode-electrolyte interface. This reduced model includes surface reactions but does not resolve the charge layer at the interface. Instead it is locally electroneutral and consistently incorporates all features of the double layer into a set of interface conditions. In the context of this reduced model we are able to derive a general Butler-Volmer equation. We discuss the application of the new Butler-Volmer equations to different scenarios like electron transfer reactions at metal electrodes, the intercalation process in lithium-iron-phosphate electrodes and adsorption processes. We illustrate the theory by an example of electroplating.

*Appeared in*

- Phys. Chem. Chem. Phys., 18 (2016), pp. 24966--24983, DOI 10.1039/C6CP04142F .

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# Semiconductor mode-locked lasers with coherent dual mode optical injection: Simulations, analysis and experiment

*Authors*

- Arkhipov, Rostislav M.
- Habruseva, Tatiana
- Pimenov, Alexander
- Radziunas, Mindaugas
- Huyet, Guillaume
- Vladimirov, Andrei G.

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px, 42.60.Fc, 42.60.Lh

*Keywords*

- Passively mode-locked semiconductor lasers, dual frequency optical injection, locking

*Abstract*

Using a delay differential equations model we study the dynamics of a passively mode-locked semiconductor laser with dual frequency coherent optical injection. The locking regions where the laser pulse repetition rate is synchronized to the separation of the two injected frequencies were calculated numerically and measured experimentally. Asymptotic analysis performed in the limit of the small injection field amplitude revealed the dependence of the locking regions on the model parameters, such as optical bandwith, absorber recovery time and linear losses.

*Appeared in*

- J. Opt. Soc. Amer. B Opt. Phys., 33 (2016) pp. 351-359.

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# A review of variational multiscale methods for the simulation of turbulent incompressible flows

*Authors*

- Ahmed, Naveed

ORCID: 0000-0002-9322-0373 - Rebollo, Tomás Chacón
- John, Volker

ORCID: 0000-0002-2711-4409 - Rubino, Samuele

*2010 Mathematics Subject Classification*

- 65M60 76F65

*Keywords*

- ncompressible Navier--Stockes equations, turbulent flows, variational multiscale methods (VMS), finite element methods, numerical Analysis

*Abstract*

Various realizations of variational multiscale (VMS) methods for simulating turbulent incompressible flows have been proposed in the past fifteen years. All of these realizations obey the basic principles of VMS methods: They are based on the variational formulation of the incompressible Navier--Stokes equations and the scale separation is defined by projections. However, apart from these common basic features, the various VMS methods look quite different. In this review, the derivation of the different VMS methods is presented in some detail and their relation among each other and also to other discretizations is discussed. Another emphasis consists in giving an overview about known results from the numerical analysis of the VMS methods. A few results are presented in detail to highlight the used mathematical tools. Furthermore, the literature presenting numerical studies with the VMS methods is surveyed and the obtained results are summarized.

*Appeared in*

- Arch. Comput. Methods Eng., 24 (2017) pp. 115--164.

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# Coupling rate-independent and rate-dependent processes: Existence results

*Authors*

- Rossi, Riccarda
- Thomas, Marita

*2010 Mathematics Subject Classification*

- 35A15 35A23 35A35 35G61 49J53 35Q74 74H20 74C05 74C10

*Keywords*

- Rate-independent processes, gradient systems, inertia, energetic solutions, existence results, time-discretization, generalized standard solids

*Abstract*

We address the analysis of an abstract system coupling a rate-independet process with a second order (in time) nonlinear evolution equation. We propose suitable weak solution concepts and obtain existence results by passing to the limit in carefully devised time-discretization schemes. Our arguments combine techniques from the theory of *gradient systems* with the toolbox for *rate-independent* evolution, thus reflecting the mixed character of the problem. Finally, we discuss applications to a class of rate-independent processes in visco-elastic solids with inertia, and to a recently proposed model for damage with plasticity.

*Appeared in*

- SIAM J. Math. Anal., 49 (2017), pp. 1419--1494.

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# Large deviations in relay-augmented wireless networks

*Authors*

- Hirsch, Christian
- Jahnel, Benedikt

ORCID: 0000-0002-4212-0065 - Keeler, Paul

ORCID: 0000-0002-2063-1075 - Patterson, Robert I. A.

ORCID: 0000-0002-3583-2857

*2010 Mathematics Subject Classification*

- 60F10 60K35

*Keywords*

- large deviations, relay, wireless network, signal-to-interference ratio, symmetry breaking

*Abstract*

We analyze a model of relay-augmented cellular wireless networks. The network users, who move according to a general mobility model based on a Poisson point process of continuous trajectories in a bounded domain, try to communicate with a base station located at the origin. Messages can be sent either directly or indirectly by relaying over a second user. We show that in a scenario of an increasing number of users, the probability that an atypically high number of users experiences bad quality of service over a certain amount of time, decays at an exponential speed. This speed is characterized via a constrained entropy minimization problem. Further, we provide simulation results indicating that solutions of this problem are potentially non-unique due to symmetry breaking. Also two general sources for bad quality of service can be detected, which we refer to as isolation and screening.

*Appeared in*

- Queueing Syst., 88 (2018), pp. 349--387 (published online on 28.10.2017).

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# Global-in-time existence of weak solutions to Kolmogorov's two-equation model of turbulence

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Naumann, Joachim

*2010 Mathematics Subject Classification*

- 35K45 35Q30 76D03 76F99

*Keywords*

- Navier-Stokes equation, Kolmogorov's turbulence model, turbulent kinetic energy, global existence for weak solutions, defect measure, scaling laws, maximum principle

*Abstract*

We consider Kolmogorov's model for the turbulent motion of an incompressible fluid in ℝ^{3}. This model consists in a Navier-Stokes type system for the mean flow ** u** and two further partial differential equations: an equation for the frequency

*ω*and for the kinetic energy

*k*each. We investigate this system of partial differential equations in a cylinder Ω x ]0,T[ (Ω ⊂ ℝ

^{3}cube, 0 < T < +∞) under spatial periodic boundary conditions on ∂Ω x ]0,T[ and initial conditions in Ω x {0}. We present an existence result for a weak solution {

*} to the problem under consideration, with*

**u**, ω, k*ω, k*obeying the inequalities and .

*Appeared in*

- C. R. Math. Acad. Sci. Paris, 353 (2015) pp. 321--326.

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# Analysis of a diffuse interface model of multispecies tumor growth

*Authors*

- Dai, Mimi
- Feireisl, Eduard
- Rocca, Elisabetta
- Schimperna, Giulio
- Schonbek , Maria E.

*2010 Mathematics Subject Classification*

- 35B25 35D30 35K35 35K57 35Q92 74G25 78A70 92C17

*Keywords*

- tumor growth, diffuse interface model, Cahn-Hilliard equation, reaction-diffusion equation, Darcy law, existence of weak solutions, singular limits

*Abstract*

We consider a diffuse interface model for tumor growth recently proposed in citecwsl. In this new approach sharp interfaces are replaced by narrow transition layers arising due to adhesive forces among the cell species. Hence, a continuum thermodynamically consistent model is introduced. The resulting PDE system couples four different types of equations: a Cahn-Hilliard type equation for the tumor cells (which include proliferating and dead cells), a Darcy law for the tissue velocity field, whose divergence may be different from 0 and depend on the other variables, a transport equation for the proliferating (viable) tumor cells, and a quasi-static reaction diffusion equation for the nutrient concentration. We establish existence of weak solutions for the PDE system coupled with suitable initial and boundary conditions. In particular, the proliferation function at the boundary is supposed to be nonnegative on the set where the velocity $vu$ satisfies $vucdotnu>0$, where $nu$ is the outer normal to the boundary of the domain. We also study a singular limit as the diffuse interface coefficient tends to zero.

*Appeared in*

- Nonlinearity, 30:4 (2017).

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# Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities

*Authors*

- Frigeri, Sergio Pietro

*2010 Mathematics Subject Classification*

- 76T99 35Q30 35Q35 76D03 76D03 76D05 76D27

*Keywords*

- Diffuse interface model, Incompressible viscous binary fluids, Navier--Stokes system, nonlocal Cahn--Hilliard equation

*Abstract*

We consider a diffuse interface model for an incompressible isothermal mixture of two viscous Newtonian fluids with different densities in a bounded domain in two or three space dimensions. The model is the nonlocal version of the one recently derived by Abels, Garcke and Grün and consists of a Navier-Stokes type system coupled with a convective nonlocal Cahn-Hilliard equation. The density of the mixture depends on an order parameter. For this nonlocal system we prove existence of global dissipative weak solutions for the case of singular double-well potentials and non degenerate mobilities. To this goal we devise an approach which is completely independent of the one employed by Abels, Depner and Garcke to establish existence of weak solutions for the local Abels et al. model.

*Appeared in*

- Math. Models Methods Appl. Sci., 26 (2016), pp. 1957--1993.

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# Global strong solutions of the full Navier--Stokes and $Q$-tensor system for nematic liquid crystal flows in $2D$: Existence and long-time behavior

*Authors*

- Cavaterra, Cecilia
- Rocca, Elisabetta
- Wu, Hao
- Xu, Xiang

*2010 Mathematics Subject Classification*

- 35B44 35D30 35K45 35Q30 76A15

*Keywords*

- Nematic liquid crystal flow, Q-tensor system, global strong solution, uniqueness of asymptotic limit

*Abstract*

We consider a full Navier-Stokes and $Q$-tensor system for incompressible liquid crystal flows of nematic type. In the two dimensional periodic case, we prove the existence and uniqueness of global strong solutions that are uniformly bounded in time. This result is obtained without any smallness assumption on the physical parameter $xi$ that measures the ratio between tumbling and aligning effects of a shear flow exerting over the liquid crystal directors. Moreover, we show the uniqueness of asymptotic limit for each global strong solution as time goes to infinity and provide an uniform estimate on the convergence rate.

*Appeared in*

- SIAM J. Math. Anal., 48 (2016) pp. 1368--1399.

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# Plane-like minimizers for a non-local Ginzburg--Landau-type energy in a periodic medium

*Authors*

- Cozzi, Matteo
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R11 35A15 35B08 82B26 35B65

*Keywords*

- Non-local energies, phase transitions, plane-like minimizers, fractional Laplacian

*Abstract*

We consider a non-local phase transition equation set in a periodic medium and we construct solutions whose interface stays in a slab of prescribed direction and universal width. The solutions constructed also enjoy a local minimality property with respect to a suitable non-local energy functional.

*Appeared in*

- J. Ec. polytech. Math., 4 (2017), pp. 337-388.

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# Continuity and density results for a one-phase nonlocal free boundary problem

*Authors*

- Dipierro, Serena
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R35 49N60 35R11, 35A15

*Keywords*

- Free boundary problems, nonlocal minimalsurfaces, fractional operators, regularity theory, fractional harmonic replacement

*Abstract*

We consider a one-phase nonlocal free boundary problem obtained by the superposition of a fractional Dirichlet energy plus a nonlocal perimeter functional. We prove that the minimizers are continuous and the free boundary has positive density from both sides. For this, we also introduce a new notion of fractional harmonic replacement in the extended variables and we study its basic properties.

*Appeared in*

- Ann. Inst. H. Poincare Anal. Non Lineaire, 34 (2017), pp. 1387-1428.

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# Chaotic orbits for systems of nonlocal equations

*Authors*

- Dipierro, Serena
- Patrizi, Stefania
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R11 34C28

*Keywords*

- homoclinic and heteroclinic connections, chaotic orbits, symbolic dynamics, fractional operators

*Abstract*

We consider a system of nonlocal equations driven by a perturbed periodic potential. We construct multibump solutions that connect one integer point to another one in a prescribed way. In particular, heteroclinc, homoclinic and chaotic trajectories are constructed.

*Appeared in*

- Comm. Math. Phys., (2016) pp. 1--44.

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# A posteriori error control for stationary coupled bulk-surface equations

*Authors*

- Eigel, Martin
- Müller, Rüdiger

*2010 Mathematics Subject Classification*

- 65N30 65N15 58J32 65M15 74S05 65M60

*Keywords*

- finite element method, a posteriori error estimator, adaptive algorithm, surface finite elements, bulk-surface elliptic equations

*Abstract*

We consider a system of two coupled elliptic equations, one defined on a bulk domain and the other one on the boundary surface. Problems of this kind are relevant for applications in engineering, chemistry and in biology like e.g. biological signal transduction. For the a posteriori error control of the coupled system, a residual error estimator is derived which takes into account the approximation errors due to the finite element discretisation in space as well as the polyhedral approximation of the surface. An adaptive refinement algorithm controls the overall error. Numerical experiments illustrate the performance of the a posteriori error estimator and the adaptive algorithm with several benchmark examples.

*Appeared in*

- IMA J. Numer. Anal., 38 (2018), pp. 271--298, DOI 10.1093/imanum/drw080 .

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# A weak formulation for a rate-independent delamination evolution with inertial and viscosity effects subjected to unilateral constraint

*Authors*

- Scala, Riccardo

*2010 Mathematics Subject Classification*

- 35L04 74C05 74D10 74R99 47H05

*Keywords*

- Second order parabolic equation, viscoelasticity, energetic formulation, delamination, adhesion, rate-independent system, unilateral constraint

*Abstract*

We consider a system of two viscoelastic bodies attached on one edge by an adhesive where a delamination process occurs. We study the dynamic of the system subjected to external forces, suitable boundary conditions, and an unilateral constraint on the jump of the displacement at the interface between the bodies. The constraint arises in a graph inclusion, while the delamination coeficient evolves in a rate-independent way. We prove the existence of a weak solution to the corresponding system of PDEs.

*Appeared in*

- Interfaces Free Bound., 19 (2017) pp. 79--107.

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# A contact problem for viscoelastic bodies with inertial effects and unilateral boundary constraints

*Authors*

- Scala, Riccardo
- Schimperna, Giulio

*2010 Mathematics Subject Classification*

- 35L10 74D10 47H05 46A20

*Keywords*

- second order parabolic equation, viscoelasticity, weak formulation, contact problem, adhesion, mixed boundary conditions, duality

*Abstract*

We consider a three-dimensional viscoelastic body subjected to external forces. Inertial effects are considered; hence the equation for the displacement field is of hyperbolic type. The equation is complemented with Dirichlet and Neuman conditions on a part the boundary, while on another part the body is in adhesive contact with a solid support. The boundary forces acting on the latter part due to the action of elastic stresses are responsible for delamination, i.e., progressive failure of adhesive bonds. This phenomenon is mathematically represented by a nonlinear ODE which describes the evolution of the delamination order parameter z. Following the lines of a new approach introduced by the authors in a preceding paper and based on duality methods in Sobolev-Bochner spaces, we define a suitable concept of weak solutions to the resulting PDE system. Correspondingly, we prove an existence result on finite time intervals of arbitrary length.

*Appeared in*

- Eur. J. App. Math, (2016), pp. 1--32, DOI 10.1017/S0956792516000097 .

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# Lagrange multiplier and singular limit of double obstacle problems for Allen--Cahn equation with constraint

*Authors*

- Farshbaf Shaker, Mohammad Hassan
- Fukao, Takeshi
- Yamazaki, Noriaki

*2010 Mathematics Subject Classification*

- 35K57 35R35 35B25

*Keywords*

- Allen-Cahn equation, constraint, double obstacle, singular limit, Lagrange multiplier, subdifferential, numerical experiments

*Abstract*

We consider an Allen--Cahn equation with a constraint of double obstacle-type. This constraint is a subdifferential of an indicator function on the closed interval, which is a multivalued function. In this paper we study the properties of the Lagrange multiplier to our equation. Also, we consider the singular limit of our system and clarify the limit of the solution and the Lagrange multiplier to our double obstacle problem. Moreover, we give some numerical experiments of our problem by using the Lagrange multiplier.

*Appeared in*

- Math. Methods Appl. Sci., 40 (2017) pp. 5--21.

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# A one-dimensional symmetry result for solutions of an integral equation of convolution type

*Authors*

- Hamel, François
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 45A05 47G10 47B34

*Keywords*

- Integral operators, convolution kernels, one-dimensional symmetry, De Giorgi Conjecture

*Abstract*

We consider an integral equation in the plane, in which the leading operator is of convolution type, and we prove that monotone (or stable) solutions are necessarily one-dimensiona.l

*Appeared in*

- Ann. Inst. H. Poincare Anal. Non Lineaire 34 (2017) pp. 469--482, added authors, changed title: A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane.

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# A note on the extremal process of the supercritical Gaussian free field

*Authors*

- Chiarini, Alberto
- Cipriani, Alessandra
- Hazra, Rajat Subhra

*2010 Mathematics Subject Classification*

- 60G15 60G30 60G55 60G70 82B41

*Keywords*

- Gaussian free field, Poisson approximation, point processes, extremal process, Stein--Chen method

*Abstract*

We consider both the infinite-volume discrete Gaussian Free Field (DGFF) and the DGFF with zero boundary conditions outside a finite boxin dimension larger or equal to 3. We show that the associated extremal process converges to a Poisson point process. The result follows from an application of the Stein-Chen method from Arratia et al. (1989).

*Appeared in*

- Electron. Comm. Probab., 20 (2015) pp. 74/1--74/10.

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# Mean-field interaction of Brownian occupation measures. II: A rigorous construction of the Pekar process

*Authors*

- Bolthausen, Erwin
- König, Wolfgang
- Mukherjee, Chiranjib

*2010 Mathematics Subject Classification*

- 60J65 60J55 60F10

*Keywords*

- Polaron problem, Gibbs measures, large deviations, Coulomb functional, tightness

*Abstract*

We consider mean-field interactions corresponding to Gibbs measures on interacting Brownian paths in three dimensions. The interaction is self-attractive and is given by a singular Coulomb potential. The logarithmic asymptotics of the partition function for this model were identified in the 1980s by Donsker and Varadhan [DV83] in terms of the Pekar variational formula, which coincides with the behavior of the partition function corresponding to the polaron problem under strong coupling. Based on this, Spohn ([Sp87]) made a heuristic observation that the strong coupling behavior of the polaron path measure, on certain time scales, should resemble a process, named as the itPekar process, whose distribution could somehow be guessed from the limiting asymptotic behavior of the mean-field measures under interest, whose rigorous analysis remained open. The present paper is devoted to a precise analysis of these mean-field path measures and convergence of the normalized occupation measures towards an explicit mixture of the maximizers of the Pekar variational problem. This leads to a rigorous construction of the aforementioned Pekar process and hence, is a contribution to the understanding of the ``mean-field approximation" of the polaron problem on the level of path measures. The method of our proof is based on the compact large deviation theory developed in [MV14], its extension to the uniform strong metric for the singular Coulomb interaction carried out in [KM15], as well as an idea inspired by a itpartial path exchange argument appearing in [BS97]

*Appeared in*

- Comm. Pure Appl. Math., 70 (2017), pp. 1598--1629.

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# Detection and storage of multivariate temporal sequences by spiking pattern reverberators

*Authors*

- Lücken, Leonhard
- Yanchuk, Serhiy

*2010 Mathematics Subject Classification*

- 37N20 92B25

*2008 Physics and Astronomy Classification Scheme*

- 05.45Xt, 87.19La, 87.19.lj, 87.19.lp, 85.40.Bh

*Keywords*

- Coincidence detector, recurrent neural network, reverberation, sustained activity, spatio-temporal pattern formation, boolean networks

*Abstract*

We consider networks of spiking coincidence detectors in continuous time. A single detector is a finite state machine that emits a pulsatile signal whenever the number incoming inputs exceeds a threshold within a time window of some tolerance width. Such finite state models are well-suited for hardware implementations of neural networks, as on integrated circuits (IC) or field programmable arrays (FPGAs) but they also reflect the natural capability of many neurons to act as coincidence detectors. We pay special attention to a recurrent coupling structure, where the delays are tuned to a specific pattern. Applying this pattern as an external input leads to a self-sustained reverberation of the encoded pattern if the tuning is chosen correctly. In terms of the coupling structure, the tolerance and the refractory time of the individual coincidence detectors, we determine conditions for the uniqueness of the sustained activity, i.e., for the funcionality of the network as an unambiguous pattern detector. We also present numerical experiments, where the functionality of the proposed pattern detector is demonstrated replacing the simplistic finite state models by more realistic Hodgkin-Huxley neurons and we consider the possibility of implementing several pattern detectors using a set of shared coincidence detectors. We propose that inhibitory connections may aid to increase the precision of the pattern discrimination.

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# Boundary behavior of nonlocal minimal surfaces

*Authors*

- Dipierro, Serena
- Savin, Ovidiu
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 49Q05 35R11 53A10

*Keywords*

- Nonlocal minimal surfaces, boundary regularity, barriers

*Abstract*

We consider the behavior of the nonlocal minimal surfaces in the vicinity of the boundary. By a series of detailed examples, we show that nonlocal minimal surfaces may stick at the boundary of the domain, even when the domain is smooth and convex. This is a purely nonlocal phenomenon, and it is in sharp contrast with the boundary properties of the classical minimal surfaces. In particular, we show stickiness phenomena to half-balls when the datum outside the ball is a small half-ring and to the side of a two-dimensional box when the oscillation between the datum on the right and on the left is large enough. When the fractional parameter is small, the sticking effects may become more and more evident. Moreover, we show that lines in the plane are unstable at the boundary: namely, small compactly supported perturba- tions of lines cause the minimizers in a slab to stick at the boundary, by a quantity that is proportional to a power of the perturbation. In all the examples, we present concrete estimates on the stickiness phenomena. Also, we construct a family of compactly supported barriers which can have independent interest.

*Appeared in*

- J. Funct. Anal. 272 (2017) pp. 1791-1851.

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# Attractor properties for irreversible and reversible interacting particle systems

*Authors*

- Jahnel, Benedikt

ORCID: 0000-0002-4212-0065 - Külske, Christof

*2010 Mathematics Subject Classification*

- 82C20 60K35

*Keywords*

- Interacting particle systems, non-equilibrium, non-reversibility, attractor property, relative entropy, gibbs measures

*Abstract*

We consider translation-invariant interacting particle systems on the lattice with finite local state space admitting at least one Gibbs measure as a time-stationary measure. The dynamics can be irreversible but should satisfy some mild non-degeneracy conditions. We prove that weak limit points of any trajectory of translation-invariant measures, satisfying a non-nullness condition, are Gibbs states for the same specification as the time-stationary measure. This is done under the additional assumption that zero entropy loss of the limiting measure w.r.t. the time-stationary measure implies that they are Gibbs measures for the same specification.We also give an alternate version of the last condition such that the non-nullness requirement can be dropped. For dynamics admitting a reversible Gibbs measure the alternative condition can be verified, which yields the attractor property for such dynamics. This generalizes convergence results using relative entropy techniques to a large class of dynamics including irreversible and non-ergodic ones. We use this to show synchronization for the rotation dynamics exhibited in citeJaKu12 possibly at low temperature, and possibly non-reversible. We assume the additional regularity properties on the dynamics: 1 There is at least one stationary measure which is a Gibbs measure. 2 Zero loss of relative entropy density under dynamics implies the Gibbs property.

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# Nonlocal Delaunay surfaces

*Authors*

- Davila, Juan
- del Pino, Manuel
- Dipierro, Serena
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R11 49Q05 49Q20

*Keywords*

- Delaunay surfaces, nonlocal perimeter, minimization problem

*Abstract*

We construct codimension 1 surfaces of any dimension that minimize a nonlocal perimeter functional among surfaces that are periodic, cylindrically symmetric and decreasing. These surfaces may be seen as a nonlocal analogue of the classical Delaunay surfaces (onduloids). For small volume, most of their mass tends to be concentrated in a periodic array and the surfaces are close to a periodic array of balls (in fact, we give explicit quantitative bounds on these facts).

*Appeared in*

- Nonlinear Anal., 137 (2016) pp. 357--380.

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# A discontinuous skeletal method for the viscosity-dependent Stokes problem

*Authors*

- Ern, Alexandre
- di Pietro, Daniele
- Linke, Alexander

ORCID: 0000-0002-0165-2698 - Schieweck, Friedhelm

*2010 Mathematics Subject Classification*

- 65N12 65N30 76D07

*2008 Physics and Astronomy Classification Scheme*

- 47.11.Fg

*Keywords*

- Stokes problem, mixed methods, curl-free forces, higher-order reconstruction, superconvergence, hybrid discontinuous Galerkin method, static condensation

*Abstract*

We devise and analyze arbitrary-order nonconforming methods for the discretization of the viscosity-dependent Stokes equations on simplicial meshes. We keep track explicitly of the viscosity and aim at pressure-robust schemes that can deal with the practically relevant case of body forces with large curl-free part in a way that the discrete velocity error is not spoiled by large pressures. The method is inspired from the recent Hybrid High-Order (HHO) methods for linear elasticity. After elimination of the auxiliary variables by static condensation, the linear system to be solved involves only discrete face-based velocities, which are polynomials of degree k >=0, and cell-wise constant pressures. Our main result is a pressure-independent energy-error estimate on the velocity of order (k+1). The main ingredient to achieve pressure-independence is the use of a divergence-preserving velocity reconstruction operator in the discretization of the body forces. We also prove an L2-pressure estimate of order (k+1) and an L2-velocity estimate of order (k+2), the latter under elliptic regularity. The local mass and momentum conservation properties of the discretization are also established. Finally, two- and three-dimensional numerical results are presented to support the analysis.

*Appeared in*

- Comput. Methods Appl. Mech. Engrg., 306 (2016) pp. 175--195.

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# Optimal transport in competition with reaction: The Hellinger--Kantorovich distance and geodesic curves

*Authors*

- Liero, Matthias

ORCID: 0000-0002-0963-2915 - Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Savaré, Giuseppe

*2010 Mathematics Subject Classification*

- 28A33 46E27 49K21 49Q20 58E30

*Keywords*

- Dissipation distance, geodesic curves, cone space, optimal transport, Onsager operator, reaction-diffusion equations

*Abstract*

We discuss a new notion of distance on the space of finite and nonnegative measures on Ω ⊂ ℝ ^{d}, which we call Hellinger-Kantorovich distance. It can be seen as an inf-convolution of the well-known Kantorovich-Wasserstein distance and the Hellinger-Kakutani distance. The new distance is based on a dynamical formulation given by an Onsager operator that is the sum of a Wasserstein diffusion part and an additional reaction part describing the generation and absorption of mass. We present a full characterization of the distance and some of its properties. In particular, the distance can be equivalently described by an optimal transport problem on the cone space over the underlying space Ω. We give a construction of geodesic curves and discuss examples and their general properties.

*Appeared in*

- SIAM J. Math. Anal., 48 (2016), pp. 2869--2911.

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# Towards doping optimization of semiconductor lasers

*Authors*

- Peschka, Dirk

ORCID: 0000-0002-3047-1140 - Rotundo, Nella
- Thomas, Marita

*2010 Mathematics Subject Classification*

- 82D37 49J20 49M15

*Keywords*

- semiconductor design, optimal control, optoelectronics

*Abstract*

We discuss analytical and numerical methods for the optimization of optoelectronic devices by performing optimal control of the PDE governing the carrier transport with respect to the doping profile. First, we provide a cost functional that is a sum of a regularization and a contribution, which is motivated by the modal net gain that appears in optoelectronic models of bulk or quantum-well lasers. Then, we state a numerical discretization, for which we study optimized solutions for different regularizations and for vanishing weights.

*Appeared in*

- J. Comput. Theor. Transp., 45 (2016), pp. 410--423.

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# Deriving effective models for multiscale systems via evolutionary $Gamma$-convergence

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 34E13 35R15 35K57 47J35 74Qxx

*Keywords*

- Reaction-diffusion systems, homogenization, gradient systems, evolutionary variational inequality, energy-dissipation principle, amplitude equations

*Abstract*

We discuss possible extensions of the recently established theory of evolutionary Gamma convergence for gradient systems to nonlinear dynamical systems obtained by perturbation of a gradient systems. Thus, it is possible to derive effective equations for pattern forming systems with multiple scales. Our applications include homogenization of reaction-diffusion systems, the justification of amplitude equations for Turing instabilities, and the limit from pure diffusion to reaction-diffusion. This is achieved by generalizing the Gamma-limit approaches based on the energy-dissipation principle or the evolutionary variational estimate.

*Appeared in*

- Control of Self-Organizing Nonlinear Systems, E. Schöll, S. Klapp, P. Hövel, eds., Understanding Complex Systems, Springer, 2016, pp. 235--251

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# Pohozaev identities for anisotropic integro-differential operators

*Authors*

- Ros-Oton, Xavier
- Serra, Joaquim
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R09 47G20 35A01

*Keywords*

- Pohozaev identity, stable Lévy processes, nonlocal operator.

*Abstract*

We establish Pohozaev identities and integration by parts type formulas for anisotropic integro-differential operators of order 2s, with s ϵ (0, 1). These identities involve local boundary terms, in which the quantity u/ds ∂Ω plays the role that ∂u/∂v plays in the second order case. Here, u is any solution to Lu = f (x, u) in Ω, with u = 0 in Rn \ Ω , and d is the distance to ∂Ω.

*Appeared in*

- Comm. Partial Differential Equations, 42 (2017), pp. 1290-1321.

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# Solutions of the fractional Allen--Cahn equation which are invariant under screw motion

*Authors*

- Cinti, Eleonora
- Davila, Juan
- Del Pino, Manuel

*2010 Mathematics Subject Classification*

- 35J61 35J20 35B08 47J30

*Keywords*

- Fractional Laplacian, entire solutions, nonlocal perimeter

*Abstract*

We establish existence and non-existence results for entire solutions to the fractional Allen--Cahn equation in R3 , which vanish on helicoids and are invariant under screw-motion. In addition, we prove that helicoids are surfaces with vanishing nonlocal mean curvature.

*Appeared in*

- J. London Math. Soc. (2), 94 (2016) pp. 295--313.

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# Large deviations for empirical measures generated by Gibbs measures with singular energy functionals

*Authors*

- Dupuis, Paul
- Laschos, Vaios
- Ramanan, Kavita

*2010 Mathematics Subject Classification*

- 60F10 60K35 60B20

*Keywords*

- Large deviations principle, empirical measures, Gibbs measures, interacting particle systems, singular potential, rate function, weak topology, Wasserstein topology, relative entropy, Coulomb gases, random matrices

*Abstract*

We establish large deviation principles (LDPs) for empirical measures associated with a sequence of Gibbs distributions on n-particle configurations, each of which is defined in terms of an inverse temperature b_{n} and an energy functional that is the sum of a (possibly singular) interaction and confining potential. Under fairly general assumptions on the potentials, we establish LDPs both with speeds (b_{n})/(n) ® ¥, in which case the rate function is expressed in terms of a functional involving the potentials, and with the speed b_{n} =n, when the rate function contains an additional entropic term. Such LDPs are motivated by questions arising in random matrix theory, sampling and simulated annealing. Our approach, which uses the weak convergence methods developed in ``A weak convergence approach to the theory of large deviations", establishes large deviation principles with respect to stronger, Wasserstein-type topologies, thus resolving an open question in ``First-order global asymptotics for confined particles with singular pair repulsion". It also provides a common framework for the analysis of LDPs with all speeds, and includes cases not covered due to technical reasons in previous works.

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# Multilevel interpolation of divergence-free vector fields

*Authors*

- Farrell, Patricio

ORCID: 0000-0001-9969-6615 - Gillow, Kathryn
- Wendland, Holger

*2010 Mathematics Subject Classification*

- 65D15 65D05

*Keywords*

- meshfree methods, multilevel approximation, divergence-free, radial basis functions

*Abstract*

We introduce a multilevel technique for interpolating scattered data of divergence-free vector fields with the help of matrix-valued compactly supported kernels. The support radius at a given level is linked to the mesh norm of the data set at that level. There are at least three advantages of this method: no grid structure is necessary for the implementation, the multilevel approach is computationally cheaper than solving a large one-shot system and the interpolant is guaranteed to be analytically divergence-free. Furthermore, though we will not pursue this here, our multiscale approach is able to represent multiple scales in the data if present. We will prove convergence of the scheme, stability estimates and give a numerical example.

*Appeared in*

- IMA J. Numer. Anal., 37 (2017) pp. 332--353, DOI 10.1093/imanum/drw006 .

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# Prescribed conditions at infinity for fractional parabolic and elliptic equations with unbounded coefficients

*Authors*

- Punzo, Fabio
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35K61 35K67 35B99 35B40 35B51

*Keywords*

- Existence and uniqueness results, fractional parabolic and elliptic equations

*Abstract*

We investigate existence and uniqueness of solutions to a class of fractional parabolic equations satisfying prescribed pointwise conditions at infinity (in space), which can be time-dependent. Moreover, we study the asymptotic behaviour of such solutions. We also consider solutions of elliptic equations satisfying appropriate conditions at infinity.

*Appeared in*

- ESAIM: COCV, 24 (2018) pp. 105--127.

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# Sharp thresholds for Gibbs-non-Gibbs transition in the fuzzy Potts models with a Kac-type interaction

*Authors*

- Jahnel, Benedikt

ORCID: 0000-0002-4212-0065 - Külske, Christof

*2010 Mathematics Subject Classification*

- 82B20 82B26

*Keywords*

- Potts models, Kac model, fuzzy Lac-Potts model, Gibbs versus non-Gibbs, large deviation principles, diluted large deviation principles

*Abstract*

We investigate the Gibbs properties of the fuzzy Potts model on the $d$-dimensional torus with Kac interaction. We use a variational approach for profiles inspired by that of Fernández, den Hollander and Martínez citeFeHoMa14 for their study of the Gibbs-non-Gibbs transitions of a dynamical Kac-Ising model on the torus. As our main result, we show that the mean-field thresholds dividing Gibbsian from non-Gibbsian behavior are sharp in the fuzzy Kac-Potts model. On the way to this result we prove a large deviation principle for color profiles with diluted total mass densities and use monotocity arguments

*Appeared in*

- Bernoulli, 23 (2017) pp. 2808--2827.

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# Narrowing of the far field in spatially modulated edge-emitting broad area semiconductor amplifiers

*Authors*

- Radziunas, Mindaugas
- Herrero, Ramon
- Botey, Muriel
- Staliunas, Kestutis

*2010 Mathematics Subject Classification*

- 35Q60 35B27 78A60 78A45 42B37

*2008 Physics and Astronomy Classification Scheme*

- 42.60.By 42.60.Da 42.60.Fc 42.60.Jf

*Keywords*

- semiconductor amplifier, traveling wave model, coupled mode approach, periodic structure, anisotropy, far field, spatial filtering

*Abstract*

We perform a detailed theoretical analysis of the far field narrowing in broad-area edgeemitting semiconductor amplifiers that are electrically injected through the contacts periodically modulated in both, longitudinal and transverse, directions. The beam propagation properties within the semiconductor amplifier are explored by a (1+2)-dimensional traveling wave model and its coupled mode approximation. Assuming a weak field regime, we analyze the impact of different parameters and modulation geometry on the narrowing of the principal far field component.

*Appeared in*

- J. Opt. Soc. Amer. B Opt. Phys., 32 (2015) pp. 993--1000.

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# A comparative numerical study of meshing functionals for variational mesh adaptation

*Authors*

- Huang, Weizhang
- Kamenski, Lennard
- Russell, Robert D.

*2010 Mathematics Subject Classification*

- 65N5 65K10

*Keywords*

- variational mesh adaptation, mesh adaptation, moving mesh, equidistribution, alignment, mesh quality measures

*Abstract*

We present a comparative numerical study for three functionals used for variational mesh adaptation. One of them is a generalization of Winslow's variable diffusion functional while the others are based on equidistribution and alignment. These functionals are known to have nice theoretical properties and work well for most mesh adaptation problems either as a stand-alone variational method or combined within the moving mesh framework. Their performance is investigated numerically in terms of equidistribution and alignment mesh quality measures. Numerical results in 2D and 3D are presented.

*Appeared in*

- J. Math. Study, 48 (2015) pp. 168--186.

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# Error estimates of B-spline based finite-element method for the wind-driven ocean circulation

*Authors*

- Rotundo, Nella
- Kim, Tae-Yeon
- Jiang, Wen
- Heltai, Luca
- Fried, Eliot

*2010 Mathematics Subject Classification*

- 65M15 65D07 74S05

*Keywords*

- Error estimates, B-splines, geostrophic equations, ocean circulation, Nitsche's method

*Abstract*

We present the error analysis of a B-spline based finite-element approximation of the stream-function formulation of the large scale wind-driven ocean circulation. In particular, we derive optimal error estimates for h-refinement using a Nitsche-type variational formulations of the two simplied linear models of the stationary quasigeostrophic equations, namely the Stommel and Stommel--Munk models. Numerical results on rectangular and embedded geometries confirm the error analysis.

*Appeared in*

- J. Sci. Comput., 69 (2016), pp. 430--459.

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# A tweezer for chimeras in small networks

*Authors*

- Omelchenko, Iryna
- Omel'chenko, Oleh

ORCID: 0000-0003-0526-1878 - Zakharova, Anna
- Wolfrum, Matthias
- Schöll, Eckehard

*2010 Mathematics Subject Classification*

- 34H10 34C15

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Xt, 05.45.Ra, 89.75.-k

*Keywords*

- nonlinear systems, dynamical networks, coherence, spatial chaos

*Abstract*

We propose a control scheme which can stabilize and fix the position of chimera states in small networks. Chimeras consist of coexisting domains of spatially coherent and incoherent dynamics in systems of nonlocally coupled identical oscillators. Chimera states are generically difficult to observe in small networks due to their short lifetime and erratic drifting of the spatial position of the incoherent domain. The control scheme, like a tweezer, might be useful in experiments, where usually only small networks can be realized.

*Appeared in*

- Phys. Rev. Lett., 116 (2016) pp. 114101/1--114101/5.

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# Timing jitter of passively mode-locked semiconductor lasers subject to optical feedback; a semi-analytic approach

*Authors*

- Jaurigue, Lina
- Pimenov, Alexander
- Rachinskii, Dmitrii
- Schöll, Eckehard
- Lüdge, Kathy
- Vladimirov, Andrei G.

*2008 Physics and Astronomy Classification Scheme*

- 42.60.Fc 05.45.-a 42.60.Mi 42.55.Px

*Keywords*

- Timing jitter, semiconductor lasers, optical feedback, passive mode-locking

*Abstract*

We propose a semi-analytical method of calculating the timing fluctuations in mode-locked semiconductor lasers and apply it to study the effect of delayed coherent optical feedback on pulse timing jitter in these lasers. The proposed method greatly reduces computation times and therefore allows for the investigation of the dependence of timing fluctuations over greater parameter domains. We show that resonant feedback leads to a reduction in the timing jitter and that a frequency-pulling region forms about the main resonances, within which a timing jitter reduction is observed. The width of these frequency-pulling regions increases linearly with short feedback delay times. We derive an analytic expression for the timing jitter, which predicts a monotonic decrease in the timing jitter for resonant feedback of increasing delay lengths, when timing jitter effects are fully separated from amplitude jitter effects. For long feedback cavities the decrease in timing jitter scales approximately as $1/tau$ with the increase of the feedback delay time $tau$.

*Appeared in*

- Phys. Rev. A, 92 (2015) pp. 053807/1--053807/11.

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# On a nonlocal Cahn--Hilliard equation with a reaction term

*Authors*

- Melchionna, Stefano
- Rocca, Elisabetta

*2010 Mathematics Subject Classification*

- 34B10 35K57 35D35

*Keywords*

- nonlocal Cahn-Hilliard, reaction terms, global-in-time existence, uniqurness, regularity, separation properties

*Abstract*

We prove existence, uniqueness, regularity and separation properties for a nonlocal Cahn- Hilliard equation with a reaction term. We deal here with the case of logarithmic potential and degenerate mobility as well an uniformly lipschitz in u reaction term g(x, t, u).

*Appeared in*

- Adv. Math. Sci. Appl., 24 (2014) pp. 461--497.

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# Interpolation inequalities in pattern formation

*Authors*

- Cinti, Eleonora
- Otto, Felix

*2010 Mathematics Subject Classification*

- 49J40 47J20

*Keywords*

- Interpolation inequalities, optimal transport, pattern formation

*Abstract*

We prove some interpolation inequalities which arise in the analysis of pattern formation in physics. They are the strong version of some already known estimates in weak form that are used to give a lower bound of the energy in many contexts (coarsening and branching in micromagnetics and superconductors). The main ingredient in the proof of our inequalities is a geometric construction which was first used by Choksi, Conti, Kohn, and one of the authors in citeCCKO in the study of branching in superconductors.

*Appeared in*

- J. Funct. Anal., 271 (2016) pp. 1043--1376.

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# On the evolution of the empirical measure for hard-sphere dynamics

*Authors*

- Simonella, Sergio
- Pulvirenti, Mario

*2010 Mathematics Subject Classification*

- 82C05 82C05 82C40 35Q20

*Keywords*

- BBGKY hierarchy, hard sphere, empirical measure, marginal, Enskog equation

*Abstract*

We prove that the evolution of marginals associated to the empirical measure of a finite system of hard spheres is driven by the BBGKY hierarchical expansion. The usual hierarchy of equations for L^1 measures is obtained as a corollary. We discuss the ambiguities arising in the corresponding notion of microscopic series solution to the Boltzmann-Enskog equation.

*Appeared in*

- Bull. Inst. Math. Acad. Sin., 10 (2015) pp. 171--204.

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# Zero-one law for directional transience of one-dimensional random walks in dynamic random environments

*Authors*

- Orenshtein, Tal
- Soares dos Santos, Renato

*2010 Mathematics Subject Classification*

- 60F20 60K37 82B41 82C44

*Keywords*

- random walk, dynamic random environment, zero-one law, directional transience, recurrence

*Abstract*

We prove the trichotomy between transience to the right, transience to the left and recurrence of one-dimensional nearest-neighbour random walks in dynamic random environments under fairly general assumptions, namely: stationarity under space-time translations, ergodicity under spatial translations, and a mild ellipticity condition. In particular, the result applies to general uniformly elliptic models and also to a large class of non-uniformly elliptic cases that are i.i.d. in space and Markovian in time.

*Appeared in*

- Electron. Comm. Probab., 21 (2016), pp. 15/1--15/11.

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# Existence, numerical convergence, and evolutionary relaxation for a rate-independent phase-transformation model

*Authors*

- Heinz, Sebastian
- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 49S05 35Q74 74C05 74N15

*Keywords*

- Energetic solution, mutual recovery sequences, H-measures, laminates, two-phase material model, evolutionary Gamma-convergence

*Abstract*

We revisit the two-well model for phase transformation in a linearly elastic body introduced and studied in A. Mielke, F. Theil, and V.I. Levita ``A variational formulation of rate--independent phase transformations using an extremum principle", *Arch. Rational Mech. Anal.*, 162, 137-177, 2002 ([MTL02]). This energetic rate-independent model is posed in terms of the elastic displacement and an internal variable that gives the phase portion of the second phase. We use a new approach based on *mutual recovery sequences*, which are adjusted to a suitable energy increment plus the associated dissipated energy and, thus, enable us to pass to the limit in the construction of energetic solutions. We give three distinct constructions of mutual recovery sequences which allow us (i) to generalize the existence result in [MTL02], (ii) to establish the convergence of suitable numerical approximations via space-time discretization, and (iii) to perform the evolutionary relaxation from the pure-state model to the relaxed mixture model. All these results rely on weak converge and involve the H-measure as an essential tool.

*Appeared in*

- Phil. Trans. R. Soc. A, 374 (2016), pp. 20150171/1--20150171/23, DOI 10.1098/rsta.2015.0171 .

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# Delay-induced patterns in a two-dimensional lattice of coupled oscillators

*Authors*

- Kantner, Markus
- Schöll, Eckehard
- Yanchuk, Serhiy

*2008 Physics and Astronomy Classification Scheme*

- 05.45.-a, 02.30.Ks, 89.75.Kd

*Keywords*

- Delay-coupled systems, Pattern formation, Bifurcations, Stability analysis

*Abstract*

We show how a variety of stable spatio-temporal periodic patterns can be created in 2D-lattices of coupled oscillators with non-homogeneous coupling delays. The results are illustrated using the FitzHugh-Nagumo coupled neurons as well as coupled limit cycle (Stuart-Landau) oscillators. A "hybrid dispersion relation" is introduced, which describes the stability of the patterns in spatially extended systems with large time-delay.

*Appeared in*

- Sci. Rep., 5 (2015) pp. 8522/1--8522/9.

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# Extremes of the supercritical Gaussian free field

*Authors*

- Chiarini, Alberto
- Cipriani, Alessandra
- Hazra, Rajat Subhra

*2010 Mathematics Subject Classification*

- 60K35 60G15 60G60 60G70

*Keywords*

- discrete Gaussian free field, extreme value theory, Gumbel distribution, Stein-Chen method

*Abstract*

We show that the rescaled maximum of the discrete Gaussian Free Field (DGFF) in dimension larger or equal to 3 is in the maximal domain of attraction of the Gumbel distribution. The result holds both for the infinite-volume field as well as the field with zero boundary conditions. We show that these results follow from an interesting application of the Stein-Chen method from Arratia et al. (1989).

*Appeared in*

- ALEA Lat. Am. J. Probab. Math. Stat., 13 (2016), pp. 711--724.

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# Mesh smoothing: An MMPDE approach

*Authors*

- Huang, Weizhang
- Kamenski, Lennard
- Si, Hang

*2010 Mathematics Subject Classification*

- 65N50 65K10

*Keywords*

- mesh smoothing, moving mesh method, tetrahedral meshes

*Abstract*

We study a mesh smoothing algorithm based on the moving mesh PDE (MMPDE) method. For the MMPDE itself, we employ a simple and efficient direct geometric discretization of the underlying meshing functional on simplicial meshes. The nodal mesh velocities can be expressed in a simple, analytical matrix form, which makes the implementation of the method relatively easy and simple. Numerical examples are provided.

*Appeared in*

- Research Notes, 24th International Meshing Roundtable (2015)

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# Fractional elliptic problems with critical growth in the whole of (R^n)

*Authors*

- Dipierro, Serena
- Medina, Maria
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35A15 35B40 35D30 35J20 35R11 49N60

*Keywords*

- Fractional equation, critical problem, concentration-compactness principle, Mountain Pass Theorem

*Abstract*

We study a nonlinear and nonlocal elliptic equation. The problem has a variational structure, and this allows us to find a positive solution by looking at critical points of a suitable energy functional. In particular, in this paper, we find a local minimum and a mountain pass solution of this functional. One of the crucial ingredient is a Concentration-Compactness principle. Some difficulties arise from the nonlocal structure of the problem and from the fact that we deal with an equation in the whole of the space (and this causes lack of compactness of some embeddings). We overcome these difficulties by looking at an equivalent extended problem.

*Appeared in*

- Lecture Notes (Scuola Normale Superiore) 2017.

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# Analysis of p(x)-Laplace thermistor models describing the electrothermal behavior of organic semiconductor devices

*Authors*

- Glitzky, Annegret
- Liero, Matthias

ORCID: 0000-0002-0963-2915

*2010 Mathematics Subject Classification*

- 35J92 35Q79 35B65 80A20 35J57

*Keywords*

- Thermistor model, p(x)-Laplacian, nonlinear coupled system, existence and boundedness, regularity theory, Caccioppoli estimates, organic light emitting diode, self-heating, Arrhenius-like conductivity law

*Abstract*

We study a stationary thermistor model describing the electrothermal behavior of organic semiconductor devices featuring non-Ohmic current-voltage laws and self-heating effects. The coupled system consists of the current-flow equation for the electrostatic potential and the heat equation with Joule heating term as source. The self-heating in the device is modeled by an Arrhenius-like temperature dependency of the electrical conductivity. Moreover, the non-Ohmic electrical behavior is modeled by a power law such that the electrical conductivity depends nonlinearly on the electric field. Notably, we allow for functional substructures with different power laws, which gives rise to a $p(x)$-Laplace-type problem with piecewise constant exponent. We prove the existence and boundedness of solutions in the two-dimensional case. The crucial point is to establish the higher integrability of the gradient of the electrostatic potential to tackle the Joule heating term. The proof of the improved regularity is based on Caccioppoli-type estimates, Poincaré inequalities, and a Gehring-type Lemma for the $p(x)$-Laplacian. Finally, Schauder's fixed-point theorem is used to show the existence of solutions.

*Appeared in*

- Nonlinear Anal. Real World Appl., 34 (2017), pp. 536--562.

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# Regular and irregular patterns of self-localized excitation in arrays of coupled phase oscillators

*Authors*

- Wolfrum, Matthias
- Omel'chenko, Oleh

ORCID: 0000-0003-0526-1878 - Sieber, Jan

*2010 Mathematics Subject Classification*

- 34C15 37N20 37N25

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Xt 89.75.Kd

*Keywords*

- Coupled oscillators, regular and irregular patterns, self-localized excitation, chimera states, route to chaos

*Abstract*

We study a system of phase oscillators with non-local coupling in a ring that supports self-organized patterns of coherence and incoherence, called chimera states. Introducing a global feedback loop, connecting the phase lag to the order parameter, we can observe chimera states also for systems with a small number of oscillators. Numerical simulations show a huge variety of regular and irregular patterns composed of localized phase slipping events of single oscillators. Using methods of classical finite dimensional chaos and bifurcation theory, we can identify the emergence of chaotic chimera states as a result of transitions to chaos via period doubling cascades, torus breakup, and intermittency. We can explain the observed phenomena by a mechanism of self-modulated excitability in a discrete excitable medium.

*Appeared in*

- Chaos, 25 (2015) pp. 053113/1--053113/7.

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# Large-deviation principles for connectable receivers in wireless networks

*Authors*

- Hirsch, Christian
- Jahnel, Benedikt

ORCID: 0000-0002-4212-0065 - Keeler, Paul

ORCID: 0000-0002-2063-1075 - Patterson, Robert I. A.

ORCID: 0000-0002-3583-2857

*2010 Mathematics Subject Classification*

- 60F10 60K35

*Keywords*

- wireless network, signal-to-interference-and-noise ratio, large-deviation principle, importance sampling

*Abstract*

We study large-deviation principles for a model of wireless networks consisting of Poisson point processes of transmitters and receivers, respectively. To each transmitter we associate a family of connectable receivers whose signal-to-interference-and-noise ratio is larger than a certain connectivity threshold. First, we show a large-deviation principle for the empirical measure of connectable receivers associated with transmitters in large boxes. Second, making use of the observation that the receivers connectable to the origin form a Cox point process, we derive a large-deviation principle for the rescaled process of these receivers as the connection threshold tends to zero. Finally, we show how these results can be used to develop importance-sampling algorithms that substantially reduce the variance for the estimation of probabilities of certain rare events such as users being unable to connect.

*Appeared in*

- Adv. Appl. Probab., 48 (2016) pp. 1061--1094.

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# On stable solutions of boundary reaction-diffusion equations and applications to nonlocal problems with Neumann data

*Authors*

- Dipierro, Serena
- Soave, Nicola
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35J62 35J92 35J93 35B53

*Keywords*

- Stability, symmetry results, classification of solution, reaction-diffusion equations, nonlocal equations

*Abstract*

We study reaction-diffusion equations in cylinders with possibly nonlinear diffusion and possibly nonlinear Neumann boundary conditions. We provide a geometric Poincar´e-type inequality and classification results for stable solutions, and we apply them to the study of an associated nonlocal problem. We also establish a counterexample in the corresponding framework for the fractional Laplacian.

*Appeared in*

- Indiana Univ. Math. J., 67:1 (2018) pp. 429--469.

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# Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Patterson, Robert I. A.

ORCID: 0000-0002-3583-2857 - Peletier, Mark A.
- Renger, D. R. Michiel

ORCID: 0000-0003-3557-3485

*2010 Mathematics Subject Classification*

- 80A30 82C35 60F10

*2008 Physics and Astronomy Classification Scheme*

- 82.20.Db

*Keywords*

- Reaction networks, non-equilibrium thermodynamics, large deviations of interacting particle systems

*DOI*

*Abstract*

We study stochastic interacting particle systems that model chemical reaction networks on the micro scale, converging to the macroscopic Reaction Rate Equation. One abstraction level higher, we study the ensemble of such particle systems, converging to the corresponding Liouville transport equation. For both systems, we calculate the corresponding large deviations and show that under the condition of detailed balance, the large deviations induce a non-linear relation between thermodynamic fluxes and free energy driving force.

*Appeared in*

- SIAM J. Appl. Math., 77 (2017), pp. 1562--1585, DOI 10.1137/16M1102240 .

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# Instabilities of stationary states in lasers with long-delay optical feedback

*Authors*

- Yanchuk, Serhiy
- Wolfrum, Matthias

*2010 Mathematics Subject Classification*

- 34K60 34K20 34K26

*Keywords*

- Lang-Kobayashi system, laser with feedback, external cavity mode, singularly perturbed delay-differential equation, stability

*Abstract*

We study the Lang-Kobayashi model in the long-delay limit, focussing our attention on the stability properties of external cavity modes (ECMs) of this system. We show that ECMs can display different types of instabilities: strong instabilities and weak modulational-type instability. We explain the origin of these instabilities and show how they affect the complicated dynamics of the Lang-Kobayashi model.

*Appeared in*

- SIAM J. Appl. Dyn. Syst. Vol. 9, No. 2 (2010), 519-535.

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# Is a nonlocal diffusion strategy convenient for biological populations in competition?

*Authors*

- Massaccesi, Annalisa
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35Q92 46N60

*Keywords*

- Fractional equations, population dynamics

*Abstract*

We study the convenience of a nonlocal dispersal strategy in a reaction-diffusion system with a fractional Laplacian operator. We show that there are circumstances - namely, a precise condition on the distribution of the resource - under which a nonlocal dispersal behavior is favored. In particular, we consider the linearization of a biological system that models the interaction of two biological species, one with local and one with nonlocal dispersal, that are competing for the same resource. We give a simple, concrete example of resources for which the equilibrium with only the local population becomes linearly unstable. In a sense, this example shows that nonlocal strategies can become successful even in an environment in which purely local strategies are dominant at the beginning, provided that the resource is sufficiently sparse. Indeed, the example considered presents a high variance of the distribu- tion of the dispersal, thus suggesting that the shortage of resources and their unbalanced supply may be some of the basic ingredients that favor nonlocal strategies.

*Appeared in*

- J. Math. Biol. 74 (2017) pp. 113-147.

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# Amplitude equations for collective spatio-temporal dynamics in arrays of coupled systems

*Authors*

- Yanchuk, Serhiy
- Perlikowski, Przemysław
- Wolfrum, Matthias
- Stefański, Andrzej
- Kapitaniak, Tomasz

*Keywords*

- coupled oscillators, amplitude equations, Ginzburg-Landau equation, spatio-temporal chaos

*Abstract*

We study the coupling induced destabilization in an array of identical oscillators coupled in a ring structure where the number of oscillators in the ring is large. The coupling structure includes different types of interactions with several next neighbors. We derive an amplitude equation of Ginzburg-Landau type, which describes the destabilization of a uniform stationary state and close-by solutions in the limit of a large number of nodes. Studying numerically an example of unidirectionally coupled Duffing oscillators, we observe a coupling induced transition to collective spatio-temporal chaos, which can be understood using the derived amplitude equations.

*Appeared in*

- Chaos, 25 (2015) pp. 033113/1--033113/8.

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# The Dirichlet problem for nonlocal operators with kernels: Convex and nonconvex domains

*Authors*

- Ros-Oton, Xavier
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35B65 35R11

*Keywords*

- Regularity theory, integro-differential equations, fractional Laplacian, anisotropic media, rough kernels

*Abstract*

We study the interior regularity of solutions to a Dirichlet problem for anisotropic operators of fractional type. A prototype example is given by the sum of one-dimensional fractional Laplacians in fixed, given directions. We prove here that an interior differentiable regularity theory holds in convex domains. When the spectral measure is a bounded function and the domain is smooth, the same regularity theory applies. In particular, solutions always possess a classical first derivative. The assumptions on the domain are sharp, since if the domain is not convex and the spectral measure is singular, we construct an explicit counterexample.

*Appeared in*

- Adv. Math., 288 (2016) pp. 732--790.

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# From large deviations to Wasserstein gradient flows in multiple dimensions

*Authors*

- Erbar, Matthias
- Maas, Jan
- Renger, D. R. Michiel

ORCID: 0000-0003-3557-3485

*2010 Mathematics Subject Classification*

- 35A15 35Q84 46N55 60F10

*Keywords*

- large deviations, gradient flows, Wasserstein calculus, Gamma convergence

*Abstract*

We study the large deviation rate functional for the empirical distribution of independent Brownian particles with drift. In one dimension, it has been shown by Adams, Dirr, Peletier and Zimmer [ADPZ11] that this functional is asymptotically equivalent (in the sense of Gamma-convergence) to the Jordan-Kinderlehrer-Otto functional arising in the Wasserstein gradient flow structure of the Fokker-Planck equation. In higher dimensions, part of this statement (the lower bound) has been recently proved by Duong, Laschos and Renger, but the upper bound remained open, since the proof in [DLR13] relies on regularity properties of optimal transport maps that are restricted to one dimension. In this note we present a new proof of the upper bound, thereby generalising the result of [ADPZ11] to arbitrary dimensions.

*Appeared in*

- Electron. Comm. Probab., 20 (2015) pp. 1--12.

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# Relaxation times for atom dislocations in crystals

*Authors*

- Patrizi, Stefania
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 82D25 35R09 74E15 35R11 47G20

*Keywords*

- Peierls-Nabarro model, nonlocal integro-differential equations, dislocation dynamics, attractive/repulsive potentials, collisions

*Abstract*

We study the relaxation times for a parabolic differential equation whose solution represents the atom dislocation in a crystal. The equation that we consider comprises the classical Peierls-Nabarro model as a particular case, and it allows also long range interactions. It is known that the dislocation function of such a model has the tendency to concentrate at single points, which evolve in time according to the external stress and a singular, long range potential. Depending on the orientation of the dislocation function at these points, the potential may be either attractive or repulsive, hence collisions may occur in the latter case and, at the collision time, the dislocation function does not disappear. The goal of this paper is to provide accurate estimates on the relaxation times of the system after collision. More precisely, we take into account the case of two and three colliding points, and we show that, after a small transition time subsequent to the collision, the dislocation function relaxes exponentially fast to a steady state. We stress that the exponential decay is somehow exceptional in nonlocal problems (for instance, the spatial decay in this case is polynomial). The exponential time decay is due to the coupling (in a suitable space/time scale) between the evolution term and the potential induced by the periodicity of the crystal.

*Appeared in*

- Calc. Var. Partial Differ. Equ., 55 (2016) pp. 71/1--71/44.

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# Mean-field interaction of Brownian occupation measures. I: Uniform tube property of the Coulomb functional

*Authors*

- König, Wolfgang
- Mukherjee, Chiranjib

*2010 Mathematics Subject Classification*

- 60J65 60J55 60F10

*Keywords*

- Gibbs measures, interacting Brownian motions, Coulomb functional, large deviations

*Abstract*

We study the transformed path measure arising from the self-interaction of a three-dimensional Brownian motion via an exponential tilt with the Coulomb energy of the occupation measures of the motion by time $t$. The logarithmic asymptotics of the partition function were identified in the 1980s by Donsker and Varadhan [DV83-P] in terms of a variational formula. Recently [MV14] a new technique for studying the path measure itself was introduced, which allows for proving that the normalized occupation measure asymptotically concentrates around the set of all maximizers of the formula. In the present paper, we show that likewise the Coulomb functional of the occupation measure concentrates around the set of corresponding Coulomb functionals of the maximizers in the uniform topology. This is a decisive step on the way to a rigorous proof of the convergence of the normalized occupation measures towards an explicit mixture of the maximizers, which will be carried out elsewhere. Our methods rely on deriving Hölder-continuity of the Coulomb functional of the occupation measure with exponentially small deviation probabilities and invoking the large-deviation theory developed in [MV14] to a certain shift-invariant functional of the occupation measures.

*Appeared in*

- Ann. Inst. H. Poincare Probab. Statist., 53 (2017), pp. 2214--2228, DOI 10.1214/16-AIHP788 .

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# Rates of convergence for extremes of geometric random variables and marked point processes

*Authors*

- Cipriani, Alessandra
- Feidt, Anne

*2010 Mathematics Subject Classification*

- 60F99 62E20

*Keywords*

- Stein-Chen method, maxima of geometric random variables, Marshall-Olkin geometric distribution, Poisson approximation, marked point process of extremes

*Abstract*

We use the Stein-Chen method to study the extremal behaviour of univariate and bivariate geometric laws. We obtain a rate for the convergence, to the Gumbel distribution, of the law of the maximum of i.i.d. geometric random variables, and show that convergence is faster when approximating by a discretised Gumbel. We similarly find a rate of convergence for the law of maxima of bivariate Marshall-Olkin geometric random pairs when approximating by a discrete limit law. We introduce marked point processes of exceedances (MPPEs), both with univariate and bivariate Marshall-Olkin geometric variables as marks and determine bounds on the error of the approximation, in an appropriate probability metric, of the law of the MPPE by that of a Poisson process with same mean measure. We then approximate by another Poisson process with an easier-to-use mean measure and estimate the error of this additional approximation. This work contains and extends results contained in the second author's PhD thesis under the supervision of Andrew D. Barbour. The thesis is available at http://arxiv.org/abs/1310.2564.

*Appeared in*

- Extremes, 19 (2016), pp. 105--138.

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# Projection methods for incompressible flow problems with WENO finite difference schemes

*Authors*

- de Frutos, Javier
- John, Volker

ORCID: 0000-0002-2711-4409 - Novo, Julia

*2010 Mathematics Subject Classification*

- 65M06

*Keywords*

- incompressible Navier-Stokes equations, finite difference, WENO schemes, non-incremental projection methods, incremental projection methods, PSPG-type stabilization

*Abstract*

Weighted essentially non-oscillatory (WENO) finite difference schemes have been recommended in a competitive study of discretizations for scalar evolutionary convection-diffusion equations [20]. This paper explores the applicability of these schemes for the simulation of incompressible flows. To this end, WENO schemes are used in several non-incremental and incremental projection methods for the incompressible Navier-Stokes equations. Velocity and pressure are discretized on the same grid. A pressure stabilization Petrov-Galerkin (PSPG) type of stabilization is introduced in the incremental schemes to account for the violation of the discrete inf-sup condition. Algorithmic aspects of the proposed schemes are discussed. The schemes are studied on several examples with different features. It is shown that the WENO finite difference idea can be transferred to the simulation of incompressible flows. Some shortcomings of the methods, which are due to the splitting in projection schemes, become also obvious.

*Appeared in*

- J. Comput. Phys., 309 (2016) pp. 368--386.

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# Properties of the solutions of delocalised coagulation and inception problems with outflow boundaries

*Authors*

- Patterson, Robert I. A.

ORCID: 0000-0002-3583-2857

*2010 Mathematics Subject Classification*

- 34G20 35A01 35A02 82C22

*Keywords*

- Coagulation, advection, existence, uniqueness, regularity, Banach ODE, propagator, boundary

*Abstract*

Well posedness is established for a family of equations modelling particle populations undergoing delocalised coagulation, advection, inflow and outflow in a externally specified velocity field. Very general particle types are allowed while the spatial domain is a bounded region of $d$-dimensional space for which every point lies on exactly one streamline associated with the velocity field. The problem is formulated as a semi-linear ODE in the Banach space of bounded measures on particle position and type space. A local Lipschitz property is established in total variation norm for the propagators (generalised semi-groups) associated with the problem and used to construct a Picard iteration that establishes local existence and global uniqueness for any initial condition. The unique weak solution is shown further to be a differentiable or at least bounded variation strong solution under smoothness assumptions on the parameters of the coagulation interaction. In the case of one spatial dimension strong differentiability is established even for coagulation parameters with a particular bounded variation structure in space. This one dimensional extension establishes the convergence of the simulation processes studied in [Patterson, textitStoch. Anal. Appl. 31, 2013] to a unique and differentiable limit.

*Appeared in*

- J. Evol. Equ., 16 (2016), pp. 261--291.

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# Outer limit of subdifferentials and calmness moduli in linear and nonlinear programming

*Authors*

- Cánovas, María J.
- Henrion, René
- López, Marco A.
- Parra, Juan

*2010 Mathematics Subject Classification*

- 90C31 49J53 94K40 90C25 90C05

*Keywords*

- calmness, local error bounds, variational analysis, linear programming, argmin mapping

*Abstract*

With a common background and motivation, the main contributions of this paper are developed in two different directions. Firstly, we are concerned with functions which are the maximum of a finite amount of continuously differentiable functions of *n* real variables, paying attention to the case of polyhedral functions. For these max-functions, we obtain some results about outer limits of subdifferentials, which are applied to derive an upper bound for the calmness modulus of nonlinear systems. When confined to the convex case, in addition, a lower bound on this modulus is also obtained. Secondly, by means of a KKT index set approach, we are also able to provide a point-based formula for the calmness modulus of the argmin mapping of linear programming problems without any uniqueness assumption on the optimal set. This formula still provides a lower bound in linear semi-infinite programming. Illustrative examples are given.

*Appeared in*

- J. Optim. Theory Appl., 169 (2016) pp. 925--952.

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