# Stochastic weighted particle methods for population balance equations

*Authors*

- Patterson, Robert I. A.

ORCID: 0000-0002-3583-2857 - Kraft, Markus

ORCID: 0000-0002-4293-8924 - Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 60J28 65C05 65C35

*Keywords*

- Monte Carlo, weighted particle, coagulation, surface growth, Smoluchowski, simulation

*DOI*

*Abstract*

A class of stochastic algorithms for the numerical treatment of population balance equations is introduced. The algorithms are based on systems of weighted particles, in which coagulation events are modelled by a weight transfer that keeps the number of computational particles constant. The weighting mechanisms are designed in such a way that physical processes changing individual particles (such as growth, or other surface reactions) can be conveniently treated by the algorithms. Numerical experiments are performed for complex laminar premixed flame systems. Two members of the class of stochastic weighted particle methods are compared to each other and to a direct simulation algorithm. One weighted algorithm is shown to be consistently better than the other with respect to the statistical noise generated. Finally, run times to achieve fixed error tolerances for a real flame system are measured and the better weighted algorithm is found to be up to three times faster than the direct simulation algorithm.

*Appeared in*

- J. Comput. Phys., 230 (2011) pp. 7456--7472 .

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# Some abstract error estimates of a finite volume scheme for a nonstationary heat equation on general nonconforming multidimensional spatial meshes

*Authors*

- Bradji, Abdallah

ORCID: 0000-0002-5889-492X - Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434

*2010 Mathematics Subject Classification*

- 65M08 65M15 35K15

*Keywords*

- Non-conforming grid, nonstationary heat equation, SUSHI scheme, implicit scheme, discrete gradient

*DOI*

*Abstract*

A general class of nonconforming meshes has been recently studied for stationary anisotropic heterogeneous diffusion problems by R. Eymard and coworkers. Thanks to these basic ideas developed for stationary problems, we derive a new discretization scheme in order to approximate the nonstationary heat problem. The unknowns of this scheme are the values at the centre of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. Although the numerical scheme stems from the finite volume method, its formulation is based on the discrete version for the weak formulation defined for the heat problem. We derive error estimates for the solution in discrete norm, and an error estimate for an approximation of the gradient, in a general framework in which the discrete bilinear form is satisfying ellipticity. We prove in particular, that, when the discrete flux is calculated using a stabilized discrete gradient, the convergence order is h+k , where h (resp. k) is the mesh size of the spatial (resp. time) discretization. This estimate is valid under the regularity assumption that the exact solution is twice continuously differentiable in time and space. These error estimates are useful because they allow us to get error estimates for the approximations of the exact solution and its first derivatives.

*Appeared in*

- Appl. Math., 58 (2013) pp. 1--38.

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# A numerical method for the simulation of an aggregation-driven population balance system

*Authors*

- Hackbusch, Wolfgang
- John, Volker

ORCID: 0000-0002-2711-4409 - Khachatryan, Aram
- Suciu, Carina

*2010 Mathematics Subject Classification*

- 45K05 44A35

*Keywords*

- Population balance systems, aggregation, integro partial differential equation, stabilised methods, convolution, calibration of parameters

*DOI*

*Abstract*

A population balance system which models the synthesis of urea is studied in this paper. The equations for the flow field, the mass and the energy balances are given in a three-dimensional domain and the equation for the particle size distribution (PSD) in a four-dimensional domain. This problem is convection-dominated and aggregation-driven. Both features require the application of appropriate numerical methods. This paper presents a numerical approach for simulating the population balance system which is based on finite element schemes, a finite difference method and a modern method to evaluate convolution integrals that appear in the aggregation term. Two experiments are considered and the numerical results are compared with experimental data. Unknown parameters in the aggregation kernel have to be calibrated. For appropriately chosen parameters, good agreements are achieved of the experimental data and the numerical results computed with the proposed method. A detailed study of the computational results reveals the influence of different parts of the aggregation kernel.

*Appeared in*

- Internat. J. Numer. Methods Fluids, 69 (2012) pp. 1646--1660.

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# A stochastic weighted particle method for coagulation-advection problems

*Authors*

- Patterson, Robert I. A.

ORCID: 0000-0002-3583-2857 - Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 6025 65C05 65C35 82C22

*Keywords*

- coagulation, advection, stochastic particle, instability

*DOI*

*Abstract*

A spatially resolved stochastic weighted particle method for inception--coagulation--advection problems is presented. Convergence to a deterministic limit is briefly studied. Numerical experiments are carried out for two problems with very different coagulation kernels. These tests show the method to be robust and confirm the convergence properties. The robustness of the weighted particle method is shown to contrast with two Direct Simulation Algorithms which develop instabilities.

*Appeared in*

- SIAM J. Sci. Comput., 34 (2012) pp. B290--B311.

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# Quasistatic damage evolution with spatial BV-regularization

*Authors*

- Thomas, Marita

*2010 Mathematics Subject Classification*

- 35K85 74R05 49J45 49S05 74R20

*Keywords*

- Partial damage, damage evolution with spatial regularization, functions of bounded variation, energetic formulation, Gamma convergence of rate-independent systems

*DOI*

*Abstract*

An existence result for energetic solutions of rate-independent damage processes is established. We consider a body consisting of a physically linearly elastic material undergoing infinitesimally small deformations and partial damage. In [ThomasMielke10DamageZAMM] an existence result in the small strain setting was obtained under the assumption that the damage variable z satisfies z∈ W^{1,r}(Ω) with r∈(1,∞) for Ω⊂R^{d}. We now cover the case r=1. The lack of compactness in W^{1,1}(Ω) requires to do the analysis in BV(Ω). This setting allows it to consider damage variables with values in 0,1. We show that such a brittle damage model is obtained as the Γ-limit of functionals of Modica-Mortola type.

*Appeared in*

- Discrete Contin. Dyn. Syst. Ser. S, 6 (2013) pp. 235--255.

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# Large deviations for cluster size distributions in a continuous classical many-body system

*Authors*

- Jansen, Sabine
- König, Wolfgang

ORCID: 0000-0002-4212-0065 - Metzger, Bernd

*2010 Mathematics Subject Classification*

- 82B21 60F10 60K35 82B31 82B05

*Keywords*

- Classical particle system, canonical ensemble, equilibrium statistical mechanics, dilute system, large deviations

*DOI*

*Abstract*

An interesting problem in statistical physics is the condensation of classical particles in droplets or clusters when the pair-interaction is given by a stable Lennard-Jones-type potential. We study two aspects of this problem. We start by deriving a large deviations principle for the cluster size distribution for any inverse temperature $betain(0,infty)$ and particle density $rhoin(0,rho_rmcp)$ in the thermodynamic limit. Here $rho_rmcp >0$ is the close packing density. While in general the rate function is an abstract object, our second main result is the $Gamma$-convergence of the rate function towards an explicit limiting rate function in the low-temperature dilute limit $betatoinfty$, $rho downarrow 0$ such that $-beta^-1logrhoto nu$ for some $nuin(0,infty)$. The limiting rate function and its minimisers appeared in recent work, where the temperature and the particle density were coupled with the particle number. In the de-coupled limit considered here, we prove that just one cluster size is dominant, depending on the parameter $nu$. Under additional assumptions on the potential, the $Gamma$-convergence along curves can be strengthened to uniform bounds, valid in a low-temperature, low-density rectangle.

*Appeared in*

- Ann. Appl. Probab., 25 (2015) pp. 930--973.

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# Path planning and collision avoidance for robots

*Authors*

- Gerdts, Matthias
- Henrion, René
- Hömberg, Dietmar
- Landry, Chantal

*2010 Mathematics Subject Classification*

- 49J15 49M25 49N90 90C30

*Keywords*

- Optimal control, collision avoidance, cooperative robots, backface culling, active set strategy

*DOI*

*Abstract*

An optimal control problem to find the fastest collision-free trajectory of a robot surrounded by obstacles is presented. The collision avoidance is based on linear programming arguments and expressed as state constraints. The optimal control problem is solved with a sequential programming method. In order to decrease the number of unknowns and constraints a backface culling active set strategy is added to the resolution technique.

*Appeared in*

- Numer. Algebra Control Optim., 2 (2012) pp. 437--463.

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# Tailoring THz radiation by controlling tunnel photoionization events in gases

*Authors*

- Babushkin, Ihar
- Skupin, Stefan
- Husakou, Anton
- Köhler, Christian
- Cabrera-Granado, Eduardo
- Bergé, Luc
- Herrmann, Joachim

*2010 Mathematics Subject Classification*

- 35Q60 78A60 76X05

*Keywords*

- THz emission, photoinduced plasma, nonlinear optics

*DOI*

*Abstract*

Applications ranging from nonlinear terahertz spectroscopy to remote sensing require broadband and intense THz radiation which can be generated by focusing two-color laser pulses into a gas. In this setup, THz radiation originates from the buildup of the electron density in sharp steps of attosecond duration due to tunnel ionization, and subsequent acceleration of free electrons in the laser field. We show that the spectral shape of the THz pulses generated by this mechanism is determined by superposition of contributions from individual ionization events. This provides a straightforward analogy with linear diffraction theory, where the ionization events play the role of slits in a grating. This analogy offers simple explanations for recent experimental observations and opens new avenues for THz pulse shaping based on temporal control of the ionization events. We illustrate this novel technique by tailoring the spectral width and position of the resulting radiation using multi-color pump pulses.

*Appeared in*

- New J. Phys., 13 (2011) pp. 123029.

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# Kramers--Kronig relations and high order nonlinear susceptibilities

*Authors*

- Brée, Carsten
- Demircan, Ayhan
- Steinmeyer, Günter

*2010 Mathematics Subject Classification*

- 78A60

*2008 Physics and Astronomy Classification Scheme*

- 42.65.-k 42.65.An 42.65.Jx 42.65.Hw

*Keywords*

- Nonlinear Optics, Femtosecond Filamentation, Pulse compression

*DOI*

*Abstract*

As previous theoretical results recently revealed, a Kramers-Kronig transform of multiphoton absorption rates allows for a precise prediction on the dispersion of the nonlinear refractive index $n_2$ in the near IR. It was shown that this method allows to reproduce recent experimental results on the importance of the higher-order Kerr effect. Extending these results, the current manuscript provides the dispersion of $n_2$ for all noble gases in excellent agreement with reference data. It is furthermore established that the saturation and inversion of the nonlinear refractive index is highly dispersive with wavelength, which indicates the existence of different filamentation regimes. While shorter laser wavelengths favor the well-established plasma clamping regime, the influence of the higher-order Kerr effect dominates in the long wavelength regime.

*Appeared in*

- Phys. Rev. A, 85 (2012) pp. 033806/1--033806/8.

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# Andronov--Hopf bifurcation of higher codimensions in a Liénard system

*Authors*

- Schneider, Klaus
- Grin, Alexander

*2010 Mathematics Subject Classification*

- 34A34 34C05 34C23 37G15

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Xt 89.75.Kd

*Keywords*

- Degenerate Hopf bifurcation, codimension three, multiple limit cycles, bifurcation diagrams

*DOI*

*Abstract*

Consider a polynominal Liènard system depending on three parameters itshape a, b, c and with the following properties: (i) The origin is the unique equilibrium for all parameters. (ii) Ifitshape a crosses zero, then the origin changes its stability, and a limit cycle bifurcates from the euqilibrium. We inverstigate analytically this bifurcation in dependence on the parameters itshape b and itshape c and establish the existence of families of limit cycles of multiplicity one, two and three bifurcating from the origin.

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# An optimization method in inverse elastic scattering for one-dimensional grating profiles

*Authors*

- Elschner, Johannes
- Hu, Guanghui

*2010 Mathematics Subject Classification*

- 35R30 74B05 78A46 35Q93

*Keywords*

- Diffraction grating, elastic waves, profile reconstruction, Tikhonov regularization, optimization method

*DOI*

*Abstract*

Consider the inverse diffraction problem to determine a two-dimensional periodic structure from scattered elastic waves measured above the structure. We formulate the inverse problem as a least squares optimization problem, following the two-step algorithm by G. Bruckner and J. Elschner (Inverse Problems (2003) 19, 315-329) for electromagnetic diffraction gratings. Such a method is based on the Kirsch-Kress optimization scheme and consists of two parts: a linear severely ill-posed problem and a nonlinear well-posed one. We apply this method to both smooth ($C^2$) and piecewise linear gratings for the Dirichlet boundary value problem of the Navier equation. Numerical reconstructions from exact and noisy data illustrate the feasibility of the method.

*Appeared in*

- Commun. Comput. Phys., 12 (2012) pp. 1434--1460.

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# Sturm--Liouville boundary value problems with operator potentials and unitary equivalence

*Authors*

- Malamud, Mark
- Neidhardt, Hagen

*2010 Mathematics Subject Classification*

- 34G10 47E05 47F05 47A20 47B25

*Keywords*

- Sturm-Liouville operators, operator potentials, elliptic partial differential operators, boundary value problems, self-adjoint extensions, unitary equivalence, direct sums of symmetric operators

*DOI*

*Abstract*

Consider the minimal Sturm-Liouville operator $A = A_rm min$ generated by the differential expression $cA := -fracd^2dt^2 + T$ in the Hilbert space $L^2(R_+,cH)$ where $T = T^*ge 0$ in $cH$. We investigate the absolutely continuous parts of different self-adjoint realizations of $cA$. In particular, we show that Dirichlet and Neumann realizations, $A^D$ and $A^N$, are absolutely continuous and unitary equivalent to each other and to the absolutely continuous part of the Krein realization. Moreover, if $infsigma_ess(T) = infgs(T) ge 0$, then the part $wt A^acE_wt A(gs(A^D))$ of any self-adjoint realization $wt A$ of $cA$ is unitarily equivalent to $A^D$. In addition, we prove that the absolutely continuous part $wt A^ac$ of any realization $wt A$ is unitarily equivalent to $A^D$ provided that the resolvent difference $(wt A - i)^-1- (A^D - i)^-1$ is compact. The abstract results are applied to elliptic differential expression in the half-space.

*Appeared in*

- J. Differential Equations, 252 (2012) pp. 5875--5922.

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# On (essentially) non-oscillatory discretizations of evolutionary convection-diffusion equations

*Authors*

- John, Volker

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

*2010 Mathematics Subject Classification*

- 65M06 65M60

*Keywords*

- Time-dependent convection-diffusion-reaction equations, under- and overshoots, FEM-FCT schemes, ENO schemes, WENO schemes

*DOI*

*Abstract*

Finite element and finite difference discretizations for evolutionary convection-diffusion-reaction equations in two and three dimensions are studied which give solutions without or with small under- and overshoots. The studied methods include a linear and a nonlinear FEM-FCT scheme, simple upwinding, an ENO scheme of order 3, and a fifth order WENO scheme. Both finite element methods are combined with the Crank--Nicolson scheme and the finite difference discretizations are coupled with explicit total variation diminishing Runge--Kutta methods. An assessment of the methods with respect to accuracy, size of under- and overshoots, and efficiency is presented, in the situation of a domain which is a tensor product of intervals and of uniform grids in time and space. Some comments to the aspects of adaptivity and more complicated domains are given. The obtained results lead to recommendations concerning the use of the methods.

*Appeared in*

- J. Comput. Phys., 231 (2012) pp. 1570--1586.

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# Directionality of THz emission from photoinduced gas plasmas

*Authors*

- Köhler, Christian
- Cabrera-Granado, Eduardo
- Babushkin, Ihar
- Bergé, Luc
- Herrmann, Joachim
- Skupin, Stefan

*2010 Mathematics Subject Classification*

- 35Q60 78A60 76X05

*Keywords*

- THz emission, photoinduced plasma, nonlinear optics

*DOI*

*Abstract*

Forward and backward THz emission by ionizing two-color laser pulses in gas is investigated by means of a simple semi-analytical model based on Jefimenko's equation and rigorous Maxwell simulations in one and two dimensions. We find the emission in backward direction having a much smaller spectral bandwidth than in forward direction and explain this by interference effects. Forward THz radiation is generated predominantly at the ionization front and is thus almost not affected by the opacity of the plasma, in excellent agreement with results obtained from a unidirectional pulse propagation model.

*Appeared in*

- Optics Letters, 36 (2011) pp. 3166--3168.

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# Catalytic branching processes via spine techniques and renewal theory

*Authors*

- Döring, Leif
- Roberts, Matthew I.

*2010 Mathematics Subject Classification*

- 60J27 60J80

*Keywords*

- Catalytic branching process, moments, renewal theory, many-to-few, spine

*DOI*

*Abstract*

In this article we contribute to the moment analysis of branching processes in catalytic media. The many-to-few lemma based on the spine technique is used to derive a system of (discrete space) partial differential equations for the number of particles in a variation of constants formulation. The long-time behavior is then deduced from renewal theorems and induction.

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# Multilevel dual approach for pricing American style derivatives

*Authors*

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

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 62L15 65C05 91B28

*Keywords*

- Optimal stopping, Dual approach, Multilevel Monte Carlo

*DOI*

*Abstract*

In this article we propose a novel approach to reduce the computational complexity of the dual method for pricing American options. We consider a sequence of martingales that converges to a given target martingale and decompose the original dual representation into a sum of representations that correspond to different levels of approximation to the target martingale. By next replacing in each representation true conditional expectations with their Monte Carlo estimates, we arrive at what one may call a multilevel dual Monte Carlo algorithm. The analysis of this algorithm reveals that the computational complexity of getting the corresponding target upper bound, due to the target martingale, can be significantly reduced. In particular, it turns out that using our new approach, we may construct a multilevel version of the well-known nested Monte Carlo algorithm of Andersen and Broadie (2004) that is, regarding complexity, virtually equivalent to a non-nested algorithm. The performance of this multilevel algorithm is illustrated by a numerical example.

*Appeared in*

- Finance Stoch., 17 (2013) pp. 717-742.

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# Nonsmooth analysis of doubly nonlinear evolution equations

*Authors*

- Mielke, Alexander

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

*2010 Mathematics Subject Classification*

- 35A15 35K50 35K85 49Q20 58E99

*Keywords*

- Doubly nonlinear equations, differential inclusions, generalized gradient flows, finite-strain elasticity

*DOI*

*Abstract*

In this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach spaces, driven by nonsmooth and nonconvex energies. We provide some general sufficient conditions, on the dissipation potential and the energy functional, for existence of solutions to the related Cauchy problem. We prove our main existence result by passing to the limit in a time-discretization scheme with variational techniques. Finally, we discuss an application to a material model in finite-strain elasticity.

*Appeared in*

- Calc. Var. Partial Differ. Equ., 46 (2013) pp. 253--310.

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# Conical diffraction by multilayer gratings: A recursive integral equations approach

*Authors*

- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 78A45 78M15 45E05 35J05

*Keywords*

- Diffraction, periodic structure, multilayer grating, singular integral formulation, recursive algorithm

*DOI*

*Abstract*

In this paper we consider an integral equation algorithm to study the scattering of plane waves by multilayer diffraction gratings under oblique incidence. The scattering problem is described by a system of Helmholtz equations with piecewise constant coefficients in $R^2$ coupled by special transmission conditions at the interfaces between different layers. Boundary integral methods lead to a system of singular integral equations, containing at least two equations for each interface. To deal with an arbitrary number of material layers we present the extension of a recursive procedure developed by Maystre for normal incidence, which transforms the problem to a sequence of equations with $2 times 2$ operator matrices on each interface. Necessary and sufficient conditions for the applicability of the algorithm are derived.

*Appeared in*

- Appl. Math., 58 (2013) pp. 279--307.

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# Primal-dual linear Monte Carlo algorithm for multiple stopping --- An application to flexible caps

*Authors*

- Balder, Sven
- Mahayni, Antje
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 60G40 91B28 65C05

*Keywords*

- Multiple stopping, dual representation, flexible caps, linear regression, Monte Carlo simulation

*DOI*

*Abstract*

In this paper we consider the valuation of Bermudan callable derivatives with multiple exercise rights. We present in this context a new primal-dual linear Monte Carlo algorithm that allows for efficient simulation of lower and upper price bounds without using nested simulations (hence the terminology). The algorithm is essentially an extension of a primal-dual Monte Carlo algorithm for standard Bermudan options proposed in Schoenmakers et al (2011), to the case of multiple exercise rights. In particular, the algorithm constructs upwardly a system of dual martingales to be plugged into the dual representation of Schoenmakers (2010). At each level the respective martingale is constructed via a backward regression procedure starting at the last exercise date. The thus constructed martingales are finally used to compute an upper price bound. At the same time, the algorithm also provides approximate continuation functions which may be used to construct a price lower bound. The algorithm is applied to the pricing of flexible caps in a Hull White (1990) model setup. The simple model choice allows for comparison of the computed price bounds with the exact price which is obtained by means of a trinomial tree implementation. As a result, we obtain tight price bounds for the considered application. Moreover, the algorithm is generically designed for multi-dimensional problems and is tractable to implement.

*Appeared in*

- Quant. Finance, 13 (2013) pp. 1003--1013.

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# Forward-backward systems for expected utility maximization

*Authors*

- Horst, Ulrich
- Hu, Ying
- Imkeller, Peter
- Réveillac, Anthony
- Zhang, Jianing

*2010 Mathematics Subject Classification*

- 60H10 93E20

*Keywords*

- Utility maximization, convex duality, maximum principle, coupled quadratic FBSDE, general utility functions

*DOI*

*Abstract*

In this paper we deal with the utility maximization problem with a general utility function. We derive a new approach in which we reduce the utility maximization problem with general utility to the study of a fully-coupled Forward-Backward Stochastic Differential Equation (FBSDE).

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# Inverse wave scattering by unbounded obstacles: Uniqueness for the two-dimensional Helmholtz equation

*Authors*

- Hu, Guanghui

*2010 Mathematics Subject Classification*

- 78A46 35R30

*Keywords*

- Inverse scattering, uniqueness, rough surface, Helmholtz equation, point sources

*DOI*

*Abstract*

In this paper we present some uniqueness results on inverse wave scattering by unbounded obstacles for the two-dimensional Helmholtz equation. We prove that an impenetrable one-dimensional rough surface can be uniquely determined by the values of the scattered field taken on a line segment above the surface that correspond to the incident waves generated by a countable number of point sources. For penetrable rough layers in a piecewise constant medium, the refractive indices together with the rough interfaces (on which the TM transmission conditions are imposed) can be uniquely identified using the same measurements and the same incident point source waves. Moreover, a Dirichlet polygonal rough surface can be uniquely determined by a single incident point source wave provided a certain condition is imposed on it.

*Appeared in*

- Appl. Anal., 91 (2012) pp. 703--717.

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# $L^infty$-estimates for divergence operators on bad domains

*Authors*

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

*2010 Mathematics Subject Classification*

- 35R05 47A60 35B65

*Keywords*

- Elliptic equations, mixed Dirichlet-Neumann conditions, $L^infty$-estimates

*DOI*

*Abstract*

In this paper, we prove $L^infty$-estimates for solutions of divergence operators in case of mixed boundary conditions. In this very general setting, the Dirichlet boundary part may be arbitrarily wild, i.e. no regularity conditions have to be imposed on it.

*Appeared in*

- Anal. Appl. (Singap.), 10 (2012) pp. 207--214.

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# Abelian theorems for stochastic volatility models with application to the estimation of jump activity of volatility

*Authors*

- Belomestny, Denis
- Panov, Vladimir

*2010 Mathematics Subject Classification*

- 62F10 60J75 60E10 62F12 60J25

*Keywords*

- affine stochastic volatility model, Abelian theorem, Blumenthal-Getoor index

*DOI*

*Abstract*

In this paper, we prove a kind of Abelian theorem for a class of stochastic volatility models $(X, V)$, where both the state process $X$ and the volatility process $V$ may have jumps. Our results relate the asymptotic behavior of the characteristic function of $X_Delta$ for some $Delta > 0$ in a stationary regime to the Blumenthal-Getoor indexes of the Lévy processes driving the jumps in $X$ and $V$ . The results obtained are used to construct consistent estimators for the above Blumenthal-Getoor indexes based on low-frequency observations of the state process $X$. We derive the convergence rates for the corresponding estimator and prove that these rates can not be improved in general.

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# Dual representations for general multiple stopping problems

*Authors*

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

ORCID: 0000-0002-4389-8266 - Zhang, Jianing

*2010 Mathematics Subject Classification*

- 60G40 65C05 91B28

*Keywords*

- General multiple stopping, Dual representations, Multiple exercise options, Volume constraints, Refraction period

*DOI*

*Abstract*

In this paper, we study the dual representation for generalized multiple stopping problems, hence the pricing problem of general multiple exercise options. We derive a dual representation which allows for cashflows which are subject to volume constraints modeled by integer valued adapted processes and refraction periods modeled by stopping times. As such, this extends the works by Schoenmakers [2010], Bender [2011a], Bender [2011b], Aleksandrov and Hambly [2010] and Meinshausen and Hambly [2004] on multiple exercise options, which either take into consideration a refraction period or volume constraints, but not both simultaneously. We also allow more flexible cashflow structures than the additive structure in the above references. For example some exponential utility problems are covered by our setting. We supplement the theoretical results with an explicit Monte Carlo algorithm for constructing confidence intervals for the price of multiple exercise options and exemplify it by a numerical study on the pricing of a swing option in an electricity market.

*Appeared in*

- Math. Finance, 25 (2015) pp. 339--370.

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# A mathematical framework for general classical systems and time irreversibility as its consequence

*Authors*

- Stephan, Holger

*2010 Mathematics Subject Classification*

- 82C03 60J25 60B05

*Keywords*

- statistical state, second law of thermodynamics, topological bidual space, Markov operator, Jensen's inequality

*DOI*

*Abstract*

It is well known that important models in statistical physics like the Fokker-Planck equation satisfy an H-theorem, i.e., have a decreasing Lyapunov function (or increasing entropy). This illustrates a symmetry break in time and reflects the second law of thermodynamics. In this paper, we show that any physically reasonable classical system has to have this property. For this purpose, we develop an abstract mathematical framework based on the theory of compact topological spaces and convex analysis. Precisely, we show:

1) Any statistical state space can be described as the convex hull of the image of the canonical embedding of the bidual space of its deterministic state space (a compact topological Hausdorff space).

2) The change of any statistical state is effected by the adjoint of a Markov operator acting in the space of observables.

3) Any Markov operator satisfies a wide class of inequalities, generated by arbitrary convex functions. As a corollary, these inequalities imply a time monotone behavior of the solution of the corresponding evolution equations.

Moreover, due to the general abstract setting, the proof of the underlying inequalities is very simple and therefore illustrates, where time symmetry breaks: A model is time reversible for any states if and only if the corresponding Markov operator is a deterministic one with dense range.

In addition, the proposed framework provides information about the structure of microscopic evolution equations, the choice of the best function spaces for their analysis and the derivation of macroscopic evolution equations.

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# On the convergence rate of grad-div stabilized Taylor--Hood to Scott--Vogelius solutions for incompressible flow problems

*Authors*

- Linke, Alexander

ORCID: 0000-0002-0165-2698 - Rebholz, Leo G.
- Wilson, Nicholas E.

*2010 Mathematics Subject Classification*

- 65M60 65N30 76D05

*Keywords*

- Navier-Stokes equations, Scott-Vogelius, Taylor-Hood, strong mass conservation, MHD, Leray-alpha

*DOI*

*Abstract*

It was recently proven that, under mild restrictions, grad-div stabilized Taylor-Hood solutions of Navier-Stokes problems converge to the Scott-Vogelius solution of that same problem. However, even though the analytical rate was only shown to be $gamma^-frac12$ (where $gamma$ is the stabilization parameter), the computational results suggest the rate may be improvable $gamma^-1$. We prove herein the analytical rate is indeed $gamma^-1$, and extend the result to other incompressible flow problems including Leray-$alpha$ and MHD. Numerical results are given that verify the theory.

*Appeared in*

- J. Math. Anal. Appl., 381 (2011) pp. 612--626.

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# Quasiconvexity equals rank-one convexity for isotropic sets of 2x2 matrices

*Authors*

- Heinz, Sebastian

*2010 Mathematics Subject Classification*

- 26B25 52A30

*Keywords*

- Quasiconvexity, rank-one convexity, lamination convexity, isotropy

*DOI*

*Abstract*

Let K be a given compact set of real 2x2 matrices that is isotropic, meaning invariant under the left and right action of the special orthogonal group. Then we show that the quasiconvex hull of K coincides with the rank-one convex hull (and even with the lamination convex hull of order 2). In particular, there is no difference between quasiconvexity and rank-one convexity for K. This is a generalization of a known result for connected sets.

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# An effective medium approach to the asymptotics of the statistical moments of the parabolic Anderson model and Lifshitz tails

*Authors*

- Metzger, Bernd

*2010 Mathematics Subject Classification*

- 60H25 82B44 82C44, 35J10, 35P20, 58J35

*Keywords*

- Random medium, random Schrödinger operators, heat equation with random potential, parabolic Anderson problem, large deviations, moment asymptotics, integrated density of states, Lifshitz tail

*DOI*

*Abstract*

Originally introduced in solid state physics to model amorphous materials and alloys exhibiting disorder induced metal-insulator transitions, the Anderson model $H_omega= -Delta + V_omega $ on $l^2(bZ^d)$ has become in mathematical physics as well as in probability theory a paradigmatic example for the relevance of disorder effects. Here $Delta$ is the discrete Laplacian and $V_omega = V_omega(x): x in bZ^d$ is an i.i.d. random field taking values in $bR$. A popular model in probability theory is the parabolic Anderson model (PAM), i.e. the discrete diffusion equation $partial_t u(x,t) =-H_omega u(x,t)$ on $ bZ^d times bR_+$, $u(x,0)=1$, where random sources and sinks are modelled by the Anderson Hamiltonian. A characteristic property of the solutions of (PAM) is the occurrence of intermittency peaks in the large time limit. These intermittency peaks determine the thermodynamic observables extensively studied in the probabilistic literature using path integral methods and the theory of large deviations. The rigorous study of the relation between the probabilistic approach to the parabolic Anderson model and the spectral theory of Anderson localization is at least mathematically less developed. We see our publication as a step in this direction. In particular we will prove an unified approach to the transition of the statistical moments $langle u(0,t) rangle$ and the integrated density of states from classical to quantum regime using an effective medium approach. As a by-product we will obtain a logarithmic correction in the traditional Lifshitz tail setting when $V_omega$ satisfies a fat tail condition.

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# Filamentary pulse self-compression: The impact of the cell windows

*Authors*

- Brée, Carsten
- Demircan, Ayhan
- Bethge, Jens
- Nibbering, Erik T. J.
- Skupin, Stefan
- Bergé, Luc
- Steinmeyer, Günter

*2010 Mathematics Subject Classification*

- 78A60

*2008 Physics and Astronomy Classification Scheme*

- 42.65.-k 42.65.Jx 42.65.Re

*Keywords*

- Nonlinear Optics, Femtosecond Filamentation, Pulse compression

*DOI*

*Abstract*

Self-compression of multi-millijoule laser pulses during filamentary propagation is usually explained by the interplay of self-focusing and defocusing effects, causing a substantial concentration of energy on the axis of the propagating optical pulse. Recently, it has been argued that cell windows may play a decisive role in the self-compression mechanism. As such windows have to be used for media other than air their presence is often unavoidable, yet they present a sudden non-adiabatic change in dispersion and nonlinearity that should lead to a destruction of the temporal and spatial integrity of the light bullets generated in the self-compression mechanism. We now experimentally prove that there is in fact a self-healing mechanism that helps to overcome the potentially destructive consequences of the cell windows. We show in two carefully conducted experiments that the cell window position decisively influences activation or inhibition of the self-healing mechanism. A comparison with a windowless cell shows that presence of this mechanism is an important prerequisite for the exploitation of self-compression effects in windowed cells filled with inert gases.

*Appeared in*

- Phys. Rev. A, 83 (2011) pp. 043803/1-043803/7.

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# Chimera states are chaotic transients

*Authors*

- Wolfrum, Matthias
- Omel'chenko, Oleh

ORCID: 0000-0003-0526-1878

*2010 Mathematics Subject Classification*

- 34C15 37D45 37N20 37N25

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Xt, 89.75.Kd

*Keywords*

- coupled phase oscillators, chaotic transients

*DOI*

*Abstract*

Spatiotemporal chaos and turbulence are universal concepts for the explanation of irregular behavior in various physical systems. Recently, a remarkable new phenomenon, called "chimera states", has been described, where in a spatially homogeneous system regions of irregular incoherent motion coexist with regular synchronized motion, forming a self organized pattern in a population of nonlocally coupled oscillators. Whereas most of the previous studies of chimera states focused their attention to the case of large numbers of oscillators employing the thermodynamic limit of infinitely many oscillators, we investigate here the properties of chimera states in populations of finite size using concepts from deterministic chaos. Our calculations of the Lyapunov spectrum show that the incoherent motion, which is described in the thermodynamic limit as a stationary behavior, in finite size systems appears as weak spatially extensive chaos. Moreover, for sufficiently small populations the chimera states reveal their transient nature: after a certain time-span we observe a sudden collapse of the chimera pattern and a transition to the completely coherent state. Our results indicate that chimera states can be considered as chaotic transients, showing the same properties as type-II supertransients in coupled map lattices.

*Appeared in*

- Phys. Rev. E, 84 (2011) pp. 015201(R)/1--015201(R)/4.

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# Efficient and accurate log-Lévy approximations to Lévy driven LIBOR models

*Authors*

- Papapantoleon, Antonis
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266 - Skovmand, David

*2010 Mathematics Subject Classification*

- 91G30 91G60 60G51

*Keywords*

- LIBOR market model, Levy processes, drift term, Picard approximation, option pricing, caps, swaptions, annuities

*DOI*

*Abstract*

The LIBOR market model is very popular for pricing interest rate derivatives, but is known to have several pitfalls. In addition, if the model is driven by a jump process, then the complexity of the drift term is growing exponentially fast (as a function of the tenor length). In this work, we consider a Lévy-driven LIBOR model and aim at developing accurate and efficient log-Lévy approximations for the dynamics of the rates. The approximations are based on truncation of the drift term and Picard approximation of suitable processes. Numerical experiments for FRAs, caps and swaptions show that the approximations perform very well. In addition, we also consider the log-Lévy approximation of annuities, which offers good approximations for high volatility regimes.

*Appeared in*

- J. Comput. Finance, 15 (2012) pp. 3--44.

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# Computational aspects of quasi-static crack propagation

*Authors*

- Knees, Dorothee
- Schröder, Andreas

*2010 Mathematics Subject Classification*

- 74R10, 49L25, 49J40, 74G65, 65M60, 65N30

*Keywords*

- rate-independent crack propagation, self-contact, global energetic model, BV-model, vanishing viscosity approach, convergence rate for energy release rates, finite elements

*DOI*

*Abstract*

The focus of this note lies on the numerical analysis of models describing the propagation of a single crack in a linearly elastic material. The evolution of the crack is modeled as a rate-independent process based on the Griffith criterion. We follow two different approaches for setting up mathematically well defined models: the global energetic approach and an approach based on a viscous regularization. We prove the convergence of solutions of fully discretized models (i.e. with respect to time and space) and derive relations between the discretization parameters (mesh size, time step size, viscosity parameter, crack increment) which guarantee the convergence of the schemes. Further, convergence rates are provided for the approximation of energy release rates by certain discrete energy release rates. Thereby we discuss both, models with self-contact conditions on the crack faces as well as models with pure Neumann conditions on the crack faces. The convergence proofs rely on regularity estimates for the elastic fields close to the crack tip and local and global finite element error estimates. Finally the theoretical results are illustrated with some numerical calculations.

*Appeared in*

- Discrete Contin. Dyn. Syst. Ser. S, 6 (2013) pp. 63--99.

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# Measurement and simulation of a droplet population in a turbulent flow field

*Authors*

- Bordás, Robert
- John, Volker

ORCID: 0000-0002-2711-4409 - Schmeyer, Ellen
- Thévenin, Dominique

*2010 Mathematics Subject Classification*

- 76F65 76T10

*Keywords*

- two-phase turbulent flow, disperse droplet population, non-intrusive measurements, population blance systems, variational multiscale method

*DOI*

*Abstract*

The interaction of a disperse droplet population (spray) in a turbulent flow field is studied by combining wind tunnel experiments with simulations based on the model of a population balance system. The behavior of the droplets is modeled numerically by a population balance equation. Velocities of the air and of the droplets are determined by non-intrusive measurements. A direct discretization of the 4D equation for the droplet size distribution is used in the simulations. Important components of the numerical algorithm are a variational multiscale method for turbulence modeling, an upwind scheme for the 4D equation and a pre-processing approach to evaluate the aggretation integrals. The simulations of this system accurately predict the modifications of the droplet size distribution from the inlet to the outlet of the measurement section. Since the employed configuration is simple and considering that all measurement data are freely available thanks to an Internet-based repository, the considered experiment is proposed as a benchmark problem for the simulation of disperse two-phase turbulent flows.

*Appeared in*

- Comput. Fluids, 66 (2012) pp. 52--62.

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# Martingale approach in pricing European options under regime-switching

*Authors*

- Milstein, Grigori N.
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 60H35 91G10

*Keywords*

- incomplete markets, martingale measure, generalized, self-financing strategy, attainability, self-financing in mean

*DOI*

*Abstract*

The paper focuses on the problem of pricing and hedging a European contingent claim for an incomplete market model, in which evolution of price processes for a saving account and stocks depends on an observable Markov chain. The pricing function is evaluated using the martingale approach. The equivalent martingale measure is introduced in a way that the Markov chain remains the historical one, and the pricing function satisfies the Cauchy problem for a system of linear parabolic equations. It is shown that any European contingent claim is attainable using a generalized self-financing replicating strategy. For such a strategy, apart from the initial endowment, some additional funds are required both step-wise at the jump moments of the Markov chain and continuously between the jump moments. It is proved that the additional funds (the additional investments and consumptions) are present in the proposed strategy in the risk-neutral manner, hence the generalized self-financing strategy is self-financing in mean. A payment for the considered option should consist of two parts: the initial endowment and a fair insurance premium in order to compensate for contributions and consumptions arising in future.

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# An assessment of discretizations for convection-dominated convection-diffusion equations

*Authors*

- Augustin, Matthias
- Caiazzo, Alfonso

ORCID: 0000-0002-7125-8645 - Fiebach, André
- Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434 - John, Volker

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

ORCID: 0000-0002-0165-2698 - Umla, Rudolf

*2010 Mathematics Subject Classification*

- 65N12 65N30 65N08

*2008 Physics and Astronomy Classification Scheme*

- 47.11.Fg 47.11.Df

*Keywords*

- dominating convection, exponentially fitted finite volume scheme, stabilized finite element methods, Hemker problem

*DOI*

*Abstract*

The performance of several numerical schemes for discretizing convection-dominated convection-diffusion equations will be investigated with respect to accuracy and efficiency. Accuracy is considered in measures which are of interest in applications. The study includes an exponentially fitted finite volume scheme, the Streamline-Upwind Petrov--Galerkin (SUPG) finite element method, a spurious oscillations at layers diminishing (SOLD) finite element method, a finite element method with continuous interior penalty (CIP) stabilization, a discontinuous Galerkin (DG) finite element method, and a total variation diminishing finite element method (FEMTVD). A detailed assessment of the schemes based on the Hemker example will be presented.

*Appeared in*

- Comp. Meth. Appl. Mech. Engrg., 200 (2011) pp. 3395--3409.

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# Spatial rocking phenomenon in broad area semiconductor lasers

*Authors*

- Radziunas, Mindaugas
- Staliunas, Kestutis

*2010 Mathematics Subject Classification*

- 78A60 35B36 37M05 78A45

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 42.65.Pc 42.60.Jf

*Keywords*

- broad area semiconductor laser, dissipative system, optical injection, bistability, pattern formation, soliton, rocking

*DOI*

*Abstract*

The spatial ``rocking'' is a dynamical effect converting a phase-invariant oscillatory system into a phase-bistable one, where the average phase of the system locks to one of two values differing by $pi$. We demonstrate theoretically the spatial rocking in experimentally accessible and practically relevant systems -- the broad area semiconductor lasers. By numerical integration of the laser model equations we show the phase bistability of the optical fields and explore the bistability area in parameter space. We also predict the spatial patterns, such as phase domain walls and phase solitons, which are characteristic for the phase-bistable spatially extended pattern forming systems.

*Appeared in*

- Europhys. Lett., 95 (2011) pp. 14002/1--14002/6.

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# Existence, local uniqueness and asymptotic approximation of spike solutions to singularly perturbed elliptic problems

*Authors*

- Omel'chenko, Oleh

ORCID: 0000-0003-0526-1878 - Recke, Lutz

*2010 Mathematics Subject Classification*

- 35B25 35C20 35J65

*Keywords*

- non-variational problem, interior spike, boundary layer, implicit function theorem

*DOI*

*Abstract*

This paper concerns general singularly perturbed second order semilinear elliptic equations on bounded domains $Omega subset R^n$ with nonlinear natural boundary conditions. The equations are not necessarily of variational type. We describe an algorithm to construct sequences of approximate spike solutions, we prove existence and local uniqueness of exact spike solutions close to the approximate ones (using an Implicit Function Theorem type result), and we estimate the distance between the approximate and the exact solutions. Here ''spike solution'' means that there exists a point in $Omega$ such that the solution has a spike-like shape in a vicinity of such point and that the solution is approximately zero away from this point. The spike shape is not radially symmetric in general and may change sign.

*Appeared in*

- Hiroshima Math. J., 45 (2015) pp. 35--89.

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# Development of a stability prediction tool for the identification of stable milling processes

*Authors*

- Hömberg, Dietmar
- Uhlmann, Eckart
- Rott, Oliver
- Rasper, Patrick

*2010 Mathematics Subject Classification*

- 74D05 35Q74 68W99

*Keywords*

- Thermo-elasticity, milling, process-structure interaction, work piece effects, stability, delay-differential equations

*DOI*

*Abstract*

This paper deals with a new mathematical model to characterise the interaction between machine and work piece in a milling process. The model consists of a multi-body system representing the milling machine and a linear thermo-elastic work piece model. An extensive experimental analysis supported the development of the governing model equations. A numerical solution strategy is outlined and complemented by simulations of stable and unstable milling processes including work piece effects. The last part covers the development of a new algorithm for the stability analysis of large milling systems.

*Appeared in*

- Process Machine Interactions. Prediction and Manipulation of Interactions between Manufacturing Processes and Machine Tool Structures, B. Denkena, F. Hollmann, eds., Lecture Notes in Production Engineering, Springer, Berlin --- Heidelberg, 2013, pp. 203--224

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# Thermodynamics of multiphase problems in viscoelasticity

*Authors*

- Paoli, Laetitia
- Petrov, Adrien

*2010 Mathematics Subject Classification*

- 35K55 74C05 74C10

*2008 Physics and Astronomy Classification Scheme*

- 02.30.Jr

*Keywords*

- Existence result, generalized standard materials, heat equation, enthalpy transformation, maximal monotone operators, doubly nonlinear equations, shape-memory alloys

*DOI*

*Abstract*

This paper deals with a three-dimensional mixture model describing materials undergoing phase transition with thermal expansion. The problem is formulated within the framework of generalized standard solids by the coupling of the momentum equilibrium equation and the flow rule with the heat transfer equation. A global solution for this thermodynamically consistent problem is obtained by using a fixed-point argument combined with global energy estimates.

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# Rigorous derivation of a plate theory in linear elastoplasticity via Gamma convergence

*Authors*

- Liero, Matthias

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

*2010 Mathematics Subject Classification*

- 35J85 35Q72 49J45 74C05 74K20

*Keywords*

- Linearized elastoplasticity, rate-independent system, Gamma convergence, Mosco convergence, hysteresis, generalized Prandtl--Ishlinskii operator

*DOI*

*Abstract*

This paper deals with dimension reduction in linearized elastoplasticity in the rate-independent case. The reference configuration of the elastoplastic body is given by a two-dimensional middle surface and a small but positive thickness. We derive a limiting model for the case in which the thickness of the plate tends to 0. This model contains membrane and plate deformations which are coupled via plastic strains. The convergence analysis is based on an abstract Gamma convergence theory for rate-independent evolution formulated in the framework of energetic solutions. This concept is based on an energy-storage functional and a dissipation functional, such that the notion of solution is phrased in terms of a stability condition and an energy balance.

*Appeared in*

- NoDEA Nonlinear Differential Equations Appl., 19 (2012) pp. 437--457.

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# Existence result for a class of generalized standard materials with thermomechanical coupling

*Authors*

- Paoli, Laetitia
- Petrov, Adrien

*2010 Mathematics Subject Classification*

- 35A01 35K55 74F05

*Keywords*

- Existence result, generalized standard materials, doubly nonlinear equations, heat equation

*DOI*

*Abstract*

This paper deals with the study of a three-dimensional model of thermomechanical coupling for viscous solids exhibiting hysteresis effects. This model is written in accordance with the formalism of generalized standard materials. It is composed by the momentum equilibrium equation combined with the flow rule, which describes some stress-strain dependance, and the heat-transfer equation. An existence result for this thermodynamically consistent problem is obtained by using a fixed-point argument and some qualitative properties of the solutions are established.

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# An asymptotic analysis for a nonstandard Cahn--Hilliard system with viscosity

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Podio-Guidugli, Paolo
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 74A15 35K55 35A05 35B40

*Keywords*

- viscous Cahn-Hilliard system, phase field model, asymptotic limit, existence of solutions

*DOI*

*Abstract*

This paper is concerned with a diffusion model of phase-field type, consisting of a parabolic system of two partial differential equations, interpreted as balances of microforces and microenergy, for two unknowns: the problem's order parameter $rho$ and the chemical potential $mu$; each equation includes a viscosity term -- respectively, $varepsilon,partial_tmu$ and $delta,partial_trho$ -- with $varepsilon$ and $delta$ two positive parameters; the field equations are complemented by Neumann homogeneous boundary conditions and suitable initial conditions. In a recent paper [5], we proved that this problem is well-posed and investigated the long-time behavior of its $(varepsilon,delta)-$solutions. Here we discuss the asymptotic limit of the system as $eps$ tends to 0. We prove convergence of $(varepsilon,delta)-$solutions to the corresponding solutions for the case $eps$ =0, whose long-time behavior we characterize; in the proofs, we employ compactness and monotonicity arguments.

*Appeared in*

- Discrete Contin. Dyn. Syst. Ser. S, 6 (2013) pp. 353--368.

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# Calculation of ultrashort pulse propagation based on rational approximations for medium dispersion

*Authors*

- Amiranashvili, Shalva

ORCID: 0000-0002-8132-882X - Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2008 Physics and Astronomy Classification Scheme*

- 42.65.-k, 42.65.Tg, 42.81.Dp, 02.30.Mv

*Keywords*

- Ultrashort pulses, Rational approximation, Padé approximant, Envelope equation

*DOI*

*Abstract*

Ultrashort optical pulses contain only a few optical cycles and exhibit broad spectra. Their carrier frequency is therefore not well defined and their description in terms of the standard slowly varying envelope approximation becomes questionable. Existing modeling approaches can be divided in two classes, namely generalized envelope equations, that stem from the nonlinear Schrödinger equation, and non-envelope equations which treat the field directly. Based on fundamental physical rules we will present an approach that effectively interpolates between these classes and provides a suitable setting for accurate and highly efficient numerical treatment of pulse propagation along nonlinear and dispersive optical media.

*Appeared in*

- Opt. Quantum Electron., 44 (2012) pp. 241--246.

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# Shape derivatives for the scattering by biperiodic gratings

*Authors*

- Rathsfeld, Andreas

*2010 Mathematics Subject Classification*

- 78A45 35J20 49J20

*Keywords*

- biperiodic grating, time-harmonic Maxwell's equation, shape gradient

*DOI*

*Abstract*

Usually, the light diffraction by biperiodic grating structures is simulated by the time-harmonic Maxwell system with a constant magnetic permeability. For the optimization of the geometry parameters of the grating, a functional is defined which depends quadratically on the efficiencies of the reflected modes. The minimization of this functional by gradient based optimization schemes requires the computation of the shape derivatives of the functional with respect to the parameters of the geometry. Using classical ideas of shape calculus, formulas for these parameter derivatives are derived. In particular, these derivatives can be computed as material derivatives corresponding to a family of transformations of the underlying domain. However, the energy space $H(rm curl)$ for the electric fields is not invariant with respect to the transformation of geometry. Therefore, the formulas are derived first for the magnetic field vectors which belong to $[H^1]^3$. Afterwards, the magnetic fields in the shape-derivative formula are replaced by their electric counter parts. Numerical tests confirm the derived formulas.

*Appeared in*

- Appl. Numer. Math., 72 (2013) pp. 19--32.

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# A vanishing viscosity approach to a rate-independent damage model

*Authors*

- Knees, Dorothee
- Rossi, Riccarda
- Zanini, Chiara

*2010 Mathematics Subject Classification*

- 35D40 74R05 74C05 35K86 49J40

*Keywords*

- rate-independent damage evolution, vanishing viscosity method, arc-length reparameterization, time discretization

*DOI*

*Abstract*

We analyze a rate-independent model for damage evolution in elastic bodies. The central quantities are a stored energy functional and a dissipation functional, which is assumed to be positively homogeneous of degree one. Since the energy is not simultaneously (strictly) convex in the damage variable and the displacements, solutions may have jumps as a function of time. The latter circumstance makes it necessary to recur to suitable notions of weak solution. However, the by-now classical concept of global energetic solution fails to describe accurately the behavior of the system at jumps. Hence, we consider rate-independent damage models as limits of systems driven by viscous, rate-dependent dissipation. We use a technique for taking the vanishing viscosity limit, which is based on arc-length reparameterization. In this way, in the limit we obtain a novel formulation for the rate-independent damage model, which highlights the interplay of viscous and rate-independent effects in the jump regime, and provides a better description of the energetic behavior of the system at jumps.

*Appeared in*

- Math. Models Methods Appl. Sci., 23 (2013) pp. 565--616.

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# Mayer and virial series at low temperature

*Authors*

- Jansen, Sabine

*2010 Mathematics Subject Classification*

- 82B05 82B26

*Keywords*

- classical statistical mechanics, Mayer and virial series, phase transitions

*DOI*

*Abstract*

We analyze the Mayer pressure-activity and virial pressure-density series for a classical system of particles in continuous configuration space at low temperature. Particles interact via a finite range potential with an attractive tail. We propose physical interpretations of the Mayer and virial series' radius of convergence, valid independently of the question of phase transition: the Mayer radius corresponds to a fast increase from very small to finite density, and the virial radius corresponds to a cross-over from monatomic to polyatomic gas. Our results have consequences for the search of a low density, low temperature solid-gas phase transition, consistent with the Lee-Yang theorem for lattice gases and with the continuum Widom-Rowlinson mode.

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# Amplifications of picosecond laser pulses in tapered semiconductor amplifiers: Numerical simulations versus experiments

*Authors*

- Tronciu, Vasile
- Schwertfeger, Sven
- Radziunas, Mindaugas
- Klehr, Andreas
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Wenzel, Hans

*2010 Mathematics Subject Classification*

- 78A60 37M05

*2008 Physics and Astronomy Classification Scheme*

- 42.60.Da 42.60.Jf

*Keywords*

- Picosecond pulse amplification, tapered semiconductor amplifier

*DOI*

*Abstract*

We apply a travelling wave model to the simulation of the amplification of laser pulses generated by Q-switched or mode-locked distributed-Bragg reflector lasers. The power amplifier monolithically integrates a ridge-waveguide section acting as pre-amplifier and a flared gain-region amplifier. The diffraction limited and spectral-narrow band pulses injected in to the pre-amplifier have durations between 10 ps and 100 ps and a peak power of typical 1 W. After the amplifier, the pulses reach a peak power of several tens of Watts preserving the spatial, spectral and temporal properties of the input pulse. We report results obtained by a numerical solution of the travelling-wave equations and compare them with experimental investigations. The peak powers obtained experimentally are in good agreement with the theoretical predictions. The performance of the power amplifier is evaluated by considering the dependence of the pulse energy as a function of different device and material parameters.

*Appeared in*

- Opt. Commun., 285 (2012) pp. 2897--2904.

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# Global existence result for phase transformations with heat transfer in shape memory alloys

*Authors*

- Paoli, Laetitia
- Petrov, Adrien

*2010 Mathematics Subject Classification*

- 35K55 74C05 74C10

*Keywords*

- Existence result, generalized standard materials, heat equation, enthalpy transformation, maximal monotone operators, doubly nonlinear equations, shape-memory alloys

*DOI*

*Abstract*

We consider three-dimensional models for rate-independent processes describing materials undergoing phase transformations with heat transfer. The problem is formulated within the framework of generalized standard solids by the coupling of the momentum equilibrium equation and the flow rule with the heat transfer equation. Under appropriate regularity assumptions on the initial data, we prove the existence a global solution for this thermodynamically consistent system, by using a fixed-point argument combined with global energy estimates.

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# Large deviations for Brownian intersection measures

*Authors*

- König, Wolfgang

ORCID: 0000-0002-4212-0065 - Mukherjee, Chiranjib

*2010 Mathematics Subject Classification*

- 60J65 60J55 60F10

*Keywords*

- intersection of Brownian paths, intersection local time, intersection measure, exponential approximation, large deviations

*DOI*

*Abstract*

We consider $p$ independent Brownian motions in $R^d$. We assume that $pgeq 2$ and $p(d-2)<d$. Let $ell_t$ denote the intersection measure of the $p$ paths by time $t$, i.e., the random measure on $R^d$ that assigns to any measurable set $Asubset R^d$ the amount of intersection local time of the motions spent in $A$ by time $t$. Earlier results of Chen citeCh09 derived the logarithmic asymptotics of the upper tails of the total mass $ell_t(R^d)$ as $ttoinfty$. In this paper, we derive a large-deviation principle for the normalised intersection measure $t^-pell_t$ on the set of positive measures on some open bounded set $BsubsetR^d$ as $ttoinfty$ before exiting $B$. The rate function is explicit and gives some rigorous meaning, in this asymptotic regime, to the understanding that the intersection measure is the pointwise product of the densities of the normalised occupation times measures of the $p$ motions. Our proof makes the classical Donsker-Varadhan principle for the latter applicable to the intersection measure. A second version of our principle is proved for the motions observed until the individual exit times from $B$, conditional on a large total mass in some compact set $Usubset B$. This extends earlier studies on the intersection measure by König and Mörters citeKM01,KM05.

*Appeared in*

- Comm. Pure Appl. Math., 66 (2013) pp. 263--306.

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# Blow-up versus boundedness in a nonlocal and nonlinear Fokker--Planck equation

*Authors*

- Dreyer, Wolfgang
- Huth, Robert
- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Rehberg, Joachim
- Winkler, Michael

*2010 Mathematics Subject Classification*

- 35K55 35Q84 35B30

*2008 Physics and Astronomy Classification Scheme*

- 82.47.Aa

*Keywords*

- energy-dissipation relation, gradient flow with time-dependent constraint, Fokker-Planck equation

*DOI*

*Abstract*

We consider a Fokker-Planck equation on a compact interval where, as a constraint, the first moment is a prescribed function of time. Eliminating the associated Lagrange multiplier one obtains nonlinear and nonlocal terms. After establishing suitable local existence results, we use the relative entropy as an energy functional. However, the time-dependent constraint leads to a source term such that a delicate analysis is needed to show that the dissipation terms are strong enough to control the work done by the constraint. We obtain global existence of solutions as long as the prescribed first moment stays in the interior of an interval. If the prescribed moment converges to a constant value inside the interior of the interval, then the solution stabilises to the unique steady state.

*Appeared in*

- ZAMP Z. Angew. Math. Phys., 66 (2015) pp. 293--315 with the title ``Global existence for a nonlocal and nonlinear Fokker--Planck equation''

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# The longest excursion of a random interacting polymer

*Authors*

- Köcher, Janine
- König, Wolfgang

ORCID: 0000-0002-4212-0065

*2010 Mathematics Subject Classification*

- 60F05 82D60

*Keywords*

- Free energy, interacting polymer, longest excursion, extreme value theory, renewal theory

*DOI*

*Abstract*

We consider a random $N$-step polymer under the influence of an attractive interaction with the origin and derive a limit law -- after suitable shifting and norming -- for the length of the longest excursion towards the Gumbel distribution. The embodied law of large numbers in particular implies that the longest excursion is of order $log N$ long. The main tools are taken from extreme value theory and renewal theory.

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# Identification of the thermal growth characteristics of coagulated tumor tissue in laser-induced thermotherapy

*Authors*

- Hömberg, Dietmar
- Liu, Jijun
- Togobytska, Nataliya

*2010 Mathematics Subject Classification*

- 35R30 74F05 74N99

*Keywords*

- Inverse problem, nonlinear equation, optimal control, iteration scheme, coupled system, numerics

*DOI*

*Abstract*

We consider an inverse problem arising in laser-induced thermotherapy, a minimally invasive method for cancer treatment, in which cancer tissues is destroyed by coagulation. For the dosage planning numerical simulation play an important role. To this end a crucial problem is to identify the thermal growth kinetics of the coagulated zone. Mathematically, this problem is a nonlinear and nonlocal parabolic heat source inverse problem. The solution to this inverse problem is defined as the minimizer of a nonconvex cost functional. The existence of the minimizer is proven. We derive the Gateaux derivative of the cost functional, which is based on the adjoint system, and use it for a numerical approximation of the optimal coefficient.

*Appeared in*

- Math. Methods Appl. Sci., 35 (2012) pp. 497--509.

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# Geodesic convexity of the relative entropy in reversible Markov chains

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 60J27 53C21 53C23 82B35

*Keywords*

- Markov chain, detailed balance, relative entropy, Onsager matrix, harmonic mean, geodesic convexity

*DOI*

*Abstract*

We consider finite-dimensional, time-continuous Markov chains satisfying the detailed balance condition as gradient systems with the relative entropy E as driving functional. The Riemannian metric is defined via its inverse matrix called the Onsager matrix K. We provide methods for establishing geodesic λ-convexity of the entropy and treat several examples including some more general nonlinear reaction systems

*Appeared in*

- Calc. Var. Partial Differ. Equ., 48 (2013) pp. 1-31.

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# Global higher integrability of minimizers of variational problems with mixed boundary conditions

*Authors*

- Fiaschi, Alice
- Knees, Dorothee
- Reichelt, Sina

*2010 Mathematics Subject Classification*

- 74C05 49N60 49S05 35B65

*Keywords*

- Higher integrability of gradients of minimizers, p-growth, mixed boundary conditions, damage, uniform Caccioppoli-like inequality

*DOI*

*Abstract*

We consider integral functionals with densities of p-growth, with respect to gradients, on a Lipschitz domain with mixed boundary conditions. The aim of this paper is to prove that, under uniform estimates within certain classes of p-growth and coercivity assumptions on the density, the minimizers are of higher integrability order, meaning that they belong to the space of first order Sobolev functions with an integrability of order p+ε for a uniform ε >0. The results are applied to a model describing damage evolution in a nonlinear elastic body and to a model for shape memory alloys.

*Appeared in*

- J. Math. Anal. Appl., 401 (2013) pp. 269--288.

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# Generalized Prandtl--Ishlinskii operators arising from homogenization and dimension reduction

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 34C55 47J40 774Qxx 74C05 74K20

*Keywords*

- Hysteresis operators, play operator, Prandtl-Ishlinskii operator, Gamma convergence, rate-independent system, homogenization, elastoplasticity, plastic plate model

*DOI*

*Abstract*

We consider rate-independent evolutionary systems over a physically domain Ω that are governed by simple hysteresis operators at each material point. For multiscale systems where ε denotes the ratio between the microscopic and the macroscopic length scale, we show that in the limit ε → 0 we are led to systems where the hysteresis operators at each macroscopic point is a * generalized Prandtl-Ishlinskii operator*

*Appeared in*

- Phys. B, 407 (2012) pp. 1330--1335.

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# Uniqueness in inverse scattering of elastic waves by three-dimensional polyhedral diffraction gratings

*Authors*

- Elschner, Johannes
- Hu, Guanghui

*2010 Mathematics Subject Classification*

- 78A46 35B27 35R30 74B05

*Keywords*

- Inverse scattering, uniqueness, three-dimensional diffraction grating, Navier equation, boundary conditions of the third (fourth) kind

*DOI*

*Abstract*

We consider the inverse elastic scattering problem of determining a three-dimensional diffraction grating profile from scattered waves measured above the structure. In general, a grating profile cannot be uniquely determined by a single incoming plane wave. We completely characterize and classify the bi-periodic polyhedral structures under the boundary conditions of the third and fourth kinds that cannot be uniquely recovered by only one incident plane wave. Thus we have global uniqueness for a polyhedral grating profile by one incident elastic plane wave if and only if the profile belongs to neither of the unidentifiable classes, which can be explicitly described depending on the incident field and the type of boundary conditions. Our approach is based on the reflection principle for the Navier equation and the reflectional and rotational invariance of the total field.

*Appeared in*

- J. Inverse Ill-Posed Probl., 19 (2011) pp. 717--768.

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# Exact solutions to the Riemann problem for compressible isothermal Euler equations for two phase flows with and without phase transition

*Authors*

- Dreyer, Wolfgang
- Hantke, Maren
- Warnecke, Gerald

*2010 Mathematics Subject Classification*

- 80A22 76T15 35L65

*2008 Physics and Astronomy Classification Scheme*

- 05.70.Fh, 05.70.Ce

*Keywords*

- Conservation laws, phase transitions, non-classical shocks, two phase flow model, exact Riemann solver, sharp interface model, thermodynamics

*DOI*

*Abstract*

We consider the isothermal Euler equations with phase transition between a liquid and a vapor phase. The mass transfer is modeled by a kinetic relation. We prove existence and uniqueness results. Further, we construct the exact solution for Riemann problems. We derive analogous results for the cases of initially one phase with resulting condensation by compression or evaporation by expansion. Further we present numerical results for these cases. We compare the results to similar problems without phase transition.

*Appeared in*

- Quart. Appl. Math., 71 (2013) pp. 509--540.

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# Parabolic Anderson model with finite number of moving catalysts

*Authors*

- Castell, Fabienne
- Gün, Onur
- Maillard, Gregory

*2010 Mathematics Subject Classification*

- 60H25 82C44 60F10 35B40

*Keywords*

- Parabolic Anderson problem, catalytic random medium, intermittency, moment Lyapunov exponents

*DOI*

*Abstract*

We consider the parabolic Anderson model (PAM) which is given by the equation $partial u/partial t = kappaDelta u + xi u$ with $ucolon, Z^dtimes [0,infty)to R$, where $kappa in [0,infty)$ is the diffusion constant, $Delta$ is the discrete Laplacian, and $xicolon,Z^dtimes [0,infty)toR$ is a space-time random environment. The solution of this equation describes the evolution of the density $u$ of a ``reactant'' $u$ under the influence of a ``catalyst'' $xi$.newlineindent In the present paper we focus on the case where $xi$ is a system of $n$ independent simple random walks each with step rate $2drho$ and starting from the origin. We study the emphannealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of $u$ w.r.t. $xi$ and show that these exponents, as a function of the diffusion constant $kappa$ and the rate constant $rho$, behave differently depending on the dimension $d$. In particular, we give a description of the intermittent behavior of the system in terms of the annealed Lyapunov exponents, depicting how the total mass of $u$ concentrates as $ttoinfty$. Our results are both a generalization and an extension of the work of Gärtner and Heydenreich citegarhey06, where only the case $n=1$ was investigated.

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# Emergence of rate-independent dissipation from viscous systems with wiggly energies

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 74N30 74D10 70K70

*Keywords*

- Gamma convergence for evolution, De Giorgi formulation, rate-independent plasticity, viscous gradient flow, wiggly energy

*DOI*

*Abstract*

We consider the passage from viscous system to rate-independent system in the limit of vanishing viscosity and for wiggly energies. Our new convergence approach is based on the (R,R^{*}) formulation by De Giorgi, where we pass to the Γ limit in the dissipation functional. The difficulty is that the type of dissipation changes from a quadratic functional to one that is homogeneous of degree 1. The analysis uses the decomposition of the restoring force into a macroscopic part and a fluctuating part, where the latter is handled via homogenization.

*Appeared in*

- Contin. Mech. Thermodyn., 24 (2012) pp. 591--606.

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# Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection

*Authors*

- Galvin, Keith
- Linke, Alexander

ORCID: 0000-0002-0165-2698 - Rebholz, Leo
- Wilson, Nicholas

*2010 Mathematics Subject Classification*

- 76D05 76M10

*Keywords*

- mixed finite elements, incompressible Navier-Stokes equations, poor mass conservation, grad-div stabilization, natural convection, Scott-Vogelius element

*DOI*

*Abstract*

We consider the problem of poor mass conservation in mixed finite element algorithms for flow problems with large rotation-free forcing in the momentum equation. We provide analysis that suggests for such problems, obtaining accurate solutions necessitates either the use of pointwise divergence-free finite elements (such as Scott-Vogelius), or heavy grad-div stabilization of weakly divergence-free elements. The theory is demonstrated in numerical experiments for a benchmark natural convection problem, where large irrotational forcing occurs with high Rayleigh numbers.

*Appeared in*

- Comput. Methods Appl. Mech. Engrg., 237--240 (2012) pp. 166--176.

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# Shifted linear systems in electromagnetics. Part II: Systems with multiple right-hand sides

*Authors*

- Schlundt, Rainer

ORCID: 0000-0002-4424-4301 - Schmückle, Franz-Josef
- Heinrich, Wolfgang

*2010 Mathematics Subject Classification*

- 35Q60 65F10 65F15 65N22 78M25

*Keywords*

- Microwave device, Maxwell's equations, Scattering matrix, Boundary value problem, PML boundary condition, Eigenvalue problem, Linear algebraic equations, Multiple shifts, Multiple right-hand sides, Krylov subspace method, Polynomial preconditioning, Initial guesses

*DOI*

*Abstract*

We consider the solution of multiply shifted linear systems for multiple right-hand sides. The coefficient matrix is symmetric, complex, and indefinite. The matrix is shifted by different multiples of the identity. Such problems arise in a number of applications, including the electromagnetic simulation in the development of microwave and mm-wave circuits and modules.

The properties of microwave circuits can be described in terms of their scattering matrix which is extracted from the orthogonal decomposition of the electric field. We discretize the Maxwell's equations with orthogonal grids using the Finite Integration Technique (FIT).

Some Krylov subspace methods have been used to solve systems with multiple right-hand sides. We use both the block-QMR method and a symmetric band Lanczos process based on coupled recurrences with polynomial preconditioning.

We present a method for providing initial guesses to a linear solver both for systems with multiple shifts and for systems with multiple right-hand sides each with a different shift.

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# Global existence result for thermoviscoelastic problems with hysteresis

*Authors*

- Paoli, Laetitia
- Petrov, Adrien

*2010 Mathematics Subject Classification*

- 35A01 35Q80 74C05

*Keywords*

- Existence result, generalized standard materials, heat equation, enthalpy transformation, maximal monotone operators, doubly nonlinear equations, plasticity, shape-memory alloys

*DOI*

*Abstract*

We consider viscoelastic solids undergoing thermal expansion and exhibiting hysteresis effects due to plasticity or phase transformations. Within the framework of generalized standard solids, the problem is described in a 3D setting by the momentum equilibrium equation, the flow rule describing the dependence of the stress on the strain history, and the heat transfer equation. Under appropriate regularity assumptions on the data, a local existence result for this thermodynamically consistent system is established, by combining existence results for ordinary differential equations in Banach spaces with a fixed-point argument. Then global estimates are obtained by using both the classical energy estimate and more specific techniques for the heat equation introduced by Boccardo and Gallouet. Finally a global existence result is derived.

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# Dispersion of nonlinear group velocity determines shortest envelope solitons

*Authors*

- Amiranashvili, Shalva

ORCID: 0000-0002-8132-882X - Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Akhmediev, Nail

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Tg 05.45.Yv 42.81.Dp

*Keywords*

- Generalized nonlinear Schrödinger equation, Nonlinear group velocity dispersion, Soliton, Cusp

*DOI*

*Abstract*

We demonstrate that a generalized nonlinear Schrödinger equation (NSE), that includes dispersion of the intensity-dependent group velocity, allows for exact solitary solutions. In the limit of a long pulse duration, these solutions naturally converge to a fundamental soliton of the standard NSE. In particular, the peak pulse intensity times squared pulse duration is constant. For short durations this scaling gets violated and a cusp of the envelope may be formed. The limiting singular solution determines then the shortest possible pulse duration and the largest possible peak power. We obtain these parameters explicitly in terms of the parameters of the generalized NSE.

*Appeared in*

- Phys. Rev. A, 84 (2011) pp. 43834/1--043834/5

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# Large deviations for the local times of a random walk among random conductances

*Authors*

- König, Wolfgang

ORCID: 0000-0002-4212-0065 - Salvi, Michele
- Wolff, Tilman

*2010 Mathematics Subject Classification*

- 60J65 60J55 60F10

*Keywords*

- continuous-time random walk, random conductances, randomly perturbed Laplace operator, large deviations, Donsker--Varadhan rate function

*DOI*

*Abstract*

We derive an annealed large deviation principle for the normalised local times of a continuous-time random walk among random conductances in a finite domain in $Z^d$ in the spirit of Donsker-Varadhan citeDV75. We work in the interesting case that the conductances may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small values of the conductances and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution. As an application, we identify the logarithmic asymptotics of the lower tails of the principal eigenvalue of the randomly perturbed negative Laplace operator in the domain.

*Appeared in*

- Electron. Comm. Probab., 17 (2012) pp. 1--11.

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# A gradient structure for systems coupling reaction-diffusion effects in bulk and interfaces

*Authors*

- Glitzky, Annegret
- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 35K45 35Q92 78A35 78A57

*Keywords*

- Gradient-flow evolution, electro-reaction-diffusion systems, interface kinetics, reversible mass action type reactions, free energy functional

*DOI*

*Abstract*

We derive gradient-flow formulations for systems describing drift-diffusion processes of a finite number of species which undergo mass-action type reversible reactions. Our investigations cover heterostructures, where material parameter may depend in a nonsmooth way on the space variable. The main results concern a gradient flow formulation for electro-reaction-diffusion systems with active interfaces permitting drift-diffusion processes and reactions of species living on the interface and transfer mechanisms allowing bulk species to jump into an interface or to pass through interfaces. The gradient flows are formulated in terms of two functionals: the free energy and the dissipation potential. Both functionals consist of a bulk and an interface integral. The interface integrals determine the interface dynamics as well as the self-consistent coupling to the model in the bulk. The advantage of the gradient structure is that it automatically generates thermodynamically consistent models.

*Appeared in*

- ZAMP Z. Angew. Math. Phys., 64 (2013) pp. 29--52.

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# The many-to-few lemma and multiple spines

*Authors*

- Harris, Simon S.
- Roberts, Matthew I.

*2010 Mathematics Subject Classification*

- 60J80

*Keywords*

- Branching processes, many-to-one, many-to-few, spine, change of measure

*DOI*

*Abstract*

We develop an extension to the spine theory of branching processes, and use it to give a simple and intuitive identity for calculating additive functionals of such processes, generalizing the well-known many-to-one lemma.

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# Condition number and eccentricity of a closed convex cone

*Authors*

- Henrion, René
- Seeger, Alberto

*2010 Mathematics Subject Classification*

- 46B10 46B20 52A41

*Keywords*

- Convex cone, circumradius, inradius, condition number, eccentricity, simplicial cones

*DOI*

*Abstract*

We discuss some extremality issues concerning the circumradius, the inradius, and the condition number of a closed convex cone in $mathbbR^n$. The condition number refers to the ratio between the circumradius and the inradius. We also study the eccentricity of a closed convex cone, which is a coefficient that measures to which extent the circumcenter differs from the incenter.

*Appeared in*

- Math. Scand., 109 (2011) pp. 285--308.

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# Optimal control of the sweeping process

*Authors*

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

*2010 Mathematics Subject Classification*

- 49J52 49J53 45K24 45M25 90C30

*Keywords*

- Sweeping process, optimal control, dissipative differential inclusions, variational analysis, generalized differentiation

*DOI*

*Abstract*

We formulate and study an optimal control problem for the sweeping (Moreau) process, where control functions enter the moving sweeping set. To the best of our knowledge, this is the first study in the literature devoted to optimal control of the sweeping process. We first establish an existence theorem of optimal solutions and then derive necessary optimality conditions for this optimal control problem of a new type, where the dynamics is governed by discontinuous differential inclusions with variable right-hand sides. Our approach to necessary optimality conditions is based on the method of discrete approximations and advanced tools of variational analysis and generalized differentiation. The final results obtained are given in terms of the initial data of the controlled sweeping process and are illustrated by nontrivial examples.

*Appeared in*

- Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19 (2012) pp. 117--159.

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# A simple path to asymptotics for the frontier of a branching Brownian motion

*Authors*

- Roberts, Matthew I.

*2010 Mathematics Subject Classification*

- 60J80

*Keywords*

- Branching Brownian motion, KPP equation, change of measure

*DOI*

*Abstract*

We give proofs of two results about the position of the extremal particle in a branching Brownian motion, one concerning the median position and another the almost sure behaviour. Our methods are based on a many-to-two lemma which allows us to estimate the effect of the branching structure on the system by considering two dependent Bessel processes.

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# Position-orientation adaptive smoothing of diffusion weighted magnetic resonance data (POAS)

*Authors*

- Becker, Saskia
- Tabelow, Karsten

ORCID: 0000-0003-1274-9951 - Voss, Henning U.
- Anwander, Alfred
- Heidemann, Robin M.
- Polzehl, Jörg

ORCID: 0000-0001-7471-2658

*2010 Mathematics Subject Classification*

- 62P10 62G05 22E99

*Keywords*

- Diffusion weighted magnetic resonance imaging, POAS, Structural adaptive smoothing, Special Euclidean motion group, Lie groups

*DOI*

*Abstract*

We introduce an algorithm for diffusion weighted magnetic resonance imaging data enhancement based on structural adaptive smoothing in both space and diffusion direction. The method, called POAS, does not refer to a specific model for the data, like the diffusion tensor or higher order models. It works by embedding the measurement space into a space with defined metric and group operations, in this case the Lie group of three-dimensional Euclidean motion SE(3). Subsequently, pairwise comparisons of the values of the diffusion weighted signal are used for adaptation. The position-orientation adaptive smoothing preserves the edges of the observed fine and anisotropic structures. The POAS-algorithm is designed to reduce noise directly in the diffusion weighted images and consequently also to reduce bias and variability of quantities derived from the data for specific models. We evaluate the algorithm on simulated and experimental data and demonstrate that it can be used to reduce the number of applied diffusion gradients and hence acquisition time while achieving similar quality of data, or to improve the quality of data acquired in a clinically feasible scan time setting.

*Appeared in*

- Medical Image Analysis, 16 (2012) pp. 1142--1155

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# An electronic model for solar cells including active interfaces and energy resolved defect densities

*Authors*

- Glitzky, Annegret

*2010 Mathematics Subject Classification*

- 35K57 35R05 35B45 78A35

*Keywords*

- Reaction-diffusion systems, drift-diffusion processes, active interfaces, energy resolved defect densities, existence, uniqueness, a priori estimates

*DOI*

*Abstract*

We introduce an electronic model for solar cells taking into account heterostructures with active interfaces and energy resolved volume and interface trap densities. The model consists of continuity equations for electrons and holes with thermionic emission transfer conditions at the interface and of ODEs for the trap densities with energy level and spatial position as parameters, where the right hand sides contain generation-recombination as well as ionization reactions. This system is coupled with a Poisson equation for the electrostatic potential. We show the thermodynamic correctness of the model and prove a priori estimates for the solutions to the evolution system. Moreover, existence and uniqueness of weak solutions of the problem are proven. For this purpose we solve a regularized problem and verify bounds of the corresponding solution not depending on the regularization level.

*Appeared in*

- SIAM J. Math. Anal., 44 (2012) pp. 3874--3900.

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# Boundary coefficient control --- A maximal parabolic regularity approach

*Authors*

- Hömberg, Dietmar
- Krumbiegel, Klaus
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 35K20 35B65 47F05 49J20 49K20

*Keywords*

- Parabolic equation, mixed boundary condition, maximal parabolic $L^p$-regularity, optimal control, sufficient optimality conditions

*DOI*

*Abstract*

We investigate a control problem for the heat equation. The goal is to find an optimal heat transfer coefficient in the Robin boundary condition such that a desired temperature distribution at the boundary is adhered. To this end we consider a function space setting in which the heat flux across the boundary is forced to be an $L^p$ function with respect to the surface measure, which in turn implies higher regularity for the time derivative of temperature. We show that the corresponding elliptic operator generates a strongly continuous semigroup of contractions and apply the concept of maximal parabolic regularity. This allows to show the existence of an optimal control and the derivation of necessary and sufficient optimality conditions.

*Appeared in*

- Appl. Math. Optim., 67 (2013) pp. 3--31 under the title ``Optimal control of a parabolic equation with dynamic boundary condition"

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# Distributed optimal control of a nonstandard system of phase field equations

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Podio-Guidugli, Paolo
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 74A15 35K55 49K20

*Keywords*

- distributed optimal control, nonlinear phase field systems, first-order necessary optimality conditions

*DOI*

*Abstract*

We investigate a distributed optimal control problem for a phase field model of Cahn-Hilliard type. The model describes two-species phase segregation on an atomic lattice under the presence of diffusion; it has been introduced recently in [4], on the basis of the theory developed in [15], and consists of a system of two highly nonlinearly coupled PDEs. For this reason, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.

*Appeared in*

- Contin. Mech. Thermodyn., 24 (2012) pp. 437--459.

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# Fermionic and bosonic Laughlin state on thick cylinders

*Authors*

- Jansen, Sabine

*2010 Mathematics Subject Classification*

- 81V70 81R40

*Keywords*

- quantum many-body theory, symmetry breaking, quasi-state decomposition, fractional quantum Hall effect, Coulomb systems, jellium, powers of Vandermonde determinants

*DOI*

*Abstract*

We investigate a many-body wave function for particles on a cylinder known as Laughlin's function. It is the power of a Vandermonde determinant times a Gaussian. Our main result is: in a many-particle limit, at fixed radius, all correlation functions have a unique limit, and the limit state has a non-trivial period in the axial direction. The result holds regardless how large the radius is, for fermions as well as bosons. In addition, we explain how the algebraic structure used in proofs relates to a ground state perturbation series and to quasi-state decompositions, and we show that the monomer-dimer function introduced in an earlier work is an exact, zero energy, ground state of a suitable finite range Hamiltonian; this is interesting because of formal analogies with some quantum spin chains.

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# Stationary solutions for two-layer lubrication equations

*Authors*

- Jachalski, Sebastian
- Huth, Robert
- Kitavtsev, Georgy
- Peschka, Dirk

ORCID: 0000-0002-3047-1140 - Wagner, Barbara

*2010 Mathematics Subject Classification*

- 76Dxx 76Txx 35B40 35C20 49Jxx

*Keywords*

- thin films, gamma-convergence, matched asymptotics, free boundaries, bilayer

*DOI*

*Abstract*

We investigate stationary solutions of flows of thin liquid bilayers in an energetic formulation which is motivated by the gradient flow structure of its lubrication approximation. The corresponding energy favors the liquid substrate to be only partially covered by the upper liquid. This is expressed by a negative spreading coefficient which arises from an intermolecular potential combining attractive and repulsive forces and leads to an ultra-thin layer of thickness ε. For the corresponding lubrication models existence of stationary solutions is proven. In the limit ε to 0 matched asymptotic analysis is applied to derive sharp-interface models and the corresponding contact angles, i.e. the Neumann triangle. In addition we use Γ-convergence and derive the equivalent sharp-interface models rigorously in this limit. For the resulting model existence and uniqueness of energetic minimizers are proven. The minimizers agree with solutions obtained by matched asymptotics.

*Appeared in*

- SIAM J. Appl. Math., 73 (2013) pp. 1183--1202.

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# The elliptic-regularization principle in Lagrangian mechanics

*Authors*

- Liero, Matthias

ORCID: 0000-0002-0963-2915 - Stefanelli, Ulisse

*2010 Mathematics Subject Classification*

- 70H03 70H30 65L10

*Keywords*

- Lagrangian mechanics, variational principle, elliptic regularization, time discretization

*DOI*

*Abstract*

We present a novel variational approach to Lagrangian mechanics based on elliptic regularization with respect to time. A class of parameter-dependent global-in-time minimization problems is presented and the convergence of the respective minimizers to the solution of the system of Lagrange's equations is ascertained. Moreover, we extend this perspective to mixed dissipative/nondissipative situations, present a finite time-horizon version of this approach, and provide related Γ-convergence results. Finally, some discussion on corresponding time-discrete versions of the principle is presented.

*Appeared in*

- J. Nonlinear Sci., 23 (2013) pp. 179--204, under the title "A new minimum principle for Lagrangian mechanics"

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# A new model for passive mode-looking in a semiconductor laser

*Authors*

- Vladimirov, Andrei
- Turaev, Dmitry
- Kozyreff, Gregory

*2010 Mathematics Subject Classification*

- 78A60 34C23

*2008 Physics and Astronomy Classification Scheme*

- 42.60.Fc,42.55.Px,42.60.Mi,42.65.Pc,42.60.Gd

*Keywords*

- semiconductor laser, mode-locking, delay differential equations, bifurcations

*DOI*

*Abstract*

We propose a new model for passive mode-locking that is a set of ordinary delay differential equations. We assume the ring cavity geometry and a Lorentzian spectral filtering of the pulses, but do not use small gain and loss and weak saturation approximations. By means of a continuation method we study mode-locking solutions and their stability. We found that stable mode-locking can exist even when the non-lasing state between pulses becomes unstable.

*Appeared in*

- Optics Letters, 29, #11, 1221-1223 (2004)

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# On the stability of periodic orbits in delay equations with large delay

*Authors*

- Sieber, Jan
- Wolfrum, Matthias
- Lichtner, Mark
- Yanchuk, Serhiy

*2010 Mathematics Subject Classification*

- 34K13 34K20 34K06

*Keywords*

- Periodic solutions, large delay, stability, asymptotic continuous spectrum, strongly unstable spectrum, Floquet multipliers

*DOI*

*Abstract*

We prove a necessary and sufficient criterion for the exponential stability of periodic solutions of delay differential equations with large delay. We show that for sufficiently large delay the Floquet spectrum near criticality is characterized by a set of curves, which we call asymptotic continuous spectrum, that is independent on the delay.

*Appeared in*

- Discrete Contin. Dyn. Syst., 33 (2013) pp. 3109--3134.

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# Linearized plasticity is the evolutionary Gamma limit of finite plasticity

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Stefanelli, Ulisse

*2010 Mathematics Subject Classification*

- 74C15 49J45

*Keywords*

- Finite-strain elastoplasticity, linearized elastoplasticity, Gamma-convergence, rate-independent processes

*DOI*

*Abstract*

We provide a rigorous justification of the classical linearization approach in plasticity. By taking the small-deformations limit, we prove via Gamma-convergence for rate-independent processes that energetic solutions of the quasi-static finite-strain elastoplasticity system converge to the unique strong solution of linearized elastoplasticity

*Appeared in*

- J. Eur. Math. Soc. (JEMS), 15 (2013) pp. 923--948.

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# Improvement of output beam quality in broad area lasers with off-axis feedback

*Authors*

- Lichtner, Mark
- Tronciu, Vasile
- Vladimirov, Andrei G.

*2010 Mathematics Subject Classification*

- 78860 37M05 78A45

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 42.60.Mi 42.60.Da 42.55.Sf

*Keywords*

- Broad area semiconductor laser, off-axis feedback, supermode

*DOI*

*Abstract*

We report a method to improve the beam quality of broad area lasers by using a V-shaped external cavity formed by two off-axis feedback mirrors that allow to select a single transverse mode with the intensity modulated in the transverse direction. We find that in the case when one of the two feedback mirrors is absent a spontaneous formation of self-induced transverse population grating leading to a reduction of the lasing threshold is observed. Most favorable conditions for stabilization of single transverse supermode and creation of a high power and high brightness plane wave traveling in the extended cavity are obtained for equal re ectivities of the two external reflectors.

*Appeared in*

- IEEE J. Quantum Electron., 48 (2012) pp. 353--360.

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# Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 35K57 80A17 82B35 35Q72 74F25 82B35

*Keywords*

- Gradient flow, Onsager system, Onsager operator, dual dissipation potential, dual entropy-production potential, thermionic emission, reversible reactions

*DOI*

*Abstract*

We show that many couplings between parabolic systems for processes in solids can be formulated as a gradient system with respect to the total free energy or the total entropy. This includes Allen-Cahn, Cahn-Hilliard, and reaction-diffusion systems and the heat equation. For this, we write the coupled system as an Onsager system (**X**,Φ,*K*) defining the evolution $dot U$= - *K*(U) DΦ(U). Here Φ is the driving functional, while the Onsager operator *K*(U) is symmetric and positive semidefinite. If the inverse *G*=*K*^{-1} exists, the triple (**X**,Φ,*G*) defines a gradient system. Onsager systems are well suited to model bulk-interface interactions by using the dual dissipation potential Ψ^{*}(U, Ξ)= ½ ⟨Ξ *K*(U) Ξ⟩. Then, the two functionals Φ and Ψ^{*} can be written as a sum of a volume integral and a surface integral, respectively. The latter may contain interactions of the driving forces in the interface as well as the traces of the driving forces from the bulk. Thus, capture and escape mechanisms like thermionic emission appear naturally in Onsager systems, namely simply through integration by parts.

*Appeared in*

- Discrete Contin. Dyn. Syst. Ser. S, 6 (2013) pp. 479--499.

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# Strong solutions for the interaction of a rigid body and a viscoelastic fluid

*Authors*

- Götze, Karoline

*2010 Mathematics Subject Classification*

- 74F10 76A10 35Q35

*Keywords*

- fluid-solid interactions, viscoelastic fluids

*DOI*

*Abstract*

We study a coupled system of equations describing the movement of a rigid body which is immersed in a viscoelastic fluid. It is shown that under natural assumptions on the data and for general goemetries of the rigid body, excluding contact scenarios, a unique local-in-time strong solution exists.

*Appeared in*

- J. Math. Fluid Mech., 15 (2013) pp. 663--688.

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# Well-posedness and long-time behavior for a nonstandard viscous Cahn--Hilliard system

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Podio-Guidugli, Paolo
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 74A15 35K55 35A05, 35B40

*Keywords*

- Cahn-Hilliard equation, phase field model, well-posedness, long-time behavior

*DOI*

*Abstract*

We study a diffusion model of phase field type, consisting of a system of two partial differential equations encoding the balances of microforces and microenergy; the two unknowns are the order parameter and the chemical potential. By a careful development of uniform estimates and the deduction of certain useful boundedness properties, we prove existence and uniqueness of a global-in-time smooth solution to the associated initial/boundary-value problem; moreover, we give a description of the relative $omega$-limit set.

*Appeared in*

- SIAM J. Appl. Math., 71 (2011) pp. 1849--1870.

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# Passing to the limit in a Wasserstein gradient flow: From diffusion to reaction

*Authors*

- Arnrich, Steffen
- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Peletier, Mark A.
- Savaré, Giuseppe
- Veneroni, Marco

*2010 Mathematics Subject Classification*

- 35K67 35B25 35B27 49S99 35K20 35K57 60F10 70F40 70G75 37L05

*Keywords*

- Fokker-Planck equation, transport equation, metric evolution, Gamma convergence

*DOI*

*Abstract*

We study a singular-limit problem arising in the modelling of chemical reactions. At finite $e>0$, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by $1/e$, and in the limit $eto0$, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier, Savaré, and Veneroni, em SIAM Journal on Mathematical Analysis, 42(4):1805--1825, 2010, using the linear structure of the equation. In this paper we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular, we make no use of the linearity, nor of the fact that it is a second-order system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the propety of being a emphcurve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradient-flow structure, we prove that the sequence of rescaled solutions is pre-compact in an appropriate topology. We then prove a Gamma-convergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the $e$-problem converge to a solution of the limiting problem.

*Appeared in*

- Calc. Var. Partial Differ. Equ., 44 (2012) pp. 419--454.

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# Strong synchronization of weakly interacting oscillons

*Authors*

- Turaev, Dmitry
- Vladimirov, Andrei G.
- Zelik, Sergey

*2010 Mathematics Subject Classification*

- 37N20 49K20 34E10

*2008 Physics and Astronomy Classification Scheme*

- 05.45.-a 42.65.Pc

*Keywords*

- Localized structures of light, light pulses, oscillons, interaction of dissipative soliton

*DOI*

*Abstract*

We study interaction of well-separated oscillating localized structures (oscillons). We show that oscillons emit weakly decaying dispersive waves, which leads to formation of bound states due to subharmonic synchronization. We also show that in optical applications the Andronov-Hopf bifurcation of stationary localized structures leads to a drastic increase in their interaction strength.

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# On a singularly perturbed initial value problem in case of a double root of the degenerate equation

*Authors*

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

*2010 Mathematics Subject Classification*

- 34A12 34E05 34E15

*Keywords*

- singularly perturbed first order ordinary differential equation, initial value problem, boundary layer, double root of degenerate equation, asymptotic expansion

*DOI*

*Abstract*

We study the initial value problem of a singularly perturbed first order ordinary differential equation in case that the degenerate equation has a double root. We construct the formal asymptotic expansion of the solution such that the boundary layer functions decay exponentially. This requires a modification of the standard procedure. The asymptotic solution will be used to construct lower and upper solutions guaranteeing the existence of a unique solution and justifying its asymptotic expansion.

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# A model for the evolution of laminates in finite-strain elastoplasticity

*Authors*

- Hackl, Klaus
- Heinz, Sebastian
- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 49Q20 49S05 74C15

*Keywords*

- Rate-independent evolution, finite plasticity, gradient Young measures, polyconvexity

*DOI*

*Abstract*

We study the time evolution in elastoplasticity within the rate-independent framework of generalized standard materials. Our particular interest is the formation and the evolution of microstructure. Providing models where existence proofs are possible is a challenging task since the presence of microstructure comes along with a lack of convexity and, hence, compactness arguments cannot be applied to prove the existence of solutions. In order to overcome this problem, we will incorporate information on the microstructure into the internal variable, which is still compatible with generalized standard materials. More precisely, we shall allow for such microstructure that is given by simple or sequential laminates. We will consider a model for the evolution of these laminates and we will prove a theorem on the existence of solutions to any finite sequence of time-incremental minimization problems. In order to illustrate the mechanical consequences of the theory developed some numerical results, especially dealing with the rotation of laminates, are presented.

*Appeared in*

- ZAMM Z. Angew. Math. Mech., 92 (2012) pp. 888--909.

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# MAC schemes on triangular Delaunay meshes

*Authors*

- Eymard, Robert
- Fuhrmann, Jürgen

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

ORCID: 0000-0002-0165-2698

*2010 Mathematics Subject Classification*

- 76D05 65N08

*Keywords*

- incompressible Navier-Stokes equations, generalized MAC discretization, unstructured Delaunay grid, finite volume method, coupled flow problem, convergence proof

*DOI*

*Abstract*

We study two classical generalized MAC schemes on unstructured triangular Delaunay meshes for the incompressible Stokes and Navier-Stokes equations and prove their convergence for the first time. These generalizations use the duality between Voronoi and triangles of Delaunay meshes, in order to construct two staggered discretization schemes. Both schemes are especially interesting, since compatible finite volume discretizations for coupled convection-diffusion equations can be constructed which preserve discrete maximum principles. In the first scheme, called tangential velocity scheme, the pressures are defined at the vertices of the mesh, and the discrete velocities are tangential to the edges of the triangles. In the second scheme, called normal velocity scheme, the pressures are defined in the triangles, and the discrete velocities are normal to the edges of the triangles. For both schemes, we prove the convergence in $L^2$ for the velocities and the discrete rotations of the velocities for the Stokes and the Navier-Stokes problem. Further, for the normal velocity scheme, we also prove the strong convergence of the pressure in $L^2$. Linear and nonlinear numerical examples illustrate the theoretical predictions.

*Appeared in*

- Numer. Methods Partial Differential Equations, 30 (2014) pp. 1397--1424.

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