# MAPLE for stochastic differential equations

*Authors*

- Cyganowski, Sasha
- Kloeden, Peter, E.
- Pohl, Thomas

*2010 Mathematics Subject Classification*

- 60H10 65H05 60H30 68Q40 93E15 93E30

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# On a class of continuous coagulation-fragmentation equations

*Authors*

- Laurencot, Philippe

*2010 Mathematics Subject Classification*

- 35F25 45K05 35B40 82C22

*Keywords*

- coagulation, fragmentation, existence of solutions, gelation, large time behaviour

*Abstract*

A model for the dynamics of particles undergoing simultaneously coalescence and breakup is considered, each particle being assumed to be fully identified by its size. Existence of solutions to the corresponding evolution integral partial differential equation is shown for product-type coagulation kernels with a weak fragmentation. The failure of density of conservation (or gelation) is also investigated in some particular cases.

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# Iterative procedure for multidimensional Euler equations

*Authors*

- Dreyer, Wolfgang
- Kunik, Matthias
- Sabelfeld, Karl
- Simonov, Nikolai
- Wilmanski, Krzysztof

*2010 Mathematics Subject Classification*

- 35L45 35L65 35L67 82B40 11K45

*Keywords*

- Initial value problems, hyperbolic systems of first order, conservation laws, shocks, kinetic theory, Monte Carlo methods

*Abstract*

A numerical iterative scheme is suggested to solve the Euler equations in two and three dimensions. The step of the iteration procedure consists of integration over the velocity which is here carried out by three different approximate integration methods, and in particular, by a special Monte Carlo technique. Regarding the Monte Carlo integration, we suggest a dependent sampling technique which ensures that the statistical errors are quite small and uniform in space and time. Comparisons of the Monte Carlo calculations with the trapezoidal rule and a gaussian integration method show good agreement.

*Appeared in*

- Monte Carlo Methods and Appl., 4 (1998), No. 3, pp. 253-271

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# One-Particle Stochastic Lagrangian Model for Turbulent Dispersion in Horizontally Homogeneous Turbulence

*Authors*

- Kurbanmuradov, Orazgeldy
- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05 76F99 65C20

*Keywords*

- Stochastic models of turbulence, Generalized Langevin stochastic differential equation, Eulerian and Lagrangian one- and two-particle stochastic modelsof turbulent transport, particle dispersion in the surface layer of the turbulent atmosphere

*Abstract*

A one-particle stochastic Lagrangian model in 2D and 3D dimensions is constructed for transport of particles in horizontaly homogeneous turbulent flows with arbitrary one-point probability density function. It is shown that in the case of anisotropic turbulence with gaussian pdf, this model essentially differs from the known Thomson's model. The results of calculations according to our model in the case of neutrally stratified atmospheric surface layer agree satisfactorily with the measurements known from the literature.

*Appeared in*

- Monte Carlo Methods and Appl., 4 (1998), No. 2, pp. 127-140

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# The convergence and stability of splitting finite difference schemes for nonlinear evolutionary type equations

*Authors*

- Ivanauskas, Feliksas
- Radziunas, Mindaugas

*2010 Mathematics Subject Classification*

- 65N06 65N12

*Keywords*

- evolutionary equations, finite difference scheme, splitting scheme

*Abstract*

A splitting finite difference scheme for an initial-boundary value problem for a two-dimensional nonlinear evolutionary type equation is considered. The problem is split into nonlinear and linear parts. The linear part is also split into locally one-dimensional equations. The convergence and stability of the scheme in $L_2$ and $C$ norms are proved.

*Appeared in*

- Lithuanian Math. Journal, 45(3), pp. 413-434, 2005.

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# Continuum--sites stepping--stone models, coalescing exchangeable partitions, and random trees

*Authors*

- Donnelly, Peter
- Evans, Steven N.
- Fleischmann, Klaus
- Kurtz, Thomas G.
- Zhou, Xiaowen

*2010 Mathematics Subject Classification*

- 60K35 60G57 60J60

*Keywords*

- coalesce, partition, right process, annihilate, dual, diffusion, exchangeable, vector measure, tree, Hausdorff dimension, packing dimension, capacity equivalence, fractal

*Abstract*

Analogues of stepping--stone models are considered where the site--space is continuous, the migration process is a general Markov process, and the type--space is infinite. Such processes were defined in previous work of the second author by specifying a Feller transition semigroup in terms of expectations of suitable functionals for systems of coalescing Markov processes. An alternative representation is obtained here in terms of a limit of interacting particle systems. It is shown that, under a mild condition on the migration process, the continuum--sites stepping--stone process has continuous sample paths. The case when the migration process is Brownian motion on the circle is examined in detail using a duality relation between coalescing and annihilating Brownian motion. This duality relation is also used to show that a random compact metric space that is naturally associated to an infinite family of coalescing Brownian motions on the circle has Hausdorff and packing dimension both almost surely equal to $frac12$ and, moreover, this space is capacity equivalent to the middle--$frac12$ Cantor set (and hence also to the Brownian zero set).

*Appeared in*

- Ann. Probab., 28(2000), pp. 1063-1110

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# Coagulation of aerosol particles in turbulent flows

*Authors*

- Kurbanmuradov, Orazgeldy
- Sabelfeld, Karl

*2010 Mathematics Subject Classification*

- 65C05 76F99

*Keywords*

- coagulation of particles, turbulent flows, random coagulation coefficient, numerical study

*Abstract*

Coagulation of particles in turbulent flows is studied. The size distribution of particles is governed by Smoluchowski equation with random coagulation coefficient. This coefficient is derived by a generalization of the approach by Saffman and Turner. A special stochastic time is introduced which drastically simplifies the analysis of the influence of the intermittency. A detailed numerical study is given for two cases with known explicit solutions of Smoluchowski equation.

*Appeared in*

- Monte Carlo Methods Appl., 6(2000), pp. 211-253

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# Finite time extinction of super-Brownian motions with catalysts

*Authors*

- Dawson, Donald A.
- Fleischmann, Klaus
- Mueller, Carl

*2010 Mathematics Subject Classification*

- 60J80 60J55 60G57

*Keywords*

- catalytic super-Brownian motion, historical superprocess, critical branching, finite time extinction, measure-valued branching, random medium, good and bad paths, stoppedmeasures, collision local time, comparison, coupling, stopped historical superprocess, branching rate functional, super-random walk, interacting Feller's branching diffusion

*Abstract*

Consider a catalytic super-Brownian motion $X=X^Gamma$ with finite variance branching. Here "catalytic" means that branching of the reactant $X$ is only possible in the presence of some catalyst. Our intrinsic example of a catalyst is a stable random measure $Gamma $ on $R$ of index $0< gamma <1$. Consequently, here the catalyst is located in a countable dense subset of $R$. Starting with a finite reactant mass $X_0$ supported by a compact set, $X$ is shown to die in finite time. Our probabilistic argument uses the idea of good and bad historical paths of reactant ``particles44 during time periods $[T_n,T_n+1)$. Good paths have a significant collision local time with the catalyst, and extinction can be shown by individual time change according to the collision local time and a comparison with Feller's branching diffusion. On the other hand, the remaining bad paths are shown to have a small expected mass at time $T_n+1$ which can be controlled by the hitting probability of point catalysts and the collision local time spent on them.

*Appeared in*

- Ann. Probab., 28(2) (2000), pp. 603-642

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# Deviations from typical type proportions in critical multitype Galton-Watson processes

*Authors*

- Fleischmann, Klaus
- Vatutin, Vladimir A.

*2010 Mathematics Subject Classification*

- 60J80 60J15

*Keywords*

- marked particle, typical type proportions, non-degenerate limit, non-extinction, deviations, asymptotic expansion

*Abstract*

Consider a critical K-type Galton-Watson process $Z(t): t=0,1,...$, and a real vector $w = (w_1,...,w_K)^top$. It is well-known that under rather general assumptions, $langle Z(t),wrangle := sum_k Z_k(t)w_k$ conditioned on non-extinction and appropriately scaled has a limit in law as $tuparrow infty$ [Vatutin (1977)]. But the limit degenerates to 0 if the vector w deviates seriously from 'typical' type proportions, i.e. if w is orthogonal to the left eigenvectors related to the maximal eigenvalue of the mean value matrix. We show that in this case (under reasonable additional assumptions on the offspring laws) there exists a better normalization which leads to a non-degenerate limit. Opposed to the finite variance case, which was already resolved in Athreya and Ney (1974) and Badalbaev and Mukhitdinov (1989), the limit law (for instance its 'index') may seriously depend on w.

*Appeared in*

- Teor. Veroyatnist. i Primenen., 45(2000), pp. 30-51

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# On the hot spots of a catalytic super-Brownian motion

*Authors*

- Delmas, Jean-Francois
- Fleischmann, Klaus

*2010 Mathematics Subject Classification*

- 60J80 60G57 60K35

*Keywords*

- Catalytic super-Brownian medium, superprocess, measure-valued process, collision local time of catalyst and reactant, two-dimensional process, catalytic medium

*Abstract*

Consider the catalytic super-Brownian motion $X^rho$ (reactant) in $R^d$, $dle 3$, which branching rates vary randomly in time and space and in fact are given by an ordinary super-Brownian motion $rho$ (catalyst). Our main object of study is the collision local time $L=L_[rho,X^rho](d[s,x])$ of catalyst and reactant. It determines the covariance measure in the martingale problem for $X^rho$ and reflects the occurence of 'hot spots' of reactant which can be seen in simulations of $X^rho$. In dimension 2, spatial marginal collision densities exist and, via self-similarity, enter as factor in the long-term random ergodic limit of $L$ (diffusiveness of the 2-dimensional model).

*Appeared in*

- Probab. Theory Relat. Fields, 121(3) (2001), pp. 389-421

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# Regular oscillations in systems with stochastic resonance

*Authors*

- Milstein, Grigori N.
- Tretyakov, Michael V.

*2010 Mathematics Subject Classification*

- 60H10 93E30

*Keywords*

- Noise-driven monostable, bistable and coupled bistable systems, periodic forcing, boundary value problems of parabolic type, numerical integration of stochastic differential equations

*Abstract*

Constructive sufficient conditions for regular oscillations in systems with stochastic resonance are given. For bistable systems, they rely on the fact that the probability of transition of a point from one well to the other with subsequent stay there during the half-period of the periodic forcing is close to 1. Using these conditions, domains of parameters corresponding to the regular oscillations are indicated. The regular oscillations are considered in bistable and monostable systems with additive and multiplicative noise. Special attention is paid to numerical methods. Algorithms based on numerical integration of stochastic differential equations turn out to be most natural both for simulation of sample trajectories and for solution of related boundary value problems of parabolic type. Results of numerical experiments are presented.

*Appeared in*

- Physica D Nonlinear Phenomena, vol./issue: 140/3-4, (2000), pp. 244-256, under new title: Numerical analysis of noise-induced regular oscillations.

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# Generalized bang-bang properties in the optimization of plates

*Authors*

- Sprekels, Jürgen
- Tiba, Dan

*2010 Mathematics Subject Classification*

- 35R30 93C20 73K40

*Keywords*

- Bang-bang principles, shape optimization of plates, optimal control, necessary conditions of optimality

*Abstract*

For a simly supported plate, we consider two optimization problems: the volume minimization and the identification of a coefficient. Via a transformation recently introduced by the authors, we obtain the optimality conditions in a qualified form, and their analysis yields the bang-bang properties for the optimal thickness.

*Appeared in*

- C. R. Acad. Sci. Paris, t. 327 serie I, 705-710 (1998)

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# Reconstruction of source terms in evolution equations by exact controllability

*Authors*

- Yamamoto, Masahiro

*2010 Mathematics Subject Classification*

- 35L55 35R30 93B05

*Keywords*

- Source term, reconstruction, operator equation of second kind, Fredholm equation of second kind, Hilbert Uniqueness method, exact controllability

*Abstract*

For fixed $rho = rho(x,t)$, we consider the solution $u(f)$ to $$ u''(x,t) + Au(x,t) = f(x)rho(x,t), quad x in Omega, thinspace t > 0 $$ $$ u(x,0) = u'(x,0) = 0, qquad x in Omega, qquad B_ju(x,t) = 0, quad x in partialOmega, thinspace t > 0, thinspace 1 le j le m, $$ where $u'= fracpartial upartial t$, $u'' = fracpartial^2 u partial t^2$, $Omega subset R^r$, $r ge 1$ is a bounded domain with smooth boundary, $A$ is a uniformly symmetric elliptic differential operator of order $2m$ with $t$-independent smooth coefficients, $B_j$, $1 le j le m$, are $t$-independent boundary differential operators such that the system $ A, B_j _1le j le m$ is well-posed. Let $ C_j _1 le j le m$ be complementary boundary differential operators of $ B_j _1 le j le m$. We consider a multidimensional linear inverse problem : for given $Gamma subset partialOmega$, $T > 0$ and $n in 1, ..., m$, determine $f(x)$, $x in Omega$ from $C_ju(f)(x,t)$, $x in Gamma$, $0

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# Boundary Layer Approximate Approximations and Cubature of Potentials in Domains

*Authors*

- Ivanov, Tjavdar
- Mazya, Vladimir
- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 65D32 65D15 65N38

*Keywords*

- Volume potentials, semi-analytic cubature formulae, approximate approximations, approximate multi-resolution

*Abstract*

In this article we present a new approach to the computation of volume potentials over bounded domains, which is based on the quasi-interpolation of the density by %linear combinations of smooth, almost locally supported basis functions for which the corresponding volume potentials are known. The quasi-interpolant is a linear combination of the basis function with shifted and scaled arguments and with coefficients explicitly given by the point values of the density. Thus, the approach results in semi-analytic cubature formulae for volume potentials, which prove to be high order approximations of the integrals. It is based on multi-resolution schemes for accurate approximations up to the boundary by applying approximate refinement equations of the basis functions and iterative approximations on finer grids. We obtain asymptotic error estimates for the quasi-interpolation and corresponding cubature formulae and provide some numerical examples.

*Appeared in*

- Adv. Comput. Math., 10 (1999) No. 3/4, pp. 311-342

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# Linear Elliptic Boundary Value Problems with Non-smooth Data: Normal Solvability on Sobolev-Campanato Spaces

*Authors*

- Griepentrog, Jens André
- Recke, Lutz

*2010 Mathematics Subject Classification*

- 35J55 46E35 47A53

*Keywords*

- Bounded measurable coefficients, Lipschitz domains, Regular sets, Non-homogeneous mixed boundary conditions, Regularity up to the boundary of weak solutions, Smoothness of the coefficient-to-solution-map, Arbitrary space dimension

*Abstract*

In this paper linear elliptic boundary value problems of second order with non-smooth data (bounded measurable coefficients, Lipschitz domains, regular sets, non-homogeneous mixed boundary conditions) are considered. It is shown that such boundary value problems generate Fredholm operators between appropriate Sobolev-Campanato spaces, that the weak solutions are Holder continuous up to the boundary and that they depend smoothly (in the sense of a Holder norm) on the coefficients and on the right hand sides of the equations and boundary conditions.

*Appeared in*

- Math. Nachr., 225 (2001) pp. 39--74.

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# Piecewise Linear Wavelet Collocation on Triangular Grids,Approximation of the Boundary Manifold and Quadrature

*Authors*

- Ehrich, Sven
- Rathsfeld, Andreas

*2010 Mathematics Subject Classification*

- 65N38 45L10 65R20

*Keywords*

- pseudo-differential equation of order 0 and -1, piecewise linear collocation, wavelet algorithm, approximation of parametrization, quadrature

*Abstract*

In this paper we consider a piecewise linear collocation method for the solution of a pseudo-differential equations of order $r=0,-1$ over a closed and smooth boundary manifold. The trial space is the space of all continuous and piecewise linear functions defined over a uniform triangular grid and the collocation points are the grid points. For the wavelet basis in the trial space we choose the three-point hierarchical basis together with a slight modification near the boundary points of the global patches of parametrization. We choose three, four, and six term linear combinations of Dirac delta functionals as wavelet basis in the space of test functionals. Though not all wavelets have vanishing moments, we derive the usual compression results, i.e. we prove that, for $N$ degrees of freedom, the fully populated stiffness matrix of $N^2$ entries can be approximated by a sparse matrix with no more than $O(N [log N ]^2.25)$ non-zero entries. The main topic of the present paper, however, is to show that the parametrization can be approximated by low order piecewise polynomial interpolation and that the integrals in the stiffness matrix can be computed by quadrature, where the quadrature rules are combinations of product integration applied to non analytic factors of the integrand and of high order Gauss rules applied to the analytic parts. The whole algorithm for the assembling of the matrix requires no more than $O(N [log N ]^4.25)$ arithmetic operations, and the error of the collocation approximation, including the compression, the approximative parametrization, and the quadratures, is less than $O(N^-1[log N]^2)$. Note that, in contrast to well-known algorithms by v.Petersdorff, Schwab, and Schneider, only a finite degree of smoothness is required.

*Appeared in*

- Electronic Transaction on Numerical Analysis 12, 2001, pp. 149-192

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# Electro-reaction-diffusion systems including cluster reactions of higher order

*Authors*

- Glitzky, Annegret
- Huenlich, Rolf

*2010 Mathematics Subject Classification*

- 35K57 78A35 35D05 35B45

*Keywords*

- Reaction-diffusion systems, higher order reactions, drift-diffusion processes, motion of charged particles, global estimates, existence, uniqueness, asymptotic behaviour

*Abstract*

In this paper we consider electro-reaction-diffusion systems modelling the transport of charged species in two-dimensional heterostructures. Our aim is to investigate the case that besides of reactions with source terms of at most second order so called cluster reactions of higher order are involved. We prove the unique solvability of the model equations and show the global boundedness and asymptotic properties of the solution. In order to get necessary a priori estimates we apply an anisotropic iteration scheme followed by usual Moser iterations. Then existence is obtained by cutting off the reaction terms.

*Appeared in*

- Math. Nachr., 216(2000), pp. 95-118

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# Minimax detection of a signal for qn-balls with l_pn balls removed

*Authors*

- Ingster, Yuri I.

*2010 Mathematics Subject Classification*

- 62G10 62G20

*Keywords*

- minimax hypotheses testing, asymptotics of error probabilities, infinitely divisible distributions

*Abstract*

In this paper we continue the researches of hypothesis testing problems leading to infinitely divisible distributions which have been started in the papers by Ingster, 1996a, 1997. Let the $n$-dimensional Gaussian random vector $x=xi+v$ is observed where $xi$ is a standard $n$-dimensional Gaussian vector and $vin R_n$ is an unknown mean. We consider the minimax hypothesis testing problem $H_0: v=0$ versus alternatives $H_1: vin V_n$, where $V_n$ is $l^n_q$-ball of radius $R_1,n$ with $l^n_p$-balls of radius $R_2,n$ removed. We are interesting in the asymptotics (as $n toinfty$) of the minimax second kind error probability $beta_n(alpha)=beta_n(alpha,p,q,R_ 1,n,R_2,n)$ where $alpha in (0,1)$ is a level of the first kind error probability. Close minimax estimation problem had been studied by Donoho and Johnstone (1994). We show that the asymptotically least favorably priors in the problem of interest are of the product type: $pi^n=pi_n times cdots times pi_n$. Here $pi_n= (1-h_n)delta_0+frach_n2(delta_-b_n+delta_b-n)$ are the three-point measures with some $h_n=h_n(p,q,R_1,n,R_2,n$ and $b_n=b_n(p,q,R_1,n,R_2,n$. This reduces the problem of interest to Bayssian hypothesis testing problems where the asymptotics of error probabilities had been studied by Ingster, 1996a, 1997. In particularly, if $ pleq q$, then the asymptotics of $beta_nalpha$ are of Gaussian type, but if $p>q$ then its are either Gaussian or degenerate or belong to a special class of infinitely divisible distributions which was described in Ingster, 1996a, 1997.

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# A class of time discrete schemes for a phase-field system of Penrose-Fife type

*Authors*

- Klein, Olaf

ORCID: 0000-0002-4142-3603

*2010 Mathematics Subject Classification*

- 65M12 35K50 80A22 35K60 35R35 65M15

*Keywords*

- Phase-field model, Penrose-Fife model, semidiscretization, convergence, error estimates

*Abstract*

In this paper, a phase field system of Penrose--Fife type with non-conserved order parameter is considered. A class of time-discrete schemes for an initial--boundary value problem for this phase-field system is presented. In three space dimensions, convergence is proved and an error estimate is derived. For one scheme, this error estimate is linear with respect to the time-step size.

*Appeared in*

- M2AN Math. Model. Numer. Anal., 33 (1999), no. 1, pp. 1261-1292

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# Local Estimation for an Integral Equation of First Kind with Analytic Kernel

*Authors*

- Cheng, Jin
- Prößdorf, Siegfried
- Yamamoto, Masashiro

*2010 Mathematics Subject Classification*

- 45A05 45H05 45M10

*Keywords*

- First kind integral equation, Riesz kernel, analytickernel, pointwise estimate for the solution severly ill-posed, Cauchy problem, Laplace equation.problem, Laplace equation

*Abstract*

In this paper, an integral equation of the first kind with Riesz kernel is discussed. Since the kernel of this integral equation is analytic, this problem is severe ill-posed. We prove that, for solutions of the integral equation, a local conditional pointwise estimate holds at a point if the solution has some additional smoothness properties in a neighbourhood of this point.

*Appeared in*

- J. Inverse and Ill-Posed Problems, 6 (1998), pp. 115-126

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# Tikhonov regularization for an integral equation of the first kind with logarithmic kernel

*Authors*

- Bruckner, Gottfried
- Cheng, Jin

*2010 Mathematics Subject Classification*

- 45A05 65R30

*Keywords*

- integral equation of the first kind, analytic kernel, Tikhonov regularization, convergence rate

*Abstract*

In this paper, we discuss stability and Tikhonov regularization for the integral equation of the first kind with logarithmic kernel. Since the kernel is analytic in our case, the problem is severely ill-posed. We prove a convergence rate for the regularized solution and describe a method for its numerical calculation.

*Appeared in*

- J. Inverse Ill-posed Problems, 8(2000), pp. 1-11

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# Nonlinear equations in non-reflexive Banach spaces and strongly nonlinear differential equations

*Authors*

- Soltanov, Kamal N.
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35J65 35K60 35A05

*Keywords*

- Fully nonlinear PDEs, strongly nonlinear elliptic and parabolic problems, nonlinear equations in non-reflexive spaces, existence results, weak solutions

*Abstract*

In this paper, we study strongly nonlinear degenerate elliptic and parabolic equations of the form $,F (x,u,Du,ldots,D^(2,m,-,1)u,Lu) = 0,$ and $,u_t = F (x,t,u, Du,ldots,D^(2,m,-,1)u,Lu),$, respectively, where $,L,$ is a linear operator of the derivatives of highest (i.e., of $,2,m$--th) order. Under very weak restrictions on the growth of $,F,$ with respect to the derivatives of $,u,$, existence results for weak solutions are proved. These existence results are based on general solvability results for nonlinear operator equations in Banach spaces which will be proved in this paper.

*Appeared in*

- Adv. Math. Sci. Appl., 9 (1999), No. 2, pp. 939-972

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# Singular behaviour of finite approximations to the addition model

*Authors*

- Laurencot, Philippe

*2010 Mathematics Subject Classification*

- 82C22 34A34

*Keywords*

- Addition model, coagulation, non-existence

*Abstract*

Instantaneous gelation in the addition model with superlinear rate coefficients is investigated. The conjectured post-gelation solution is shown to arise naturally as the limit of solutions to some finite approximations as the number of equations grows to infinity. Non-existence of continuous solutions to the addition model is also established in that case.

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# Stability of solutions to chance constrained stochastic programs

*Authors*

- Henrion, René
- Römisch, Werner

*2010 Mathematics Subject Classification*

- 90C15 90C31

*Keywords*

- stochastic programming, chance constraints, $r$-concave measures, quantitative stability

*Abstract*

Perturbations of convex chance constrained stochastic programs are considered the underlying probability distributions of which are $r$-concave. Verifiable sufficient conditions are established guaranteeing Hoelder continuity properties of solution sets with respect to variations of the original distribution. Examples illustrate the potential, sharpness and limitations of the results.

*Appeared in*

- J. Guddat, R. Hirabayashi, H.Th. Jongen and F. Twilt (eds): Parametric Optimization and Related Topics V, Peter Lang, Frankfurt a.M., 2000, pp. 95-114.

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# Phase-field models with hysteresis

*Authors*

- Krejčí, Pavel
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K55 47H30 80A22

*Keywords*

- Phase-field systems, phase transitions, hysteresis operators, thermodynamic consistency

*Abstract*

Phase-field systems as mathematical models for phase transitions have drawn increasing attention in recent years. However, while capable of capturing many of the experimentally observed phenomena, they are only of restricted value in modelling hysteresis effects occurring during phase transition processes. To overcome this shortcoming of existing phase-field theories, the authors have recently proposed a new approach to phase-field models which is based on the mathematical theory of hysteresis operators developed in the past fifteen years. In particular, they have proved well-posedness and thermodynamic consistency for hysteretic phase field models which are related to the Caginalp and Penrose-Fife models. In this paper, these results are extended into different directions: we admit temperature-dependent relaxation coefficients and relax the growth conditions for the hysteresis operators considerably

*Appeared in*

- J. Math. Anal., 252 (2000), pp. 198-219

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# Hysteresis operators in phase-field models ofPenrose-Fife type

*Authors*

- Krejčí, Pavel
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K55 80A22 47H30

*Keywords*

- Phase-field systems, phase transitions, hysteresis operators, well-posedness of parabolic systems, thermodynamic consistency, Penrose-Fife model

*Abstract*

Phase-field systems as mathematical models for phase transitions have drawn a considerable interest in recent years. However, while they are capable of capturing many of the experimentally observed phenomena, they are only of restricted value in modelling hysteresis effects occuring during phase transition processes. To overcome this shortcoming of existing phase-field theories, the authors recently proposed a new approach to phase-field models which is based on the mathematical theory of hysteresis operators developed in the past fifteen years. Well-posedness and thermodynamic consistency were proved for a phase-field system with hysteresis which is closely related to the model advanced by Caginalp in a series of papers. In this note the more difficult case of a phase-field system of Penrose-Fife type with hysteresis is investigated. Under slightly more restrictive assumptions than in the Caginalp case it is shown that the system is well-posed and thermodynamically consistent.

*Appeared in*

- Appl. Math. 43 (1998), pp.207-222

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# On polynomial mixing and convergence rate for stochastic difference and differential equations

*Authors*

- Veretennikov, Alexander Yu.

*2010 Mathematics Subject Classification*

- 60F15 60J05 60J60

*Keywords*

- Mixing, recurrence, Markov process, SDE, polynomial convergence

*Abstract*

Polynomial bounds for $beta$-mixing and for the rate of convergence to the invariant measure are established for discrete time Markov processes and solutions of SDEs under weak stability assumptions.

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# Optimal Nonparametric Testing of Qualitative Hypotheses

*Authors*

- Duembgen, Lutz
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G10

*Keywords*

- adaptivity, convexity, Levy modulus of continuity, monotonicity, positivity

*Abstract*

Suppose one observes a process $Y$ on the unit interval, where $dY = f + n^-1/2 d W$ with an unknown function parameter $f$, given scale parameter $n geq 1$ and standard Brownian motion $W$. We propose two classes of tests of qualitative nonparametric hypotheses about $f$ such as monotonicity or convexity. These tests are asymptotically optimal and adaptive with respect to two different criteria. As a by-product we obtain an extension of Levys modulus of continuity of Brownian motion. It is of independent interest because of its potential applications to simultaneous confidence intervals in nonparametric curve estimation.

*Appeared in*

- Ann. Statist., 29, no. 1 (2001), pp. 124 - 152 under the title: Multiscale testing of qualitative hypotheses.

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# Lyapunov functions for cocycle attractors in nonautonomous difference equations

*Authors*

- Kloeden, Peter E.

*2010 Mathematics Subject Classification*

- 93D30 34C35

*Keywords*

- Nonautonomous system, pull-back attraction, cocycle attractor, Lyapunov function

*Abstract*

The construction of a Lyapunov function characterizing the pullback attraction of a cocycle attractor of a nonautonomous discrete time dynamical system involving Lipschitz continuous mappings is presented.

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# On energy estimates for electro-diffusion equations arising in semiconductor technology

*Authors*

- Glitzky, Annegret
- Huenlich, Rolf

*2010 Mathematics Subject Classification*

- 35K57 78A35 35B40 35K45

*Keywords*

- Reaction-diffusion systems, drift-diffusion processes, pair diffusion models, steady states, asymptotic behaviour

*Abstract*

The design of modern semiconductor devices requires the numerical simulation of basic fabrication steps. We investigate some electro-reaction-diffusion equations which describe the redistribution of charged dopants and point defects in semiconductor structures and which the simulations should be based on. Especially, we are interested in pair diffusion models. We present new results concerned with the existence of steady states and with the asymptotic behaviour of solutions which are obtained by estimates of the corresponding free energy and dissipation functionals.

*Appeared in*

- Research Notes in Mathematics vol. 406, 2000, pp.158-174

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# On maximum of Gaussian non-centered fieldsindexed on smooth manifolds

*Authors*

- Piterbarg, Vladimir

*2010 Mathematics Subject Classification*

- 60G15 60G60

*Keywords*

- Gaussian fields, large excursions, maximum tail distribution, exact asymptotics

*Abstract*

The double sum method of evaluation of probabilities of large deviations for Gaussian processes with non-zero expectations is developed. Asymptotic behaviors of the tail of non-centered locally stationary Gaussian fields indexed on smooth manifold are evaluated. In particular, smooth Gaussian fields on smooth manifolds are considered.

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# On the characterization of self-regularizationproperties of a fully discrete projectionmethod for Symm's integral equation

*Authors*

- Pereverzev, Sergei V.
- Prößdorf, Siegfried

*2010 Mathematics Subject Classification*

- 65R30 65R20 45L10

*Keywords*

- Symm's integral equation, ill-posed problem, fully discrete projection method, perturbations in the data, error analysis, self-regularization

*Abstract*

The influence of small perturbations in the kernel and the right-hand side of Symm's boundary integral equation, considered in an ill-posed setting, is analyzed. We propose a modification of a fully discrete projection method which is more economical in the sense of complexity and allows to obtain the optimal order of accuracy in the power scale with respect to the level of the noise in the kernel or in the parametric representation of the boundary.

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# Standard Chase on Black Swans and Canards

*Authors*

- Shchepakina, Elena A.
- Sobolev, Vladimir A.

*2010 Mathematics Subject Classification*

- 34C34 34E15 34D15

*Keywords*

- Integral manifolds, duck-trajectories, singularlyperturbed systems

*Abstract*

The paper is devoted to the study of the relationship between integral manifolds of ordinary differential equations and duck--trajectories. We derive sufficient conditions for the existence of continuous slow integral surfaces that are devided into stable and unstable parts and propose a method of construction of surfaces consisting of duck--trajectories.

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# The work of Vladimir Maz'ya on integral and pseudodifferential operators

*Authors*

- Elschner, Johannes

*2010 Mathematics Subject Classification*

- 31-02 35-02 45-02 58-02

*Keywords*

- Integral operators, pseudodifferntial operators, obliquederivative problem, differential operators in the half--space, characteristic Cauchy problem, multiplier theory, boundary integral equations

*Abstract*

The paper presents an outline of Vladimir Maz'ya's important and influential contributions to the solvability theory of integral and pseudodifferential equations.

*Appeared in*

- The Mazya anniversary collection, Vol. 1, Op. Theory: Adv. Appl., Vol. 109, pp. 35-52, Birkhaeuser, Basel, 1999

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# On Quasi--interpolation with non--uniformly distributed centers on Domains and Manifolds

*Authors*

- Mazya, Vladimir
- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 41A30 41A63 41A25

*Keywords*

- Approximate approximations, multivariate quasi-interpolation, nonregular centers, manifolds

*Abstract*

The paper studies quasi-interpolation by scaled shifts of a smooth and rapidly decaying function. The centers are images of a smooth mapping of the $h Z^n$-lattice in $R^s$, $s ge n$, and the scaling parameters are proportional to $h$. We show that for a large class of generating functions the quasi-interpolants provide high order approximations up to some prescribed accuracy. Although the approximants do not converge as $h$ tends to zero, this is not feasible in computations if a scalar parameter is suitably chosen. The lack of convergence is compensated for by more flexibility in the choice of generating functions used in numerical methods for solving operator equations.

*Appeared in*

- J. Approx. Th. 10 (2001), pp. 125-145

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# Tricomi's composition formula and the analysis of multiwavelet approximation methods for boundary integral equations

*Authors*

- Prößdorf, Siegfried

*2010 Mathematics Subject Classification*

- 41A15 41A17 41A35 41A63 45B05 45E05 45E10 45L10 45M10 47A50 47A75 65N12 65N35 65N38

*Keywords*

- Tricomi's composition formula, collocation methods, defected splines, Galerkin-Petrov methods, multidimensional singular integral operators, multiscaling functions, multiwavelets, numerical symbol, pseudodifferential operators, stability conditions, superapproximation, symbol calculus

*Abstract*

The present paper is mainly concerned with the convergence analysis of Galerkin-Petrov methods for the numerical solution of periodic pseudodifferential equations using wavelets and multiwavelets as trial functions and test functionals. Section 2 gives an overview on the symbol calculus of multidimensional singular integrals using Tricomi's composition formula. In Section 3 we formulate necessary and sufficient stability conditions in terms of the so-called numerical symbols and demonstrate applications to the Dirchlet problem for the Laplace equation.

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# Numerical methods for nonlinear parabolic equations with small parameter based on probability approach

*Authors*

- Milstein, Grigori N.
- Tretyakov, Michael V.

*2010 Mathematics Subject Classification*

- 35K55 60H10 60H30 65M99

*Keywords*

- Semilinear parabolic equations, reaction-diffusion systems, probabilistic representations for equations of mathematical physics, stochastic differential equations with small noise

*Abstract*

The probabilistic approach is used for constructing special layer methods to solve the Cauchy problem for semilinear parabolic equations with small parameter. In spite of the probabilistic nature these methods are nevertheless deterministic. The algorithms are tested by simulating the Burgers equation with small viscosity and the generalized KPP-equation with a small parameter.

*Appeared in*

- Mathematics of Computation, vol. 60 (2000), no. 229, pp. 237-267, under new title: Numerical algorithms for semilinear parabolic equations with small parameter based on weak approximation of stochastic differential equations.

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# Optical Imaging: 3D Approximation and Perturbation Approaches for Time--domain Data

*Authors*

- Huenlich, Rolf
- Model, Regine
- Orlt, Matthias
- Walzel, Monika

*2010 Mathematics Subject Classification*

- 35R30

*Keywords*

- Near-infrared imaging, diffusion model, time-resolved data, image reconstruction, perturbation approach, finite-element method

*Abstract*

The reconstruction method presented here is based on the diffusion approximation for the light propagation in turbid media and on a minimization strategy for the output-least-squares problem. A perturbation approach is introduced for the optical properties. Here, the number of free variables of the inverse problem can be strongly reduced by exploiting a priori information such as the search for single inhomogeneities within a relatively homogeneous object, a typical situation for breast cancer detection. Higher accuracy and a considerable reduction of the computational effort are achieved by solving a parabolic differential equation for a perturbation density, i.e. the difference between the photon density in an inhomogeneous object and the density in the homogeneous case being given by an analytic expression. The calculations are performed by a 2D FEM algorithm, however, as a time-dependent correction factor is applied, the 3D situation is well approximated. The method was successfully tested by the University of Pennsylvania standard data set. Data noise was generated and taken into account in a modified data set. The influence of different noise on the reconstruction results is discussed.

*Appeared in*

- Applied Optics, 37 (1998), pp. 7968-7976

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# Improved Numerical Methods for the Simulation of MicrowaveCircuits

*Authors*

- Hebermehl, Georg
- Heinrich, Wolfgang
- Schlundt, Rainer

ORCID: 0000-0002-4424-4301 - Zscheile, Horst

*2010 Mathematics Subject Classification*

- 35Q60 65F10 65F15 65N22

*Keywords*

- Microwave device simulation, scattering matrix, Maxwellianequations, boundary value problem, finite-volume method, eigenvalue problem, system of linear algebraic equations

*Abstract*

The scattering matrix describes monolithic microwave integrated circuits that are connected to transmission lines in terms of their wave modes. Using a finite-volume method the corresponding boundary value problem of Maxwell's equations can be solved by means of a two-step procedure. An eigenvalue problem for non-symmetric matrices yields the wave modes. The eigenfunctions determine the boundary values at the ports of the transmission lines for the calculation of the fields in the three dimensional structure. The electromagnetic fields and the matrix elements are achieved by the solution of large-scale systems of linear equations with indefinite symmetric matrices. Improved numerical solutions for the time and memory consuming problems are treated in this paper. Numerical results are discussed for real life problems. The numerical effort could be reduced considerably. This paper is a revised version of the preprint No. 378.

*Appeared in*

- Surv. Math.Ind., vol 9 (1999), no. 2, pp. 117-129

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# Some Analytic Solutions for Stochastic Reactor Models Based on the Joint Composition PDF

*Authors*

- Fey, Harald
- Kraft, Markus

ORCID: 0000-0002-4293-8924

*2010 Mathematics Subject Classification*

- 60K40 35L99

*Keywords*

- Probability density function transport equation, analytic solution, stochastic reactor models, partially stirred reactor, partially stirred plug flow reactor

*Abstract*

The stochastic reactor models Partially Stirred Reactor (PaSR) and Partially Stirred Plug Flow Reactor (PaSPFR) have been investigated. These models are based on a simplified joint composition PDF transport equation. Analytic solutions for five different Cauchy problems for the PDF transport equation as given by the stochastic reactor models are presented. In all cases, molecular mixing in the stochastic reactor models is described by the LMSE mixing model. The analytic solutions have been found by combining the method of characteristics with a set of ordinary differential equations for the statistical moments to account for the functional dependence of the coefficients in the corresponding PDF transport equation. For each case an example problem is discussed to illustrate the behavior of the analytic solution.

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# Initial and Boundary Value Problems of Hyperbolic Heat Conduction

*Authors*

- Dreyer, Wolfgang
- Kunik, Matthias

*2010 Mathematics Subject Classification*

- 80-99 35L15 35L20 35L65 35L67

*Keywords*

- heat transfer, initial and boundary value problems for a hyperbolic system, shock waves, kinetic theory

*Abstract*

This is a study on the initial and boundary value problem of a symmetric hyperbolic system which is related to the conduction of heat in solids at low temperatures. The nonlinear system consists of a conservation equation for the energy density $e$ and a balance equation for the heat flux $Q_i$, where $e$ and $Q_i$ are the four basic fields of the theory. The initial and boundary value problem that uses exclusively prescribed boundary data for the energy density $e$ is solved by a new kinetic approach. This method includes the formation of shock fronts and the broadening of heat pulses. These effects cannot be observed in the linearized theory, as it is described in citeD/Str. noindent The kinetic representations of the initial and boundary value problem reveal a peculiar phenomenon. To the solution there contribute integrals containing the initial fields $e_0 (x), Q_0 (x)$ as well as integrals that need knowledge on energy em and heat flux at a boundary. However, only one of these quantities can be controlled in an experiment. When this ambiguity is removed by a continuity condition, it will turn out that after some very short time energy and heat flux are related to each other according to the Rankine Hugoniot relation.

*Appeared in*

- Initial and boundary value problems of hyperbolic heat conduction, Contin. Mech. Thermodyn., 11 (1999), No. 4, pp. 227-245

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# A temperature time counter scheme for the Boltzmann equation

*Authors*

- Rjasanow, Sergej
- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 65C05 76P05 82C80

*Keywords*

- Boltzmann equation, stochastic particle method, time counter, local temperature, numerical experiments

*Abstract*

This paper gives a rigorous derivation of a new stochastic particle method for the Boltzmann equation and illustrates its numerical efficiency. Using estimates based on the local temperature of the simulation cells, any truncation error related to large velocities is avoided. Moreover, time steps between collisions are larger than in the standard direct simulation method. This fact and an efficient modelling procedure for the index distribution of the collision partners lead to a considerable reduction of computational effort in certain applications.

*Appeared in*

- SIAM J. Numer. Anal.., 37(2000), pp. 1800-1819

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# Existence, Uniqueness and Regularity for Solutions of theConical Diffraction Problem

*Authors*

- Elschner, Johannes
- Hinder, Rainer
- Penzel, F.
- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 78A45 35J50 35J20 35B65

*Keywords*

- Maxwell equations, system of Helmholtz equations, transmission problem, strongly ellipticvariational formulation, singularities of solutions

*Abstract*

This paper is devoted to the analysis of two Helmholtz equations in $R^2$ coupled via quasiperiodic transmission conditions on a set of piecewise smooth interfaces. The solution of this system is quasi--periodic in one direction and satisfies outgoing wave conditions with respect to the other direction. It is shown that Maxwell's equations for the diffraction of a time--harmonic oblique incident plane wave by periodic interfaces can be reduced to problems of this kind. The analysis is based on a strongly elliptic variational formulation of the differential problem in a bounded periodic cell involving nonlocal boundary operators. We obtain existence and uniqueness results for solutions corresponding to electromagnetic fields with locally finite energy. Special attention is paid to the regularity and leading asymptotics of solutions near the edges of the interface.

*Appeared in*

- Math. Models Meth. Appl. Sci. 10 (2000) pp. 317-341

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# Travelling Wave Equations for Semiconductor Lasers with Gain Dispersion

*Authors*

- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Sieber, Jan
- Wenzel, Hans
- Wolfrum, Matthias
- Wuensche, Hans-Jochen

*2010 Mathematics Subject Classification*

- 78A60 78A50 78-99

*Keywords*

- Semiconductor laser modelling, gain dispersion of semiconductors, DFB Lasers

*Abstract*

This paper modifies the coupled mode model for semiconductor lasers, taking into account the gain dispersion of the optical waveguide. Fitting the true gain curve by a Lorentzian, we obtain a correction for the dielectric function of the waveguide. A review of the derivation of the coupled mode model from the Maxwell Equations, including the corrected dielectric function, leads to an extended set of model equations. This extended model consists of the modified coupled mode equations and additional polarization equations and reflects spectral selectivity due to the geometry (waveguide dispersion) as well as the material properties (material dispersion). Although it is mathematically more complex, it does not increase the computational effort for the dynamical simulation essentially and, thus, it should replace the original model at least for numerical calculations.

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# A study of the coarsening in tin/lead solders

*Authors*

- Dreyer, Wolfgang
- Mueller, Wolfgang H.

*2010 Mathematics Subject Classification*

- 76R50 82B26 82B24 35K30 35K35

*Keywords*

- Diffusion, phase transision, interface problems, initial value problems for higher order parabolic equations, boundary value problems for higher order parabolic equations

*Abstract*

This paper presents a model, which is capable to simulate the coarsening process observed during thermo-mechanical treatment of binary tin-lead solders. Fourier transforms and spectral theory are used for the numerical treatment of the thermo-elastic as well as of the diffusion problem encountered during phase separation in these alloys. More specifically, the analysis is based exclusively on continuum theory, first, relies on the numerical computation of the local stresses and strains in a representative volume element (RVE). Second, this information is used in an extended diffusion equation to predict the local concentrations of the constituents of the solder. Besides the classical driving forces for phase separation, as introduced by Fick and Cahn-Hilliard, this equation contains an additional term which links the mechanical to the thermodynamical problem. It connects internal and external stresses, strains, temperature, as well as concentrations and allows for a comprehensive study of the coarsening and aging process.

*Appeared in*

- Internat. J. Solids Structures 37 (2000), no. 28, 3841-3871

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# A stochastic method for solving Smoluchowski's coagulation equation

*Authors*

- Kolodko, Anastasia
- Sabelfeld, Karl
- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 65C05 76F99

*Keywords*

- coagulation equation, Monte Carlo estimators, spatially inhomogeneous case, isotropic turbulent flow

*Abstract*

This paper studies a stochastic particle method for the numerical treatment of Smoluchowski equation governing the coagulation of particles in a host gas. Convergence in probability is established for the Monte Carlo estimators, when the number of particles tends to infinity. The deterministic limit is characterized as the solution of a discrete in time version of the Smoluchowski equation. Under some restrictions it is shown that this stochastic finite difference scheme is convergent to the solution of the original Smoluchowski equation. Extensions on a nonhomogeneous Smoluchowski equation are given, and in particular, a coagulation process in an isotropic fully developed turbulent flow is studied.

*Appeared in*

- Math. Comput. Simulation 49(1-2) (1999), pp. 57-79

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# Approximative solution of the coagulation-fragmentation equation by stochastic particle systems

*Authors*

- Eibeck, Andreas
- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 60K35 65C05 82C22

*Keywords*

- coagulation-fragmentation equation, interacting particle systems, convergence, existence of solutions, simulation algorithm

*Abstract*

This paper studies stochastic particle systems related to the coagulation-fragmentation equation. For a certain class of unbounded coagulation kernels and fragmentation rates, relative compactness of the stochastic systems is established and weak accumulation points are characterized as solutions. These results imply a new existence theorem. Finally a simulation algorithm based on the particle systems is proposed.

*Appeared in*

- Stochastic Anal. Appl., 18(2000), pp. 921-948

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# Parallel Modular Dynamic Process Simulation

*Authors*

- Borchardt, Jürgen
- Ehrhardt, Klaus
- Grund, Friedrich
- Horn, Dietmar

*2010 Mathematics Subject Classification*

- 65Y05 80A30 65L05 65H10 65F50

*Keywords*

- Systems of differential-algebraic equations, Parallelization, Block partitioned systems, Newton-type methods, Sparse matrix techniques, Chemical process simulation, Dynamic Simulation of distillation plants

*Abstract*

To meet the needs of plant wide dynamic process simulation of today's complex, highly interconnected chemical production plants, parallelizable numerical methods using divide and conquer strategies are considered. The large systems of differential algebraic equations (DAE's) arising from an unit oriented modular modeling of chemical and physical processes in a chemical plant are partitioned into blocks. Using backward differentiation formulas (BDF), a partitioned system of nonlinear equations has to be solved at each discretization point of time. By formally extending these systems, block--structured Newton--type methods are applied for their solution. These methods enable a coarse grain parallelization and imply an adaptive relaxation decoupling between blocks. The resulting linear subsystems with sparse and unsymmetric coefficient matrices are solved with a Gaussian elimination method using pseudo code techniques for an efficient multiple refactorization and solution. Results from dynamic simulation runs for industrial distillation plants on parallel computers are given.

*Appeared in*

- Scientific Computing in Chemical Engineering II, vol. 2 (Keil, F., Mackens, W., Voss, H., and Werther, J., eds.) Springer-Verlag Berlin Heidelberg New York, 1999, pp. 152-159

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# Existence and uniqueness of solutions of certain systems of algebraic equations with off diagonal nonlinearity

*Authors*

- Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434

*2010 Mathematics Subject Classification*

- 65M12 65H10 15A48 65E99

*Keywords*

- systems of algebraic equations, positive matrices, viscous conservation laws, finite volume methods, nonlinear parabolic PDEs

*Abstract*

Using inverse positivity properties and Brouwer's fixed point theorem, we derive existence and uniqueness results for certain nonlinear systems of equations with off diagonal nonlinearity. The type of systems considered arises from stable finite volume discretizations of viscous nonlinear conservation laws and has a wide range of applications.

*Appeared in*

- Applied Numerical Mathematics, 37 (3):357-368, 2001

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# Domain separation by means of sign changing eigenfunctions of $p$-Laplacians

*Authors*

- Gaertner, Klaus
- Gajewski, Herbert

*2010 Mathematics Subject Classification*

- 35J20 58E12

*Keywords*

- $p$-Laplacian, eigenfunctions, separators

*Abstract*

We are interested in algorithms for constructing surfaces $Gamma$ of possibly small measure that separate a given domain $Omega$ into two regions of equal measure. Using the integral formula for the total gradient variation, we show that such separators can be constructed approximatively by means of sign changing eigenfunctions of the $p$-Laplacian, $p rightarrow 1$, under homogeneous Neumann boundary conditions. These eigenfunctions are proven to be limits of continuous and discrete steepest descent methods applied to suitable norm quotients.

*Appeared in*

- Appl. Anal., 79 (2001) pp. 483--501.

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# Wavelet method and asymptotically minimax estimation of regression

*Authors*

- Golubev, Georgii

*2010 Mathematics Subject Classification*

- 62G05 62G20

*Keywords*

- Minimax risk, entire analytic functions, wavelet method, hard thresholding

*Abstract*

We attempt to recover a regression function from noisy data. It is assumed that the underlying function is a piecewise entire analytic function. Types and the number of singularities are assumed to be unknown. We show how to chose smoothing parameters and a wavelet basis to achieve the asymptotically minimax risk up to the constant.

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# A diffusion representation of the nonparametric iid experiment on a interval

*Authors*

- Genon-Catalot, Valentine
- Laredo, Catherine
- Nussbaum, Michael

*2010 Mathematics Subject Classification*

- 62B 62M05 62G07

*Keywords*

- Nonparametric experiments, asymptotic equivalence, diffusion process, discretization, level crossing inverse Gaussian regression, signal in white noise, iid model, density estimation

*Abstract*

We consider a diffusion model of small variance type with positive drift function varying in a nonparametric set. We investigate discrete versions of this continuous model with respect to statistical equivalence, in the sense of the asymptotic theory of experiments. It is shown that the collection of level crossing times for a uniform grid of levels is asymptotically equivalent to the continuous model in the sense of Le Cam's deficiency distance, when the discretization step decreases with the noise intensity epsilon. It follows that in the continuous diffusion model, the statistic of level crossing times is asymptotically sufficient. Since the level crossing times obey a nonparametric regression model with independent data, a further asymptotic equivalence can be established, leading to a simple Gaussian signal-in-white noise problem. When the drift density f is also a probability density, this in turn is asymptotically equivalent to i.i.d. data with density f on the unit interval.

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# On an operator equation with noise in the operator and the right-hand side with application to an inverse vibration problem

*Authors*

- Bruckner, Gottfried
- Yamamoto, Masahiro

*2010 Mathematics Subject Classification*

- 65J20 35R30 35L05

*Keywords*

- regularization, linear operator equation, uncertain operator, noisy right-hand side, wave equation, point source reconstruction

*Abstract*

We consider a linear operator equation with noise in the operator and the right-hand side. As a concept, the ill-posedness of the problem is composed of the ill-posedness with respect to the operator and the ill-posedness with respect to the right-hand side, and in both the cases the ill-posedness can be characterized by an embedding operator. Starting at a numerical procedure for exact data, in the case of noisy data a numerical procedure and error estimates are given. As an example, a Volterra integral equation of the first kind is investigated and finally applied to a point source reconstruction problem for the wave equation.

*Appeared in*

- ZAMM 80, (2000), pp. 377-388

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# Singularly perturbed boundary value problems for systems of Tichonov's type in case of exchange of stabilities

*Authors*

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

*2010 Mathematics Subject Classification*

- 34B15 34E20

*Keywords*

- exchange of stabilities, singularly perturbed boundary

*Abstract*

We consider a system of ordinary differential equations consisting of a singularly perturbed scalar differential equation of second order and a scalar differential equation of first or second order and study a Neuman-Cauchy or a Neuman-Dirichlet problem. We assume that the degenerate equation has two intersecting solutions such that the standard theory for systems of Tichonov's type cannot be applied. We introduce the notation of a composed stable solution. By means of the technique of ordered lower and upper solutions we prove the existence of a solution of our problems near the composed stable solution for sufficiently small $varepsilon$ and determine its asymptotic behavior in $varepsilon$.

*Appeared in*

- J. Differential Equations, 159 (1999), No. 2, pp. 427-446

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# On Large Deviation Probabilities in Ergodic Theorem for Singularly Perturbed Stochastic Systems

*Authors*

- Pergamenshchikov, S.M

*2010 Mathematics Subject Classification*

- 60F10

*Keywords*

- fast and slow components, large deviations, stochastic differential, singular perturbations, ergodic theorem

*Abstract*

We consider a two scale system of stochastic differential equations. We study asymptotic properties of integral functionals of slow component of this system and establish some large deviation type estimations for these functionals.

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# Determination of point wave sources by pointwise observations: stability and reconstruction

*Authors*

- Bruckner, Gottfried
- Yamamoto, Masahiro

*2010 Mathematics Subject Classification*

- 35L05 35R30 65R30

*Keywords*

- Point source reconstruction, wave equation, uniqueness, stability, regularization

*Abstract*

We consider a wave equation with point source terms: $$ leftaligned fracpartial^2 upartial t^2(x,t) & = fracpartial^2 upartial x^2 (x,t) + lambda(t)sum^N_k=1alpha_kdelta(x-x_k), qquad 0

*Appeared in*

- Inverse Problems, 16(2000), pp. 723-748

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# Qualitative Stability of Convex Programs with Probabilistic Constraints

*Authors*

- Henrion, René

*2010 Mathematics Subject Classification*

- 90C15 90C31

*Keywords*

- stochastic programming, probabilistic constraints, qualitative stability, r-concave measures

*Abstract*

We consider convex stochastic optimization problems with probabilistic constraints which are defined by so-called $r$-concave probability measures. Since the true measure is unknown in general, the problem is usually solved on the basis of estimated approximations, hence the issue of perturbation analysis arises in a natural way. For the solution set mapping and for the optimal value function, stability results are derived. In order to include the important class of empirical estimators, the perturbations are allowed to be arbitrary in the space of probability measures (in contrast to the convexity property of the original measure). All assumptions relate to the original problem. Examples show the necessity of the formulated conditions and illustrate the sharpness of results in the respective settings.

*Appeared in*

- In V.H. Nguyen, J. -J. Strodiot and P. Tossings (eds.): Optimization (Lect. Notes in Economics and Mathematical Systems, Vol. 481), (2000), pp. 164-180

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# Direct Linear Solvers for Vector and Parallel Computers

*Authors*

- Grund, Friedrich

*2010 Mathematics Subject Classification*

- 65F05 65F50 65Y05 80A30 92E20

*Keywords*

- Gaussian elimination, Sparse-matrix techniques, Vectorization, Parallelization, Chemical process simulation

*Abstract*

We consider direct methods for the numerical solution of linear systems with unsymmetric sparse matrices. Different strategies for the determination of the pivots are studied. For solving several linear systems with the same pattern structure we generate a pseudo code, that can be interpreted repeatedly to compute the solutions of these systems. The pseudo code can be advantageously adapted to vector and parallel computers. For that we have to find out the instructions of the pseudo code which are independent of each other. Based on this information, one can determine vector instructions for the pseudo code operations (vectorization) or spread the operations among different processors (parallelization). The methods are successfully used on vector and parallel computers for the circuit simulation of VLSI circuits as well as for the dynamic process simulation of complex chemical production plants.

*Appeared in*

- Jose M. L. M. Palma, Jack Dongarra and Vicente Hernandez, editors: Vector and Parallel Processing - VECPAR'98, Third International Conference, Porto, Portugal, June 1998, volume 1573 of Lecture Notes in Computer Science, pages 114-127. Springer-Verlag Berlin Heidelberg New York, 1999

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# Higher order asymptotic optimality in testing problemswith nuisance parameters

*Authors*

- Bening, Vladimir E.
- Chibisov, Dimitrii M.

*2010 Mathematics Subject Classification*

- 62F05

*Keywords*

- Hypothesis testing, nuisance parameters, asymptotic methods, deficiency

*Abstract*

We consider testing hypotheses about the location parameter of a symmetric distribution when a finite-dimensional nuisance parameter is present. For local alternatives, we study the power loss of asymptotically efficient tests in this problem, which is the difference between the power of the most powerful test for a given value of the nuisance parameter (as if it were known) and the power of the test at hand. The power loss is typically of order $n^-1$ and is closely related to the deficiency of the test. In particular, we obtain the lower bound for the power loss in a locally asymptotically minimax sense similar to that used in the estimation theory and indicate a test on which this bound is attained. This bound corresponds to the envelope power function obtained by Pfanzagl and Wefelmeyer (1978) for test statistics of a specific structure.

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# Stability for a continuous SOS-interface modelin a randomly perturbed periodic potential

*Authors*

- Kuelske, Christof

*2010 Mathematics Subject Classification*

- 82B44 82B28 82B41 60K35

*Keywords*

- Disordered Systems, Continuous Spins, Interfaces, SOS-Model, Contour Models, Cluster Expansions, Renormalization Group

*Abstract*

We consider the Gibbs-measures of continuous-valued height configurations on the $d$-dimensional integer lattice in the presence a weakly disordered potential. The potential is composed of Gaussians having random location and random depth, it becomes periodic under shift of the interface perpendicular to the base-plane for zero disorder. We prove that there exist localized interfaces with probability one in dimensions $dgeq 3+1$, in a `low-temperature' regime. The proof extends the method of continuous-to-discrete single-site coarse graining that was previously applied by the author for a double-well potential to the case of a non-compact image space. This allows to utilize parts of the renormalization group analysis developed for the treatment of a contour representation of a related integer-valued SOS-model in [BoK1]. We show that, for a.e. fixed realization of the disorder, the infinite volume Gibbs measures then have a representation as superpositions of massive Gaussian fields with centerings that are distributed according to the infinite volume Gibbs measures of the disordered integer-valued SOS-model with exponentially decaying interactions.

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# Asymptotic equivalence of spectral density and regression estimation

*Authors*

- Golubev, Georgii
- Nussbaum, Michael

*2010 Mathematics Subject Classification*

- 62G07 62G20

*Keywords*

- Stationary Gaussian process, spectral density, Le Cam's distance, asymptotic equivalence, local.limit theorem, signal in Gaussian white noise

*Abstract*

We consider the statistical experiment given by a sample $y(1),ldots, y(n)$ of a stationary Gaussian process with an unknown smooth spectral density. Asymptotic equivalence with a nonparametric regression in discrete Gaussian white noise is established. The key is a local limit theorem for an increasing number of empirical covariance coefficients.

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# A Metric Theory of Gravity with Condensed Matter Interpretation

*Authors*

- Schmelzer, Ilja

*2010 Mathematics Subject Classification*

- 83D05 83C45

*Keywords*

- quantum gravity

*Abstract*

We define a metric theory of gravity with preferred Newtonian frame $(X^i(x),T(x))$ by [L = L_GR + Xi g^munudelta_ijX^i_,muX^j_,nu - Upsilon g^munuT_,muT_,nu ] It allows a condensed matter interpretation which generalizes LET to gravity. The $Xi$-term influences the age of the universe. $Upsilon>0$ allows to avoid big bang singularity and black hole horizon formation. This solves the horizon problem without inflation. An atomic hypothesis solves the ultraviolet problem by explicit regularization. We give a prediction for cutoff length.

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# Ornstein-Zernike behaviour and analyticity of shapes for self-avoiding walks on $Bbb Z^d$

*Authors*

- Ioffe, Dmitry

*2010 Mathematics Subject Classification*

- 82B41 60K35 82B41

*Keywords*

- Ornstein-Zernike behaviour, self-avoiding random, local limit theorems, renewal relations

*Abstract*

We derive precise Ornstein-Zernike asymptotics for the decay of the two-point function in any direction of the simple self-avoiding walk on the integer lattice $Bbb Z^d$ in any dimension $dge 2$ and for any super-critical value of the parameter $beta>beta_c (d)$. The related geometry of the equi-decay level sets is studied as well.

*Appeared in*

- Mark. Proc. Rel. Fields, 4 (1998), pp. 323-350

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# Libor model with expiry-wise stochastic volatility and displacement

*Authors*

- Ladkau, Marcel
- Schoenmakers, John G. M.

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

*2010 Mathematics Subject Classification*

- 91G30 91G60 60H10

*Keywords*

- Displaced Libor models, stochastic volatility, calibration to cap-strike-maturity matrix, swaption pricing

*Abstract*

We develop a multi-factor stochastic volatility Libor model with displacement, where each individual forward Libor is driven by its own square-root stochastic volatility process. The main advantage of this approach is that, maturity-wise, each square-root process can be calibrated to the corresponding cap(let)vola-strike panel at the market. However, since even after freezing the Libors in the drift of this model, the Libor dynamics are not affine, new affine approximations have to be developed in order to obtain Fourier based (approximate) pricing procedures for caps and swaptions. As a result, we end up with a Libor modeling package that allows for efficient calibration to a complete system of cap/swaption market quotes that performs well even in crises times, where structural breaks in vola-strike-maturity panels are typically observed.

*Appeared in*

- Int. J. Portfolio Analysis and Management, 1 (2013) pp. 224--249.

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# On the Optimal Kernels in Nonparametric Curve Estimation

*Authors*

- Leonov, Sergei

*2010 Mathematics Subject Classification*

- 62G07 62G20

*Keywords*

- Minimax risk, exact constants, Hoelder function class, optimal recovery

*Abstract*

We discuss extremal problems which arise in various nonparametric statistical settings with Hölder function classes. We establish a new property of the solution of an optimal recovery problem that leads to the exact constants for asymptotic minimax risks.

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# On parameter estimation for ergodic Markov chains withunbounded loss functions

*Authors*

- Veretennikov, Alexander Yu.

*2010 Mathematics Subject Classification*

- 62-02 62F12

*Keywords*

- Asymptotic normality, efficiency, polynomial lossfunction, maximum likelihood estimation, Hajek - Le Cam efficiency

*Abstract*

We establish the Hajek - Le Cam asymptotic efficiency of maximum likelihood estimators for "polynomially ergodic" Markov regular experiments in the class of loss functions with a polynomial growth.

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# Model reduction by extended quasi-steady-state approximation

*Authors*

- Schneider, Klaus R.
- Wilhelm, Thomas

*2010 Mathematics Subject Classification*

- 34E15 34E05 92E20

*Keywords*

- Quasi-steady-state approximation, singularly perturbed systems, trimolecular autocatalator

*Abstract*

We extend the quasi-steady state approximation (QSSA) as well with respect to the class of differential systems as with respect to the order of approximation. As an application we prove that the trimolecular autocatalator can be approximated by a fast bimolecular reaction system. Finally we describe a class of singularly perturbed systems for which the first order QSSA can easily be obtained.

*Appeared in*

- J. Math. Biol., 408(2000), pp. 443-450

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# IMPORTANCE SAMPLING FOR LARGE ANDMODERATE LARGE DEVIATION SIMULATION OF TESTS AND ESTIMATORS

*Authors*

- Ermakov, Mikhail S.

*2010 Mathematics Subject Classification*

- 62E25 60F10 65C05

*Keywords*

- Importance sampling, Large deviations, Empiricalmeasure, Hadamard-type derivative, Statistical simulation

*Abstract*

We find the effective importance sampling procedures for the simulation of large and moderate large deviations of tests and estimators. The computational burden of these effective procedures has no exponential rate as in the direct simulation. The results are applied to the simulation of large and moderate large deviations of L,M,R-statistics and omega-square tests.

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# Evolution variational inequalities and multidimensionalhysteresis operators.

*Authors*

- Krejci, Pavel

*2010 Mathematics Subject Classification*

- 58E35 47H30 73E05

*Keywords*

- Variational inequality, hysteresis operators

*Abstract*

We give an overview of the theory of multidimensional hysteresis operators defined as solution operators of rate-independent variational inequalities in a Hilbert space $X$ with given convex constraints. Emphasis is put on analytical properties of these operators in the space of functions of bounded variation with values in $X$, in Sobolev spaces and in the space of continuous functions. We discuss in detail the influence of the geometry of the convex constraint on the input-output behavior. It is shown how multidimensional hysteresis operators arise naturally in constitutive laws of rate-independent plasticity and concrete examples of application of the above theory in material sciences are given.

*Appeared in*

- Nonlinear Differential Equations (Drabete, P, Krejci, P., Takac, P., eds.) Boca Raton: Chapman & Hall, 1999, CRC Res. Notes Math. 404, pp. 47--110

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# Stability and existence of solutions of time-implicit finite volumeschemes for viscous nonlinear conservation laws

*Authors*

- Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434 - Langmach, Hartmut

*2010 Mathematics Subject Classification*

- 65E99 65M12

*Keywords*

- viscous conservation laws, finite volume methods, nonlinear parabolic PDEs, groundwater flow

*Abstract*

We introduce a time-implicite Voronoi box based finite volume discretization for the initial-boundary value problem of a scalar nonlinear viscous conservation law in a one, two- or threedimensional domain. Using notations from the theory of explicit finite volume methods for hyperbolic problems and results from the Perron-Frobenius theory of nonnegative matrices, we establish various existence, stability and uniqueness results for the discrete problem. The class of schemes introduced covers as well hyperbolic problems as well as nonlinear diffusion problems. To clarify our results, we provide numerical examples, and we show the practical relevance of our considerations with a groundwater flow example.

*Appeared in*

- Applied Numerical Mathematics, 37 91-2): 201-230, 2001

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# The continuous spin random field model:Ferromagnetic ordering in $dgeq 3$

*Authors*

- Kuelske, Christof

*2010 Mathematics Subject Classification*

- 82B44 82B26 82B28

*Keywords*

- Disordered Systems, Contour Models, Cluster Expansions, Renormalization Group, Random Field Model

*Abstract*

We investigate the Gibbs-measures of ferromagnetically coupled continuous spins in double-well potentials subjected to a random field (our specific example being the $phi^4$ theory), showing ferromagnetic ordering in $dgeq 3$ dimensions for weak disorder and large energy barriers. We map the random continuous spin distributions to distributions for an Ising-spin system by means of a single-site coarse-graining method described by local transition kernels. We derive a contour-representation for them with notably positive contour activities and prove their Gibbsianness. This representation is shown to allow for application of the discrete-spin renormalization group developed by Bricmont/Kupiainen implying the result in $dgeq 3$.

*Appeared in*

- Rev. Math. Phys. 11, No. 10, (1999), pp. 1269-1314

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# Fluctuations in the Hopfield model at the critical temperature

*Authors*

- Gentz, Barbara
- Loewe, Matthias

*2010 Mathematics Subject Classification*

- 60K35 60F05 82C32

*Keywords*

- Hopfield model, spin glasses, neural networks, random disorder, limit theorems, non-Gaussian fluctuations, critical temperature

*Abstract*

We investigate the fluctuations of the order parameter in the Hopfield model of spin glasses and neural networks at the critical temperature. The number of patterns M(N) is allowed to grow with the number N of spins but the growth rate is subject to the constraint that M(N)^15/N tends to zero. As the system size N increases, on a set of large probability the distribution of the appropriately scaled order parameter under the Gibbs measure comes arbitrarily close (in a metric which generates the weak topology) to a non-Gaussian measure which depends on the realization of the random patterns. This random measure is given explicitly by its (random) density.

*Appeared in*

- Markov Process. Related Fields 5 (1999), no. 4. 423-449

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# Adaptation in minimax nonparametric hypothesis testing for ellipsoids and Besov bodies

*Authors*

- Ingster, Yuri, I.

*2010 Mathematics Subject Classification*

- 62G10 62G20

*Keywords*

- nonparametric hypotheses testing, minimax hypotheses testing, adaptive hypotheses testing, asymptotics of error probabilities

*Abstract*

We observe an infinitely dimensional Gaussian random vector $x=xi+v$ where $xi$ is a sequence of standard Gaussian variables and $vin l_2$ is an unknown mean. Let $V_varepsilon(tau,rho_varepsilon)subset l_2$ be sets which correspond to $l_q$-ellipsoids %of the radiuses $R/varepsilon$ of power semi-axes $a_i=i^-sR/varepsilon$ with $l_p$-ellipsoid %of the radiuses $rho_varepsilon/varepsilon$ and of semi-axes $b_i=i^-rrho_varepsilon/varepsilon$ removed or to similar Besov bodies $B_q,t,s(R/varepsilon)$ with Besov bodies $B_p,h,r(rho_varepsilon/varepsilon)$ removed. Here $tau =(kappa,R)$ or $tau =(kappa,h,t,R), kappa=(p,q,r,s)$ are the parameters which define the sets $V_varepsilon$ for given radiuses $rho_varepsilonto 0$, $0

0, varepsilonto 0$ is asymptotical parameter. For the case $tau$ is known hypothesis testing problem $H_0: v=0$ versus alternatives $H_varepsilon,tau:vin V_varepsilon(tau, rho_varepsilon)$ have been considered by Ingster and Suslina [11] in minimax setting. It was shown that there is a partition of the set of $kappa$ on to regions with different types of asymptotics: classical, trivial, degenerate and Gaussian (of two main and some "boundary" types). Also there is essential dependence of the structure of asymptotically minimax tests on the parameter $kappa$ for the case of Gaussian asymptotics . In this paper we consider alternative $H_varepsilon,Gamma:vin V_varepsilon(Gamma)$ for sets $V_varepsilon(Gamma)= bigcup_tauin GammaV_varepsilon(tau,rho_varepsilon(tau))$. This corresponds to adaptive setting: $tau$ is unknown, $tauin Gamma$ for a compact $Gamma=Ktimes Delta, Delta=[c, C]subset R_+^1, Ksubset Xi_G_1cup Xi_G_2$ where $ Xi_G_2$ and $ Xi_G_2$ are regions of main tapes of Gaussian asymptotics . First the problems of such types were studied by Spokoiny [16, 17]. For ellipsoidal case we study sharp asymptotics of minimax second kind errors $beta_varepsilon(alpha, Gamma)=beta(alpha, V_varepsilon(Gamma))$ and construct asymptotically minimax tests. % $psi_alpha,varepsilon,Gamma$. These asymptotics are analogous to degenerate type. For Besov bodies case we obtain exact rates and construct minimax consistent tests. Analogous exact rates are obtained in a signal detection problem for continuous variant of white Gaussian noise model: alternatives correspond to Besov or Sobolev balls with Sobolev or Besov balls removed. The study is based on results [11] and on an extension of methods of this paper for degenerate case.

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# Note on the notion of incompressibility in theories of porous and granular materials

*Authors*

- Wilmanski, Krzysztof

*2010 Mathematics Subject Classification*

- 73B30

*Keywords*

- incompressibility, porous and granular materials, second law of thermodynamics

*Abstract*

We present a simple two-component model of a porous material based on the constraint assumption that the so-called true components are incompressible. In my previous work on this subject [1] I pointed out that many such models are not thermodynamically admissible. Namely the second law of thermodynamics led to the conclusion that an additional field of reaction force on the constraint cannot be introduced, and, consequently, the set of field equations was overdetermined. However I speculated as well that an extension of the set of variables may lead to thermodynamic admissibility. Indeed an example presented in this paper supports this speculation. According to results of this work it seems to be neccessary to introduce higher gradients to multicomponent models with constraints in order to satisfy the second law of thermodynamics.

*Appeared in*

- Z. Angew. Mathem. Mech., 81(2001), pp. 37-42

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# A Transient Model for the Sublimation Growth of Silicon Carbide Single Crystals

*Authors*

- Bubner, Nikolaus
- Klein, Olaf

ORCID: 0000-0002-4142-3603 - Philip, Peter
- Sprekels, Jürgen
- Wilmanski, Krzysztof

*2010 Mathematics Subject Classification*

- 80A20 65M99 80A15 65C20 35L65

*Keywords*

- Modelling, sublimation growth, heat and mass transfer, numerical sublimation, conservation laws, partial differential equations

*Abstract*

We present a transient model for the em Modified Lely Method for the sublimation growth of SiC single crystals which consists of all conservation laws including reaction--diffusion equations. The model is based on a mixture theory for the gas phase. First numerical results illustrate the influence of the geometrical set--up inside the reactor on the evolution of the temperature distribution.

*Appeared in*

- Journal of Crystal Growth Vol 205 (1999), pp 294-304

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# An axisymmetric steady-state flow through a poroelastic medium under large deformations

*Authors*

- Albers, Bettina

ORCID: 0000-0003-4460-9152 - Wilmanski, Krzysztof

*2010 Mathematics Subject Classification*

- 76S05 76R50 73G99

*Keywords*

- large deformations of porous materials, diffusion and changes of porosity, filtration problems, perturbation method

*Abstract*

We present an example of the solution of a boundary value problem for a two-component porous material with large deformations of the skeleton. This example demonstrates the application of a consistent Lagrangian description of porous materials which has been proposed by K. Wilmanski. Simultaneously we demonstrate the important role of the balance equation of porosity which is an essential part of the thermodynamical model of porous materials proposed by K. Wilmanski. We show as well that a modified set of boundary conditions for permeable boundaries yields a solution of field equations which agrees qualitatively with expectations for the problem of axisymmetric stationary filtration. On the basis of a numerical evaluation of solution we indicate the existence of an instability of the model for very large porosities which could not be explained in this work.

*Appeared in*

- Arch. Appl. Mech., 69, pp 121-132 (1999)

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# Direct estimation of the index coefficients in a single-index model

*Authors*

- Hristache, Marian
- Juditsky, Anatoli
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G05 62H40 62H40

*Keywords*

- single-index model, index coefficients, link function, direct estimation, iterations

*Abstract*

We propose a new method of estimating the index coefficients in a single index model which is based on iterative improvement of the average derivative estimator. The resulting estimate is $sqrtn$--consistent under mild assumptions on the model.

*Appeared in*

- Ann. Statist., 29, no. 6 (2001), pp. 1537 - 1566

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# Adaptive weights smoothing withapplications to image segmentation

*Authors*

- Polzehl, Jörg

ORCID: 0000-0001-7471-2658 - Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62G05 62G25 62P10

*Keywords*

- locally constant approximation, adaptive weights, edge restoration, image segmentation, magnetic resonance imaging

*Abstract*

We propose a new method of nonparametric estimation which is based on locally constant smoothing with an adaptive choice of weights for every pair of data-points. Some theoretical properties of the procedure are investigated. Then we demonstrate the performance of the method on some simulated univariate and bivariate examples and compare it with other nonparametric methods. Finally we discuss applications of this procedure to Magnetic Resonance Imaging.

*Appeared in*

- J. of Royal Stat. Soc., Sers B, 335-354. under the title: Adaptive weights smooting with applications to image restoration.

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# Sharp bounds on perfect retrieval in the Hopfield model

*Authors*

- Bovier, Anton

*2010 Mathematics Subject Classification*

- 82B44 60K35 82C32

*Keywords*

- Hopfield model, storage capacity, gradient dynamic, sequential dynamic

*Abstract*

We prove a sharp upper bound on the number of patterns that can be stored in the Hopfield model if the stored patterns are required to be fixed points of the gradient dynamic. We also show corresponding bounds on the one-step convergence of the sequential gradient dynamic. The bounds coincide with the known lower bounds and confirm the heuristic expectations. The proof is based on a crucial idea of Loukianova [L] to use the negative association properties of some random variables arising in the analysis.

*Appeared in*

- J. Appl. Probab., 36 (1999), pp. 941-950.

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# A note on the decay of correlations under delta-pinning

*Authors*

- Ioffe, Dmitry
- Velenik, Yvan

*2010 Mathematics Subject Classification*

- 60K35 60J15 82A41

*Keywords*

- Effective interface model, random walks, decay of correlations

*Abstract*

We prove that for a class of massless $nablaphi$ interface models on $Bbb Z^2$ an introduction of an arbitrary small pinning self-potential leads to exponential decay of correlation, or, in other words, to creation of mass.

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# Stochastic symmetry-breaking in a Gaussian Hopfield model

*Authors*

- Bovier, Anton
- Enter, Aernout C. D. van
- Niederhauser, Beat

*2010 Mathematics Subject Classification*

- 82B44 82C32

*Keywords*

- Hopfield model, Gaussian disorder, metastates, chaotic size-dependence, extrema of gaussian processes

*Abstract*

We study a "two-pattern" Hopfield model with Gaussian disorder. We find that there are infinitely many pure states at low temperatures in this model, and we find that the metastate is supported on an infinity of symmetric pairs of pure states. The origin of this phenomen is the random breaking of a rotation symmetry of the distribution of the disorder variables.

*Appeared in*

- J. Stat. Phys. 95 (1999), pp. 181-213

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# Metastability in stochastic dynamics of disordered mean-field models

*Authors*

- Bovier, Anton
- Eckhoff, Michael
- Gayrard, Veronique
- Klein, Markus

*2010 Mathematics Subject Classification*

- 82C44 60K32

*Keywords*

- Metastability, stochastic dynamics, Markov chains, Wentzell-Freidlin theory, disordered systems, mean field models, random field Curie-Weiss model

*Abstract*

We study a class of Markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures. Our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding Gibbs measures. We derive the analog of the Wentzell-Freidlin theory in this case, showing that any transition can be decomposed, with probability exponentially close to one, into a deterministic sequence of ``admissible transitions''. For these admissible transitions we give upper and lower bounds on the expected transition times that differ only by a factor $sqrt N$, where $N$ denotes the volume of the system. The distribution rescaled transition times are shown to converge to the exponential distribution. We exemplify our results in the context of the random field Curie-Weiss model.

*Appeared in*

- Propab. Theory Related Fields, 119 (2001), pp. 99-161

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# On deterministic and stochastic sliding modes via small diffusion approximation

*Authors*

- Milstein, Grigiri N.
- Veretennikov, Alexander Yu.

*2010 Mathematics Subject Classification*

- 34A60 60H10

*Keywords*

- Differential equations with discontinuous right-hand sides, sliding modes, stochastic differential equations, stochastic averaging principle, stochastic differential equations with small noise

*Abstract*

We study solutions of a system of ordinary differential equations with discontinuity of its vector field on a smooth surface via small additive diffusion perturbations. When a diffusion term tends to zero, one obtains limiting sliding modes on the surface with explicit representation for its motion law. Stochastic sliding modes are also established.

*Appeared in*

- Markov Processes and Related Fields, 6 (2000), no. 3, pp.371-395

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# Some analytical properties of the multidimensional continuous Mroz model of plasticity

*Authors*

- Brokate, Martin
- Krejci, Pavel
- Rachinskii, Dmitrii

*2010 Mathematics Subject Classification*

- 47H30 73E05 47H15

*Keywords*

- Kinematic hardening, Mróz model, constitutive operator, memory, equations with hysteresis

*Abstract*

We study the geometrical structure of memory induced by the continuous multidimensional Mroz model of plasticity. The results are used for proving the thermodynamic consistency of the model and composition and inversion formulas for input - memory state - output operators. We also show an example of nonuniqueness of solutions to a simple initial value problem involving the Mroz operator.

*Appeared in*

- Control Cybernet., 27 (1998), 199-215

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