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116
Sickel: Optimal approximation of elliptic problems by linear and nonlinear mappings III
 Triebel, Function Spaces, Entropy Numbers, Differential Operators
, 1996
"... We study the optimal approximation of the solution of an operator equation A(u) = f by four types of mappings: a) linear mappings of rank n; b) nterm approximation with respect to a Riesz basis; c) approximation based on linear information about the right hand side f; d) continuous mappings. We co ..."
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Cited by 135 (28 self)
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We study the optimal approximation of the solution of an operator equation A(u) = f by four types of mappings: a) linear mappings of rank n; b) nterm approximation with respect to a Riesz basis; c) approximation based on linear information about the right hand side f; d) continuous mappings. We consider worst case errors, where f is an element of the unit ball of a Sobolev or Besov space Br q(Lp(Ω)) and Ω ⊂ Rd is a bounded Lipschitz domain; the error is always measured in the Hsnorm. The respective widths are the linear widths (or approximation numbers), the nonlinear widths, the Gelfand widths, and the manifold widths. As a technical tool, we also study the Bernstein numbers. Our main results are the following. If p ≥ 2 then the order of convergence is the same for all four classes of approximations. In particular, the best linear approximations are of the same order as the best nonlinear ones. The best linear approximation can be quite difficult to realize as a numerical algorithm since the optimal Galerkin space usually depends on the operator and of the shape of the domain Ω. For p < 2 there is a difference, nonlinear approximations are better than linear ones. However, in this case, it turns out that linear information about the right hand side f is again optimal. Our main theoretical tool is the best nterm approximation with respect to an optimal Riesz basis and related nonlinear widths. These general results are used to study the Poisson equation in a polygonal domain. It turns out that best nterm wavelet approximation is (almost) optimal. The main results of
Error bounds for computing the expectation by Markov chain Monte Carlo
, 2009
"... We study the error of reversible Markov chain Monte Carlo methods for approximating the expectation of a function. Explicit error bounds with respect to the l2, l4 and l∞norm of the function are proven. By the estimation the well known asymptotical limit of the error is attained, i.e. there is n ..."
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Cited by 116 (2 self)
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We study the error of reversible Markov chain Monte Carlo methods for approximating the expectation of a function. Explicit error bounds with respect to the l2, l4 and l∞norm of the function are proven. By the estimation the well known asymptotical limit of the error is attained, i.e. there is no gap between the estimate and the asymptotical behavior. We discuss the dependence of the error on a burnin of the Markov chain. Furthermore we suggest and justify a specific burnin for optimizing the algorithm.
Optimally Sparse Image Representation by the Easy Path Wavelet Transform
"... The Easy Path Wavelet Transform (EPWT) [19] has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of f ..."
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Cited by 115 (8 self)
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The Easy Path Wavelet Transform (EPWT) [19] has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of function values and it exploits the local correlations of the given data in a simple appropriate manner. In this paper, we show that the EPWT leads, for a suitable choice of the pathways, to optimal Nterm approximations for piecewise Hölder continuous functions with singularities along curves.
Weak Order for the Discretization of the Stochastic Heat Equation Driven by Impulsive Noise
, 2009
"... The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distrib ..."
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Cited by 112 (7 self)
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The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distributed by the authors. Weak order for the discretization of the stochastic heat equation driven by impulsive noise
An Error Analysis of The Multiconfiguration Timedependent Hartree Method of Quantum Dynamics
 MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
, 2010
"... This paper gives an error analysis of the multiconfiguration timedependent Hartree (MCTDH) method for the approximation of multiparticle timedependent Schrödinger equations. The MCTDH method approximates the multivariate wave function by a linear combination of products of univariate functions a ..."
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Cited by 110 (0 self)
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This paper gives an error analysis of the multiconfiguration timedependent Hartree (MCTDH) method for the approximation of multiparticle timedependent Schrödinger equations. The MCTDH method approximates the multivariate wave function by a linear combination of products of univariate functions and replaces the highdimensional linear Schrödinger equation by a coupled system of ordinary differential equations and lowdimensional nonlinear partial differential equations. The main result of this paper yields an L 2 error bound of the MCTDH approximation in terms of a bestapproximation error bound in a stronger norm and of lower bounds of singular values of matrix unfoldings of the coefficient tensor. This result permits us to establish convergence of the MCTDH method to the exact wave function under appropriate conditions on the approximability of the wave function, and it points to reasons for possible failure in other cases.
Preconditioning stochastic Galerkin saddle point systems
 SIAM J. Matrix Anal. Appl
"... Abstract. Mixed finite element discretizations of deterministic secondorder elliptic partial differential equations (PDEs) lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic secondorder elliptic PDE ..."
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Cited by 110 (4 self)
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Abstract. Mixed finite element discretizations of deterministic secondorder elliptic partial differential equations (PDEs) lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic secondorder elliptic PDEs, which couple standard mixed finite element discretizations in physical space with global polynomial approximation on a probability space, also give rise to linear systems with familiar saddle point structure. For stochastically nonlinear problems, the solution of such systems presents a serious computational challenge. The blocks are sums of Kronecker products of pairs of matrices associated with two distinct discretizations and the systems are large, reflecting the curse of dimensionality inherent in most stochastic approximation schemes. Moreover, for the problems considered herein, the leading blocks of the saddle point matrices are blockdense and the cost of a matrix vector product is nontrivial. We implement a stochastic Galerkin discretization for the steadystate diffusion problem written as a mixed firstorder system. The diffusion coefficient is assumed to be a lognormal random field, approximated via a nonlinear function of a finite number of unbounded random parameters. We study the resulting saddle point systems and investigate the efficiency of blockdiagonal preconditioners of Schur complement and augmented type, for use with minres. By introducing socalled Kronecker product preconditioners we improve the robustness of cheap, meanbased preconditioners with respect to the statistical properties of the stochastically nonlinear diffusion coefficients.
A New Hybrid Method for Image Approximation using the Easy Path Wavelet Transform
"... The Easy Path Wavelet Transform (EPWT) has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of functi ..."
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Cited by 110 (4 self)
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The Easy Path Wavelet Transform (EPWT) has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of function values and exploits the local correlations of the given data in a simple appropriate manner. However, the EPWT suffers from its adaptivity costs that arise from the storage of path vectors. In this paper, we propose a new hybrid method for image compression that exploits the advantages of the usual tensor product wavelet transform for the representation of smooth images and uses the EPWT for an efficient representation of edges and texture. Numerical results show the efficiency of this procedure. Key words. sparse data representation, tensor product wavelet transform, easy path wavelet transform, linear diffusion, smoothing filters, adaptive wavelet bases, Nterm approximation AMS Subject classifications. 41A25, 42C40, 68U10, 94A08 1
Deterministic Multilevel Algorithms for INFINITEDIMENSIONAL INTEGRATION ON R^N
 PREPRINT 40, DFGSPP 1324
, 2010
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