Results 1  10
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119
A review of curvelets and recent applications
 IEEE Signal Processing Magazine
, 2009
"... Multiresolution methods are deeply related to image processing, biological and computer vision, scientific computing, etc. The curvelet transform is a multiscale directional transform which allows an almost optimal nonadaptive sparse representation of objects with edges. It has generated increasing ..."
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Cited by 128 (10 self)
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Multiresolution methods are deeply related to image processing, biological and computer vision, scientific computing, etc. The curvelet transform is a multiscale directional transform which allows an almost optimal nonadaptive sparse representation of objects with edges. It has generated increasing interest in the community of applied mathematics and signal processing over the past years. In this paper, we present a review on the curvelet transform, including its history beginning from wavelets, its logical relationship to other multiresolution multidirectional methods like contourlets and shearlets, its basic theory and discrete algorithm. Further, we consider recent applications in image/video processing, seismic exploration, fluid mechanics, simulation of partial different equations, and compressed sensing.
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 117 (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.
Weak Order for the Discretization of the Stochastic Heat Equation Driven by Impulsive Noise
, 2009
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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 111 (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
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 111 (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
, 2009
"... 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 ..."
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Cited by 110 (4 self)
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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.
Deterministic Multilevel Algorithms for INFINITEDIMENSIONAL INTEGRATION ON R^N
 PREPRINT 40, DFGSPP 1324
, 2010
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