TY - JOUR

T1 - Adaptive cross approximation for ill-posed problems

AU - Mach, T.

AU - Reichel, L.

AU - Van Barel, M.

AU - Vandebril, R.

N1 - Funding Information:
The research was partially supported by the Research Council KU Leuven , project OT/10/038 (Multi-parameter model order reduction and its applications), PF/10/002 Optimization in Engineering Centre (OPTEC), CREA-13-012 (Can Unconventional Eigenvalue Algorithms Supersede the State of the Art), OT/11/055 (Spectral Properties of Perturbed Normal Matrices and their Applications), by the Fund for Scientific Research—Flanders (Belgium), G.0828.14N (Multivariate polynomial and rational interpolation and approximation), G.0342.12N (Reestablishing Smoothness for Matrix Manifold Optimization via Resolution of Singularities), and by the Interuniversity Attraction Poles Programme , initiated by the Belgian State, Science Policy Office, Belgian Network DYSCO (Dynamical Systems, Control, and Optimization). This research also is supported in part by NSF grant DMS-1115385 .
Publisher Copyright:
© 2016 Elsevier B.V.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2016/9/1

Y1 - 2016/9/1

N2 - Integral equations of the first kind with a smooth kernel and perturbed right-hand side, which represents available contaminated data, arise in many applications. Discretization gives rise to linear systems of equations with a matrix whose singular values cluster at the origin. The solution of these systems of equations requires regularization, which has the effect that components in the computed solution connected to singular vectors associated with small singular values are damped or ignored. In order to compute a useful approximate solution typically approximations of only a fairly small number of the largest singular values and associated singular vectors of the matrix are required. The present paper explores the possibility of determining these approximate singular values and vectors by adaptive cross approximation. This approach is particularly useful when a fine discretization of the integral equation is required and the resulting linear system of equations is of large dimensions, because adaptive cross approximation makes it possible to compute only fairly few of the matrix entries.

AB - Integral equations of the first kind with a smooth kernel and perturbed right-hand side, which represents available contaminated data, arise in many applications. Discretization gives rise to linear systems of equations with a matrix whose singular values cluster at the origin. The solution of these systems of equations requires regularization, which has the effect that components in the computed solution connected to singular vectors associated with small singular values are damped or ignored. In order to compute a useful approximate solution typically approximations of only a fairly small number of the largest singular values and associated singular vectors of the matrix are required. The present paper explores the possibility of determining these approximate singular values and vectors by adaptive cross approximation. This approach is particularly useful when a fine discretization of the integral equation is required and the resulting linear system of equations is of large dimensions, because adaptive cross approximation makes it possible to compute only fairly few of the matrix entries.

KW - Adaptive cross approximation

KW - Ill-posed problem

KW - Inverse problem

KW - Regularization

KW - Sparse discretization

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U2 - 10.1016/j.cam.2016.02.020

DO - 10.1016/j.cam.2016.02.020

M3 - Article

AN - SCOPUS:84977608760

VL - 303

SP - 206

EP - 217

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

ER -