Regularization matrices for discrete ill‐posed problems in several space dimensions

Summary Many applications in science and engineering require the solution of large linear discrete ill‐posed problems that are obtained by the discretization of a Fredholm integral equation of the first kind in several space dimensions. The matrix that defines these problems is very ill conditioned...

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Bibliographic Details
Published in:Numerical linear algebra with applications Vol. 25; no. 4
Main Authors: Dykes, Laura, Huang, Guangxin, Noschese, Silvia, Reichel, Lothar
Format: Journal Article
Language:English
Published: Oxford Wiley Subscription Services, Inc 01.08.2018
Subjects:
Conditioning
discrete ill‐posed problems, Krylov subspace iterative methods, matrix nearness problems, standard form problems, Tikhonov regularization
Discretization
Error analysis
Error detection
Integral equations
Regularization
Roundoff error
ISSN:1070-5325, 1099-1506
Online Access:Get full text
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Summary:Summary Many applications in science and engineering require the solution of large linear discrete ill‐posed problems that are obtained by the discretization of a Fredholm integral equation of the first kind in several space dimensions. The matrix that defines these problems is very ill conditioned and generally numerically singular, and the right‐hand side, which represents measured data, is typically contaminated by measurement error. Straightforward solution of these problems is generally not meaningful due to severe error propagation. Tikhonov regularization seeks to alleviate this difficulty by replacing the given linear discrete ill‐posed problem by a penalized least‐squares problem, whose solution is less sensitive to the error in the right‐hand side and to roundoff errors introduced during the computations. This paper discusses the construction of penalty terms that are determined by solving a matrix nearness problem. These penalty terms allow partial transformation to standard form of Tikhonov regularization problems that stem from the discretization of integral equations on a cube in several space dimensions.
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ISSN:1070-5325
1099-1506
DOI:10.1002/nla.2163