Truncated Singular Value Decomposition Solutions to Discrete Ill-Posed Problems with Ill-Determined Numerical Rank

Tikhonov regularization is a standard method for obtaining smooth solutions to discrete ill-posed problems. A more recent method, based on the singular value decomposition (SVD), is the truncated SVD method. The purpose of this paper is to show, under mild conditions, that the success of both trunca...

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Bibliographic Details
Published in:SIAM journal on scientific and statistical computing Vol. 11; no. 3; pp. 503 - 518
Main Author: Hansen, Per Christian
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.05.1990
Subjects:
Decomposition
Exact sciences and technology
Mathematics
Probability and statistics
Regularization methods
Sciences and techniques of general use
Statistics
Perturbation theory
Ill posed problem
Smoothing
Least squares method
Regularization
ISSN:0196-5204, 1064-8275, 2168-3417, 1095-7197
Online Access:Get full text
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Summary:Tikhonov regularization is a standard method for obtaining smooth solutions to discrete ill-posed problems. A more recent method, based on the singular value decomposition (SVD), is the truncated SVD method. The purpose of this paper is to show, under mild conditions, that the success of both truncated SVD and Tikhonov regularization depends on satisfaction of a discrete Picard condition, involving both the matrix and the right-hand side. When this condition is satisfied, then both methods are guaranteed to produce smooth solutions that are very similar.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
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ISSN:0196-5204
1064-8275
2168-3417
1095-7197
DOI:10.1137/0911028