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A modified truncated singular value decomposition method for discrete ill‐posed problems
Author(s) -
Noschese Silvia,
Reichel Lothar
Publication year - 2014
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.1938
Subject(s) - mathematics , singular value decomposition , regularization (linguistics) , matrix norm , singular value , low rank approximation , matrix (chemical analysis) , rank (graph theory) , matrix decomposition , mathematical optimization , algorithm , mathematical analysis , combinatorics , hankel matrix , eigenvalues and eigenvectors , computer science , physics , materials science , quantum mechanics , artificial intelligence , composite material
SUMMARY Truncated singular value decomposition is a popular method for solving linear discrete ill‐posed problems with a small to moderately sized matrix A . Regularization is achieved by replacing the matrix A by its best rank‐ k approximant, which we denote by A k . The rank may be determined in a variety of ways, for example, by the discrepancy principle or the L‐curve criterion. This paper describes a novel regularization approach, in which A is replaced by the closest matrix in a unitarily invariant matrix norm with the same spectral condition number as A k . Computed examples illustrate that this regularization approach often yields approximate solutions of higher quality than the replacement of A by A k .Copyright © 2014 John Wiley & Sons, Ltd.