On the condition number of matrices arising in the Tikhonov regularization method
type: Item
instance of: scholarly article
title: On the condition number of matrices arising in the Tikhonov regularization method
zbMATH Open document ID: 0813.65139
DOI: 10.1007/BF01002358
author: B. Abramovitz
published in: Acta Applicandae Mathematicae
publication date: 1994-12-01T00:00:00Z
review text: An integral equation of the first kind (1) \((Kf)(s) = \int^ b_ a k(s,t) f(t)dt = g(s)\), \(K : L^ 2 [a,b] \to L^ 2 [a,b]\), \(k \in L^ 2 ([a,b] \times [a,b])\) is known to be an ill-posed problem. As an expression of this ill-posedness, for many numerical methods (e.g., Galerkin, collocation) the approximate solution of (1) fails in general to the exact solution \(f\). Even if the numerical methods converge, it happens that the rate of decrease of the singular values of \(K\) is directly related to the rate of increase of the condition numbers of the matrices representing discrete versions of \(K\). The main result of the paper is bounding the condition number from both sides, in terms of the singular values of the operator \(K\), the numerical method being a variant of finite-dimensional Tikhonov regularization. In order to obtain such estimates, a compatibility condition is needed between the operator \(K\) and the approximation spaces. Classes of operators satisfying the compatibility condition are presented and studied. The approximation spaces are spaces of spline functions, the latter being chosen because of their special approximation properties.
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 65R30
Mathematics Subject Classification ID: 45B05
zbMATH DE Number: 698638
zbMATH Keywords: ill-posed problem
zbMATH Keywords: singular values
zbMATH Keywords: condition numbers
zbMATH Keywords: Tikhonov regularization
zbMATH Keywords: compatibility condition
reviewed by: G. Dimitriu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 698638