Nonlinear ill-posed equations: Singular value decomposition and the Picard criterion
type: Item
instance of: scholarly article
title: Nonlinear ill-posed equations: Singular value decomposition and the Picard criterion
zbMATH Open document ID: 0606.65039
DOI: 10.1016/0022-247X(86)90052-1
published in: Journal of Mathematical Analysis and Applications
publication date: 1986-01-01T00:00:00Z
review text: A representation of a compact (nonlinear) operator F:A\(\to Y\) in the form \(F(x)=\sum \phi_ n(x)y_ n\) is discussed and the existence of sequences \((x_ n)\subset A\), \((y_ n)\subset Y\) and \((\lambda_ n)\subset {\mathbb{R}}\), such that \(F(x_ n)=\lambda_ ny_ n\) which is the singular value decomposition, if F is additionally linear. Furthermore it is shown that \[ Range F\subset \{y=\sum^{\infty}_{k-1}\eta_ ky_ k: \forall n\in {\mathbb{N}}\sum^{\infty}_{k=n}| \eta_ k|^ 2\leq \lambda^ 2_ n\} \] which is similar to the Picard criterion for compact linear operators.
Mathematics Subject Classification ID: 65J05
Mathematics Subject Classification ID: 47J05
zbMATH DE Number: 3980384
zbMATH Keywords: nonlinear ill-posed equations
zbMATH Keywords: range of the operator
zbMATH Keywords: Banach space
zbMATH Keywords: compact (nonlinear) operator
zbMATH Keywords: singular value decomposition
zbMATH Keywords: Picard criterion
zbMATH Keywords: compact linear operators
author: Eberhard Schock
MaRDI profile type: Publication
cites work: A numerical method for solving Fredholm integral equations of the 1st kind
cites work: Q3334886
cites work: Q3951472
cites work: Q5534403
cites work: Q4183861
cites work: Q3344608
cites work: Q5571228
cites work: Approximation numbers of bounded operators
cites work: Q3042294
cites work: Q3966999
cites work: Q5515676
Recommended article: Ill-Posed Problems: From Linear to Nonlinear and Beyond
Recommended article: Q4375103
Recommended article: Some approximate methods for solving nonlinear ill-posed problems
Recommended article: Finite dimensional approximations to the solutions of nonlinear iii-posed problems
Recommended article: Q4878900
Recommended article: A direct method for nonlinear ill-posed problems
Recommended article: Q4451484
Recommended article: Approximation of solutions with singularities of various types for linear ill-posed problems
Recommended article: Q4719196
Recommended article: On the equivalence of singular and ill-posed problems: the \(p\)-factor regularization method
description: scientific article; zbMATH DE number 3980384