On approximate solution of nonlinear operator equations
type: Item
instance of: scholarly article
title: On approximate solution of nonlinear operator equations
zbMATH Open document ID: 0919.65039
DOI: 10.1007/BF02672925
published in: Siberian Mathematical Journal
publication date: 1999-05-11T00:00:00Z
review text: Let \(A\) be an operator with domain \(D(A)\) and range \(R(A)\) which acts in a Hilbert space \(H\) and, for \(f=f_0\), there is an exact solution \(u_0\) to the operator equation of the first kind \[ Au = f, \] i.e., \(f_0\in R(A)\), but, instead of an exact value \(f_0\) of the right-hand side, only an approximate value \(f_{\delta}\) and an error \(\delta\) are given, \(\| f_{\delta} - f_0\| \leq\delta\). The problem reads: given \(f_{\delta}\), \(\delta >0\), construct an approximate solution to the equation ``close'' to the set \(M_0\) of exact solutions. The author solves this problem and exposes necessary and sufficient conditions on the operator \(A\) for convergence of the approximate solutions obtained by the regularization method.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 65J20
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 1286347
zbMATH Keywords: ill-posed problem
zbMATH Keywords: nonlinear operator equation
zbMATH Keywords: regularization method
zbMATH Keywords: Hilbert space
zbMATH Keywords: convergence
author: Vitaly Pavlovich Tanana
reviewed by: Vladimir N. Grebenev
MaRDI profile type: Publication
cites work: PROJECTIVE STRUCTURES, FLAT BUNDLES, AND KÄHLER METRICS ON MODULI SPACES
cites work: Convergence criterion of the residual method
cites work: Q3994690
description: scientific article; zbMATH DE number 1286347