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Nonconvergence results for the application of least-squares estimation to Ill-posed problems

Authors:	T I Seidman

Institution:	(1) Department of Mathematics, University of Maryland Baltimore County, Baltimore, Maryland

Abstract:	One standard approach to solvingf(x)=b is the minimization of f(x)–b² overx in $\mathop \mathfrak{X}\limits^ \sim$ , where $\mathop \mathfrak{X}\limits^ \sim$ corresponds to a parametric representation providing sufficiently good approximation to the true solutionx. Call the minimizerx=A( $\mathop \mathfrak{X}\limits^ \sim$ ). Take $\mathop \mathfrak{X}\limits^ \sim$ = $\mathfrak{X}$ _N for a sequence { $\mathfrak{X}$ _N} of subspaces becoming dense, and so determine an approximating sequences {x _NA ( $\mathfrak{X}$ _N)}. It is shown, withf linear and one-to-one, that one need not havex _Nx iff ^–1 is not continuous.This work was supported by the US Army Research Office under Grant No. DAAG-29-77-G-0061. The author is indebted to the late W. C. Chewning for suggesting the topic in connection with computing optimal boundary controls for the heat equation (Ref. 2).

Keywords:	Ill-posed problems least-squares estimation approximation parametric representation convergence nonconvergence
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