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

Thomas I. Seidman | Zendy

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

Author(s) -

Thomas I. Seidman

Publication year - 1980

Publication title -

journal of optimization theory and applications

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.109

H-Index - 91

eISSN - 1573-2878

pISSN - 0022-3239

DOI - 10.1007/bf01686719

Subject(s) - mathematics , linear subspace , sequence (biology) , combinatorics , pure mathematics , biology , genetics

One standard approach to solvingf(x)=b is the minimization of ?f(x)-b?2 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 {xN?A ( $\mathfrak{X}$ N)}. It is shown, withf linear and one-to-one, that one need not havexN?x* iff-1 is not continuous.