Arbitrary divergence speed of the least-squares method in infinite-dimensional inverse ill-posed problems

Abstract

A standard engineering procedure for approximating the solutions of an infinite-dimensional inverse problem of the form Ax = y, where A is a given compact linear operator on a Hilbert space X and y is the given data, is to find a sequence {XN} of finite-dimensional approximating subspaces of X whose union is dense in X and to construct the sequence {xN} of least-squares solutions of the problem in X N. In 1980, Seidman showed that if the problem is ill-posed, then, without any additional assumptions on the exact solution or on the sequence of approximating subspaces XN, it cannot be guaranteed that the sequence {xN} will converge to the exact solution. In this paper, this result is extended in the following sense: it is shown that if X is separable, then for any y ∈ X, y ≠ 0 and for any arbitrarily given function there exists an injective, compact linear operator A and an increasing sequence of finite-dimensional subspaces XN ⊂ X such that for all, where xN is the least-squares solution of Ax = y in XN. © 2006 IOP Publishing Ltd.