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 XN.
In 1980, Seidman (Nonconvergence Results for the Application of Least-Squares Estimation to Ill-Posed Problems, Journal of Optimization Theory and Applications, 30, 4, 535-547, 1980) 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 fxNg will converge to the exact solution. In this article this result is extended in the following sense: it is shown that if X is separable, then for any y in X, y not equal 0, and for any arbitrarily given function s defined over the natural numbers with values in R+, there exists an injective, compact linear operator A and an increasing sequence of finite-dimensional subspaces XN contained in X such that ||xN - A^{-1}y||>= s(N) for all N, where xN is the least squares solution of Ax = y in XN.
{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 XN.
In 1980, Seidman (Nonconvergence Results for the Application of Least-Squares Estimation to Ill-Posed Problems, Journal of Optimization Theory and Applications, 30, 4, 535-547, 1980) 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 fxNg will converge to the exact solution. In this article this result is extended in the following sense: it is shown that if X is separable, then for any y in X, y not equal 0, and for any arbitrarily given function s defined over the natural numbers with values in R+, there exists an injective, compact linear operator A and an increasing sequence of finite-dimensional subspaces XN contained in X such that ||xN - A^{-1}y||>= s(N) for all N, where xN is the least squares solution of Ax = y in XN.