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

Abstract

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 {x _N ≔A ($\mathfrak{X}$ _N )}. It is shown, withf linear and one-to-one, that one need not havex _N→x* iff ⁻¹ is not continuous.