On the optimality of methods for ill-posed problems
Let A be a linear operator between Hilbert spaces H and F with non-closed range, so that \(Au=f\) is an ill-posed problem. For a method P (i.e., an arbitrary operator \(P: F\to H)\), the worst-case error on a subset \(M\subseteq H\) is defined as \[ \Delta (\delta,M,P,A):=\sup \{\| Pf_{\delta}-u\| /u\in M,\quad f_{\delta}\in F,\quad \| Au- f_{\delta}\| \leq \delta \}. \] A method \(P_{\delta}\) is called ``optimal'', if it minimizes \(\Delta\) (\(\delta\),M,P,A) over all P. In this paper, optimal methods are studied on ``source sets'' \(M_{p\rho u_ 0}:=\{u\in H/u-u_ 0=(A^*A)^{p/2}v,\quad \| v\| \leq \rho \},\) candidates for optimal methods are looked for in the class of all operators of the form \(g_ r(A^*A)A^*+(I-A^*Ag_ r(A^*A))u_ 0,\) where \(g_ r(\lambda)=rg(r\lambda)\) is a family of continuous functions on \({\mathbb{R}}^+_ 0\) with \(\sup \{\lambda^ p| h(\lambda)| /\lambda \geq 0\}<\infty\) \((h(\lambda):=1-\lambda g(\lambda)).\) Since \(\inf_{p}\Delta (\delta_ k,M_{p\rho u_ 0},P,A)\geq \rho^{1/(p+1)}\delta_ k^{p/(p+1)}\) holds for a sequence \((\delta_ k)\to 0\), the aim is to construct methods and parameter choice strategies such that \(\Delta (\delta,M_{p\rho u_ 0},P,A)\leq \rho^{1/(p+1)}\delta^{p/(p+1)}.\) Conditions on g are given such that the corresponding method with the parameter choice \(r=h^{- 1}1/(p+1)(\rho /\delta)^{2/(p+1)}\) is optimal in this sense. Note that p and \(\rho\) have to be known for this parameter choice strategy. The general result is illustrated for Tikhonov regularization, iterative methods and their continuous analogues.
scientific article; zbMATH DE number 4028767