Superlinear Convergence of Krylov Subspace Methods for Self-Adjoint Problems in Hilbert Space

Abstract

The conjugate gradient and minimum residual methods for self-adjoint problems in Hilbert space are considered. Linear and superlinear convergence results both with respect to Q-and R-rates are reviewed. New results onstep Q-superlinear and R-superlinear convergence for the minimum residual method are provided and examples are considered to underscore the relevance of a Hilbert space theory.

On Krylov solutions to infinite-dimensional inverse linear problems

“…Propositions 3.9 and 3.10 above have a noticeable consequence. It is worth noticing that the self-adjoint case has always deserved a special status in this context, theoretically and in applications: the convergence of Krylov techniques for self-adjoint operators are the object of an ample literature -see, e.g., [20,7,19,31,22,18,24]. at time t > 0.…”

Some Properties of the Arnoldi-Based Methods for Linear Ill-Posed Problems

“…Since ∥f m − f ∥ → 0 for m → ∞, by (13) and since A * is still Hilbert-Schmidt we obtain the result taking…”

“…We remark that for linear equations of the type (I + λA)f = g, where A is compact and λ > 0, the analysis allows to show the superlinear convergence of the residuals ( [14]) that we are not able to show for problems like (1). Among the existing works in which the superlinear convergence of Krylov methods is studied in the continuous setting, we quote here the recent paper [13] and its wide bibliography. As for the finite dimensional case, we remember [22], where many Krylov methods are considered.…”

Analysis of Iterative Methods in Photoacoustic Tomography with Variable Sound Speed

“…Moreover, assume that T * T = αI + K where K is a compact operator. Then, the CG method is known to converge superlinearly (see, e.g., [28]).…”