Schwarz Iterative Methods: Infinite Space Splittings

TL;DR

This paper proves convergence rates for greedy and randomized Schwarz iterative methods in solving linear elliptic variational problems, demonstrating optimal decay rates for error reduction based on infinite space splittings.

Contribution

It establishes convergence and explicit error decay rates for both greedy and randomized Schwarz iterative methods using infinite space splittings.

Findings

Squared error decay rate of O((m+1)^{-1}) for greedy methods.

Expected squared error decay rate of O((m+1)^{-1}) for randomized methods.

Convergence proven for methods based on infinite space splittings.

Abstract

We prove the convergence of greedy and randomized versions of Schwarz iterative methods for solving linear elliptic variational problems based on infinite space splittings of a Hilbert space. For the greedy case, we show a squared error decay rate of $O((m+1)^{-1})$ for elements of an approximation space $\mathcal{A}_1$ related to the underlying splitting. For the randomized case, we show an expected squared error decay rate of $O((m+1)^{-1})$ on a class $\mathcal{A}_{\infty}^{\pi}\subset \mathcal{A}_1$ depending on the probability distribution.