WING: A Simple Windowed Nonorthogonalized Initial Guess Procedure for Repeated Matrix Solves

Abstract

Many numerical methods require solution of a sequence of linear systems with the same matrix and similar right-hand sides. Krylov subspace methods are a common tool for solving such linear systems, and a carefully chosen initial guess for the solution can reduce the total number of iterations, and thereby the total computational cost, required for convergence to a specified numerical tolerance. This paper introduces the WING algorithm, a modification of Fischer's second algorithm, which lowers the cost of forming an acceptably close initial guess by skipping orthogonalization and solving the possibly singular normal equations with a pseudoinverse. We demonstrate the efficacy of the new algorithm, particularly for solving linear systems with coarse relative tolerances, with numerical benchmarks based on fluid-structure interaction, mantle convection, and earthquake models.