Improved Polynomial Bounds and Acceleration of GMRES by Solving a min-max Problem on Rectangles, and by Deflating

2025/04/08 by Nicole Spillane, Daniel B. Szyld, Spillane, Nicole +1 · 1 citation
Engineering · Computer Science · #Advanced Numerical Analysis Techniques #Constraint Satisfaction and Optimization #Educational Technology and Assessment

paper · pdf · doi:10.48550/arxiv.2504.05723

Abstract

Polynomial convergence bounds are considered for left, right, and split preconditioned GMRES. They include the cases of Weighted and Deflated GMRES for a linear system Ax = b. In particular, the case of positive definite A is considered. The well-known polynomial bounds are generalized to the cases considered, and then reduced to solving a min-max problem on rectangles on the complex plane. Several approaches are considered and compared. The new bounds can be improved by using specific deflation spaces and preconditioners. This in turn accelerates the convergence of GMRES. Numerical examples illustrate the results obtained.

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• Do you precondition on the left or on the right?

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