Weighted Inner Products for GMRES and GMRES-DR
For researchers and practitioners using iterative solvers for linear systems, this work provides new insights and methods to accelerate GMRES convergence, though the benefits are problem-dependent and incremental for general cases.
The paper investigates how weighted inner products in restarted GMRES can improve convergence, showing that weighting helps for matrices with localized eigenvectors and can break cyclic convergence patterns. For differential equation problems, incorporating the discrete cosine transform (W-GMRES-DCT) significantly improves convergence.
The convergence of the restarted GMRES method can be significantly improved, for some problems, by using a weighted inner product that changes at each restart. How does this weighting affect convergence, and when is it useful? We show that weighted inner products can help in two distinct ways: when the coefficient matrix has localized eigenvectors, weighting can allow restarted GMRES to focus on eigenvalues that otherwise slow convergence; for general problems, weighting can break the cyclic convergence pattern into which restarted GMRES often settles. The eigenvectors of matrices derived from differential equations are often not localized, thus limiting the impact of weighting. For such problems, incorporating the discrete cosine transform into the inner product can significantly improve GMRES convergence, giving a method we call W-GMRES-DCT. Integrating weighting with eigenvalue deflation via GMRES-DR also can give effective solutions.