Ronald B. Morgan

Mark Embree, Ronald B. Morgan, Huy V. Nguyen

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.

Mark Embree, Jennifer A. Loe, Ronald B. Morgan

Polynomial preconditioning can improve the convergence of the Arnoldi method for computing eigenvalues. Such preconditioning significantly reduces the cost of orthogonalization; for difficult problems, it can also reduce the number of matrix-vector products. Parallel computations can particularly benefit from the reduction of communication-intensive operations. The GMRES algorithm provides a simple and effective way of generating the preconditioning polynomial. For some problems high degree polynomials are especially effective, but they can lead to stability problems that must be mitigated. A two-level "double polynomial preconditioning" strategy provides an effective way to generate high-degree preconditioners.