Numerical Linear Algebra Days (Due giorni di Algebra Lineare Numerica)

An efficient block rational Krylov solver for Sylvester equations with adaptive pole selection

Not scheduled

20m

Gran Sasso Science Institute, L'Aquila

Talk

Speaker

Angelo Alberto Casulli (Scuola Normale Superiore di Pisa)

Description

We present an algorithm for the solution of Sylvester equations with right-hand side of low rank, based on projection onto a block rational Krylov subspace. Extending the convergence analysis in [2] to the block case, we link the convergence with the problem of minimizing the norm of a small rational matrix over the spectra or field-of-values of the involved matrices. This is in contrast with the non-block case, where the minimum problem is scalar, instead of matrix-valued. Replacing the norm of the objective function with an easier to evaluate function yields several adaptive pole selection strategies, providing a theoretical analysis for known heuristics, as well as effective novel techniques.

[1] A. Casulli, L. Robol, An efficient block rational Krylov solver for Sylvester equations with adaptive pole selection. arXiv:2301.08103, 2023.
[2] B. Beckermann. An error analysis for rational Galerkin projection applied to the Sylvester equation. SIAM Journal on Numerical Analysis, 2011.