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Compress‐and‐restart block Krylov subspace methods for Sylvester matrix equations
Author(s) -
Kressner Daniel,
Lund Kathryn,
Massei Stefano,
Palitta Davide
Publication year - 2021
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2339
Subject(s) - krylov subspace , generalized minimal residual method , coefficient matrix , mathematics , block (permutation group theory) , polynomial , iterative method , mathematical optimization , matrix (chemical analysis) , linear system , sylvester matrix , algorithm , discretization , eigenvalues and eigenvectors , matrix polynomial , polynomial matrix , mathematical analysis , geometry , physics , materials science , quantum mechanics , composite material
Summary Block Krylov subspace methods (KSMs) comprise building blocks in many state‐of‐the‐art solvers for large‐scale matrix equations as they arise, for example, from the discretization of partial differential equations. While extended and rational block Krylov subspace methods provide a major reduction in iteration counts over polynomial block KSMs, they also require reliable solvers for the coefficient matrices, and these solvers are often iterative methods themselves. It is not hard to devise scenarios in which the available memory, and consequently the dimension of the Krylov subspace, is limited. In such scenarios for linear systems and eigenvalue problems, restarting is a well‐explored technique for mitigating memory constraints. In this work, such restarting techniques are applied to polynomial KSMs for matrix equations with a compression step to control the growing rank of the residual. An error analysis is also performed, leading to heuristics for dynamically adjusting the basis size in each restart cycle. A panel of numerical experiments demonstrates the effectiveness of the new method with respect to extended block KSMs.