Compact Two-Sided Krylov Methods for Nonlinear Eigenvalue Problems

Abstract

We describe a generalization of the compact rational Krylov (CORK) methods for polynomial and rational eigenvalue problems that usually but not necessarily come from polynomial or rational approximations of genuinely nonlinear eigenvalue problems. CORK is a family of one-sided methods that reformulates the polynomial or rational eigenproblem as a generalized eigenvalue problem. By exploiting the Kronecker structure of the associated pencil, it constructs a right Krylov subspace in compact form and thereby avoids the high memory and orthogonalization costs that are usually associated with linearizations of high degree matrix polynomials. CORK approximates eigenvalues and their corresponding right eigenvectors but is not suitable in its current form for the computation of left eigenvectors. Our generalization of the CORK method is based on a class of Kronecker structured pencils that include as special cases the CORK pencils, the transposes of CORK pencils, and the symmetrically structured linearizations by Robol, Vandebril and Van Dooren, 2016. This class of structured pencils facilitates the development of a general framework for the computation of both right and left-sided Krylov subspaces in compact form, and hence allows the development of two-sided compact rational Krylov methods for nonlinear eigenvalue problems. The latter are particularly efficient when the standard inner product is replaced by a cheaper to compute quasi inner product. We show experimentally that convergence results similar to CORK can be obtained for a certain quasi inner product.

3 Case studies of model order reduction for acoustics and vibrations

“…Linearizations are also used for other polynomial, rational, or even fully nonlinear dependencies on the frequency. Efficient implementations of Krylov methods (oneand two-sided) rely on a similar property as the state vector of second-order problems [53,30]. We will give more examples in Section 3.4.…”

“…This suggests that there is also a strong connection between W 1 and W 2 . Indeed, in [30] this property is exploited by expressing W 1 = ZT 1 and…”

New strategies for determining backward perturbation bound of approximate two‐sided Krylov subspaces

“…However, the two‐sided Lanczos method may suffer from numerical instability because nonorthogonal basis matrices are utilized 1,23 . The accuracy and stability of the computed bases can be improved by using the two‐sided Arnoldi method, 17,24‐28 which replaces the biorthonormal basis matrices by using two orthonormal basis matrices V m + 1 and W m + 1 (ie, ). The two‐sided Arnoldi method proposed by Ruhe, 26,27 and later as a block method by Cullum and Zhang, 24 independently generates orthonormal bases for the right search space and the left search space .…”

“…Recently, the two‐sided Arnoldi method was considered in the task of updating matrix functions f ( A ), where the matrix is subject to a low‐rank modification 17 . A two‐sided Arnoldi algorithm with Krylov‐Schur restarting was investigated in Reference 28, and a compact two‐sided Krylov method was applied to solve nonlinear eigenvalue problems in Reference 25.…”

Block minimal bases ℓ-ifications of matrix polynomials

“…There are many (in fact, infinitely many) choices available in the literature for constructing strong linearizations and strong ℓ-ifications of matrix polynomials. From a numerical analyst point of view, this situation is very desirable, since one can choose the most favorable construction in terms of various criteria, such as conditioning and backward errors [31,32], the basis in which the polynomial is represented [1], preservation of algebraic structures [33,41], exploitation of matrix structures in numerical algoritghms [38,49,47], etc However, there has not been a framework providing a way to construct and analyze all these strong linearizations and strong ℓ-ifications in a consistent manner. Providing such a framework is one of the main goals of this work.…”