Compact Rational Krylov Methods for Nonlinear Eigenvalue Problems

Abstract

We propose a new uniform framework of Compact Rational Krylov (CORK) methods for solving large-scale nonlinear eigenvalue problems: A(λ)x = 0. For many years, linearizations are used for solving polynomial and rational eigenvalue problems. On the other hand, for the general nonlinear case, A(λ) can first be approximated by a (rational) matrix polynomial and then a convenient linearization is used. However, the major disadvantage of linearization based methods is the growing memory and orthogonalization costs with the iteration count, i.e., in general they are proportional to the degree of the polynomial. Therefore, the CORK family of rational Krylov methods exploits the structure of the linearization pencils by using a generalization of the compact Arnoldi decomposition. In this way, the extra memory and orthogonalization costs due to the linearization of the original eigenvalue problem are negligible for large-scale problems. Furthermore, we prove that each CORK step breaks down into an orthogonalization step of the original problem dimension and a rational Krylov step on small matrices. We also briefly discuss implicit restarting of the CORK method and how to exploit low rank structure. The CORK method is illustrated with two large-scale examples.Keywords : linearization, matrix pencil, rational Krylov, nonlinear eigenvalue problem MSC : 65F15, 15A22. COMPACT RATIONAL KRYLOV METHODS FOR NONLINEAR EIGENVALUE PROBLEMS *ROEL VAN BEEUMEN † , KARL MEERBERGEN † , AND WIM MICHIELS † Abstract. We propose a new uniform framework of Compact Rational Krylov (CORK) methods for solving large-scale nonlinear eigenvalue problems: A(λ)x = 0. For many years, linearizations are used for solving polynomial and rational eigenvalue problems. On the other hand, for the general nonlinear case, A(λ) can first be approximated by a (rational) matrix polynomial and then a convenient linearization is used. However, the major disadvantage of linearization based methods is the growing memory and orthogonalization costs with the iteration count, i.e., in general they are proportional to the degree of the polynomial. Therefore, the CORK family of rational Krylov methods exploits the structure of the linearization pencils by using a generalization of the compact Arnoldi decomposition. In this way, the extra memory and orthogonalization costs due to the linearization of the original eigenvalue problem are negligible for large-scale problems. Furthermore, we prove that each CORK step breaks down into an orthogonalization step of the original problem dimension and a rational Krylov step on small matrices. We also briefly discuss implicit restarting of the CORK method and how to exploit low rank structure. The CORK method is illustrated with two large-scale examples.

Parallelization of the Rational Arnoldi Algorithm

“…For instance, in [27] the author refers to V j t j as the continuation vector, while in [23] the term is used to denote (ρ j A − η j I)V j t j . The terminology of "continuation combinations" is adopted in [5,27,31] for the vectors t j . With the notion of continuation pair we want to stress that there are two parts, a root and a vector, both of which are equally important.…”

“…Currently, the choices t m = e m and t m = q m appear to be dominant in the literature; see, e.g., [27,31]. Note that t m = e m may (with probability zero) not be admissible; i.e., we would not be able to expand the space with the obtained w m+1 even though the space is not yet A-invariant.…”

“…8.5]), these methods have seen an increasing number of other applications over the last two decades or so. Examples of rational Krylov applications can be found in model order reduction [11,16,17,22], matrix function approximation [10,13,19], matrix equations [3,9,24], nonlinear eigenvalue problems [20,21,31], and nonlinear rational least squares fitting [5,6].At the core of most rational Krylov applications is the rational Arnoldi algorithm, which is a Gram-Schmidt procedure for generating an orthonormal basis of a rational Krylov space. Given a matrix A ∈ C N,N , a vector b ∈ C N , and a polynomial q m of degree at most m and such that q m (A) is nonsingular, the rational Krylov space of order m is defined as…”

A compact rational Krylov method for large‐scale rational eigenvalue problems

“…Extending the ideas of a compact representation of the Krylov vectors and of the two levels of orthogonalization from polynomials of degree 2 (TOAR) to polynomials of any degree, Kressner and Roman 19 developed a memory-efficient and stable Arnoldi process for the linearizations of matrix polynomials expressed in the Chebyshev basis. In 2015, the compact rational Krylov (CORK) method for nonlinear eigenvalue problems (NLEPs) was introduced in the work of Van Beeumen et al 20 CORK considers particular NLEPs that can be expressed and linearized in certain ways, which are solved by applying a CORK method to such linearizations. A key feature of the CORK method is that it works for many kinds of linearizations involving a Kronecker structure, as the Frobenius companion form or linearizations of matrix polynomials in different bases (as Newton or Chebyshev, among others 21 ).…”

“…More precisely, they ordered the matrices P i in reverse order and interchanged the identity blocks in  and . Since we are following, in this paper, the spirit of CORK, then  and  are written in (10) in the same way as in table 1 in the work of Van Beeumen et al 20…”