NLEIGS: A Class of Fully Rational Krylov Methods for Nonlinear Eigenvalue Problems

Güttel, Stefan; Beeumen, Roel Van; Meerbergen, Karl; Michiels, Wim

doi:10.1137/130935045

Cited by 79 publications

(226 citation statements)

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“…Finally, under the assumption thatV j+1 is of full rank, (24) is a nonorthonormal RAD, equivalent to the IRAD (20). Theorem 2.5 applied to (24) asserts that the eigenvalues of H j , K j are the roots of the rational function corresponding to the vector v j+1 + f j+1 .…”

Section: Inexact Radsmentioning

confidence: 99%

“…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…”

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confidence: 99%

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Parallelization of the Rational Arnoldi Algorithm

Berljafa¹,

Güttel²

2017

SIAM J. Sci. Comput.

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Abstract. Rational Krylov methods are applicable to a wide range of scientific computing problems, and the rational Arnoldi algorithm is a commonly used procedure for computing an orthonormal basis of a rational Krylov space. Typically, the computationally most expensive component of this algorithm is the solution of a large linear system of equations at each iteration. We explore the option of solving several linear systems simultaneously, thus constructing the rational Krylov basis in parallel. If this is not done carefully, the basis being orthogonalized may become badly conditioned, leading to numerical instabilities in the orthogonalization process. We introduce the new concept of continuation pairs, which gives rise to a near-optimal parallelization strategy that allows us to control the growth of the condition number of this nonorthogonal basis. As a consequence we obtain a significantly more accurate and reliable parallel rational Arnoldi algorithm. The computational benefits are illustrated using several numerical examples from different application areas.Key words. rational Krylov, orthogonalization, parallelization AMS subject classifications. 68W10, 65F25, 65G50, 65F15 DOI. 10.1137/16M10791781. Introduction. Rational Krylov methods have become an indispensable tool of scientific computing. Invented by Ruhe for the solution of large sparse eigenvalue problems (see, e.g., [26,27]) and frequently advocated for this purpose (see, e.g., [23,32] and [2, sect. 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

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Section: Inexact Radsmentioning

confidence: 99%

mentioning

confidence: 99%

Parallelization of the Rational Arnoldi Algorithm

Berljafa¹,

Güttel²

2017

SIAM J. Sci. Comput.

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“…Matrix polynomials expressed in those bases arise either directly from applications, or as approximations when solving more general nonlinear eigenvalue problems, see for example [12,17,29,30] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Linearizations of matrix polynomials in Newton bases

Perović

Mackey

2018

Linear Algebra and its Applications

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We discuss matrix polynomials expressed in a Newton basis, and the associated polynomial eigenvalue problems. Properties of the generalized ansatz spaces associated with such polynomials are proved directly by utilizing a novel representation of pencils in these spaces. Also, we show how the family of Fiedler pencils can be adapted to matrix polynomials expressed in a Newton basis. These new Newton-Fiedler pencils are shown to be strong linearizations, and some computational aspects related to them are discussed.

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“…For the general nonlinear case, i.e., nonpolynomial eigenvalue problem, A(λ) is first approximated by a matrix polynomial [6,11,21] or rational matrix polynomial [9,17] before a convenient linearization is applied. Most linearizations used in the literature can be written in a similar form, i.e., L(λ) = A−λB, where the parts below the first block rows of A and B have the following Kronecker structure: M ⊗ I n and N ⊗ I n , respectively.…”

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confidence: 99%

Compact Rational Krylov Methods for Nonlinear Eigenvalue Problems

Beeumen¹,

Meerbergen²,

Michiels³

2015

SIAM J. Matrix Anal. & Appl.

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149

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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.

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NLEIGS: A Class of Fully Rational Krylov Methods for Nonlinear Eigenvalue Problems

Cited by 79 publications

References 39 publications

Parallelization of the Rational Arnoldi Algorithm

Parallelization of the Rational Arnoldi Algorithm

Linearizations of matrix polynomials in Newton bases

Compact Rational Krylov Methods for Nonlinear Eigenvalue Problems

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