Implicitly restarted Arnoldi with purification for the shift-invert transformation

Meerbergen, Karl; Spence, A.

doi:10.1090/s0025-5718-97-00844-2

Cited by 50 publications

(48 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…This implies that it may be difficult for Krylov methods to find the wanted eigenvalue(s) (see also the experiments with Matlab's eigs and an implementation of Krylov-Schur [29] in Section 4). Second, approaches based on matrix transformations (see, e.g., [18] for an overview, and, for instance, [19]) are not applicable: shift-and-invert Arnoldi methods are infeasible as these require a known shift (target); in our problems we do not have a shift beforehand. Third, and most importantly, our method is able to make effective use of the action with sparser (and hence cheaper) commuting matrices in the inner iterations: it computes the eigenvalues of the sparse matrix A p while using one of the much sparser matrices A x 1 , .…”

Section: A Jacobi-davidson Type Methods For Commuting Matricesmentioning

confidence: 99%

Polynomial optimization and a Jacobi–Davidson type method for commuting matrices

Bleylevens

Hochstenbach

Peeters

2013

Applied Mathematics and Computation

View full text Add to dashboard Cite

In this paper we introduce an new Jacobi-Davidson type eigenvalue solver for a set of commuting matrices, called JD-COMM, used for the global optimization of so-called Minkowski-norm dominated polynomials in several variables. The Stetter-Möller matrix method yields such a set of real non-symmetric commuting matrices since it reformulates the optimization problem as an eigenvalue problem. A drawback of this approach is that the matrix most relevant for computing the global optimum of the polynomial under investigation is usually large and only moderately sparse. However, the other matrices are generally much sparser and have the same eigenvectors because of the commutativity. This fact is used to design the JDCOMM method for this problem: the most relevant matrix is used only in the outer loop and the sparser matrices are exploited in the solution of the correction equation in the inner loop to greatly improve the efficiency of the method. Some numerical examples demonstrate that the method proposed in this paper is more efficient than approaches that work on the main matrix (standard Jacobi-Davidson and implicitly restarted Arnoldi), as well as conventional solvers for computing the global optimum, i.e., SOSTOOLS, GloptiPoly, and PHCpack.

show abstract

Section: A Jacobi-davidson Type Methods For Commuting Matricesmentioning

confidence: 99%

Polynomial optimization and a Jacobi–Davidson type method for commuting matrices

Bleylevens

Hochstenbach

Peeters

2013

Applied Mathematics and Computation

View full text Add to dashboard Cite

show abstract

“…Also see [21] [18] [10] [11] [16]. The idea is to apply an orthogonal transformation to H k that pushes the p desired Ritz values of H k to the principle p × p block.…”

Section: Implicit Restartingmentioning

confidence: 99%

The Quadratic Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem

Meerbergen¹

2009

SIAM J. Matrix Anal. & Appl.

Self Cite

View full text Add to dashboard Cite

The Quadratic Arnoldi algorithm is an Arnoldi algorithm for the solution of the quadratic eigenvalue problem, that exploits the structure of the Krylov vectors. This allows us to reduce the memory requirements by about a half. The method is an alternative to the Second Order Arnoldi method (SOAR). In the SOAR method it is not clear how to perform an implicit restart. We discuss various choices of linearizations in L and DL. We also explain how to compute a partial Schur form of the underlying linearization with respect to the structure of the Schur vectors. We also formulate some open problems. AMS subject classification. 15A18, 65F15Abstract. The Quadratic Arnoldi algorithm is an Arnoldi algorithm for the solution of the quadratic eigenvalue problem, that exploits the structure of the Krylov vectors. This allows us to reduce the memory requirements by about a half. The method is an alternative to the Second Order Arnoldi method (SOAR). In the SOAR method it is not clear how to perform an implicit restart. We discuss various choices of linearizations in L 1 and DL. We also explain how to compute a partial Schur form of the underlying linearization with respect to the structure of the Schur vectors. We also formulate some open problems.

show abstract

“…A few remarks are in order. The starting vector is chosen so that it does not contain any components [15] in the null-space of B. The orthogonality of the m Arnoldi vectors is maintained at machine precision.…”

Section: Algorithmmentioning

confidence: 99%

Massively parallel linear stability analysis with P_ARPACK for 3D fluid flow modeled with MPSalsa

Lehoucq

Salinger

1998

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. We are interested in the stability of three-dimensional fluid flows to small disturbances. One computational approach is to solve a sequence of large sparse generalized eigenvalue problems for the leading modes that arise from discretizating the differential equations modeling the flow. The modes of interest are the eigenvalues of largest real part and their associated eigenvectors. We discuss our work to develop an efficient and reliable eigensolver for use by the massively parallel simulation code MPSalsa. MPSalsa allows simulation of complex 3D fluid flow, heat transfer, and mass transfer with detailed bulk fluid and surface chemical reaction kinetics.

show abstract

Implicitly restarted Arnoldi with purification for the shift-invert transformation

Cited by 50 publications

References 14 publications

Polynomial optimization and a Jacobi–Davidson type method for commuting matrices

Polynomial optimization and a Jacobi–Davidson type method for commuting matrices

The Quadratic Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem

Massively parallel linear stability analysis with P_ARPACK for 3D fluid flow modeled with MPSalsa

Contact Info

Product

Resources

About