Large-scale Eigenvalue Problems Research Articles

A Krylov Method for the Delay Eigenvalue Problem

The Arnoldi method is currently a very popular algorithm to solve large-scale eigenvalue problems. The main goal of this paper is to generalize the Arnoldi method to the characteristic equation of a delay-differential equation (DDE), here called a delay eigenvalue problem (DEP). The DDE can equivalently be expressed with a linear infinite-dimensional operator whose eigenvalues are the solutions to the DEP. We derive a new method by applying the Arnoldi method to the generalized eigenvalue problem associated with a spectral discretization of the operator and by exploiting the structure. The result is a scheme where we expand a subspace not only in the traditional way done in the Arnoldi method. The subspace vectors are also expanded with one block of rows in each iteration. More important, the structure is such that if the Arnoldi method is started in an appropriate way, it has the (somewhat remarkable) property that it is in a sense independent of the number of discretization points. It is mathematically equivalent to an Arnoldi method with an infinite matrix, corresponding to the limit where we have an infinite number of discretization points. We also show an equivalence with the Arnoldi method in an operator setting. It turns out that with an appropriately defined operator over a space equipped with scalar product with respect to which Chebyshev polynomials are orthonormal, the vectors in the Arnoldi iteration can be interpreted as the coefficients in a Chebyshev expansion of a function. The presented method yields the same Hessenberg matrix as the Arnoldi method applied to the operator.

Deflated block Krylov subspace methods for large scale eigenvalue problems

We discuss a class of deflated block Krylov subspace methods for solving large scale matrix eigenvalue problems. The efficiency of an Arnoldi-type method is examined in computing partial or closely clustered eigenvalues of large matrices. As an improvement, we also propose a refined variant of the Arnoldi-type method. Comparisons show that the refined variant can further improve the Arnoldi-type method and both methods exhibit very regular convergence behavior.

Improving the parallel performance of a domain decomposition preconditioning technique in the Jacobi–Davidson method for large scale eigenvalue problems

Most computational work in Jacobi–Davidson [G.L.G. Sleijpen, H.A. van der Vorst, A Jacobi–Davidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl. 17 (1996) 401–425], an iterative method for large scale eigenvalue problems, is due to a so-called correction equation. In [M. Genseberger, G.L.G. Sleijpen, H.A. van der Vorst, Domain decomposition in Jacobi–Davidson for PDE related eigenvalue problems, in preparation] a strategy for the approximate solution of the correction equation was proposed. This strategy is based on a domain decomposition preconditioning technique in order to reduce wall clock time and local memory requirements. This paper discusses the aspect that the original strategy can be improved. For large scale eigenvalue problems that need a massively parallel treatment this aspect turns out to be nontrivial. The impact on the parallel performance will be shown by results of scaling experiments up to 1024 cores.

Anasazi software for the numerical solution of large-scale eigenvalue problems

Anasazi is a package within the Trilinos software project that provides a framework for the iterative, numerical solution of large-scale eigenvalue problems. Anasazi is written in ANSI C++ and exploits modern software paradigms to enable the research and development of eigensolver algorithms. Furthermore, Anasazi provides implementations for some of the most recent eigensolver methods. The purpose of our article is to describe the design and development of the Anasazi framework. A performance comparison of Anasazi and the popular FORTRAN 77 code ARPACK is given.

Restarted generalized Krylov subspace methods for solving large-scale polynomial eigenvalue problems

In this paper, we introduce a generalized Krylov subspace \({\mathcal{G}_{m}(\mathbf{A};\mathbf{u})}\) based on a square matrix sequence {Aj} and a vector sequence {uj}. Next we present a generalized Arnoldi procedure for generating an orthonormal basis of \({\mathcal{G}_{m}(\mathbf{A};\mathbf{u})}\). By applying the projection and the refined technique, we derive a restarted generalized Arnoldi method and a restarted refined generalized Arnoldi method for solving a large-scale polynomial eigenvalue problem (PEP). These two methods are applied to solve the PEP directly. Hence they preserve essential structures and properties of the PEP. Furthermore, restarting reduces the storage requirements. Some theoretical results are presented. Numerical tests report the effectiveness of these methods.

Iterative method for dynamic condensation combined with substructuring scheme

An iterated improved reduced system (IIRS) procedure combined with substructuring scheme for both undamped and nonclassically damped structures is presented. Iterated IIRS method is an efficient reduction technique because the highly accurate eigenproperties from the repeatedly updated condensed matrices can be obtained without consuming expensive computational cost. However, single domain direct approach of this method to large structures requires much computational resources and even makes analysis intractable in the case only limited computer storage is available. These problems can be overcome by combining the substructuring scheme with IIRS procedure. The newly developed IIRS method combined with a substructuring scheme can provide an efficient methodology for large-scale eigenvalue problems. The validation of the present method and the evaluation of computational efficiency are demonstrated through the numerical examples.

Algorithm 873

A MATLAB 6.0 implementation of the LSTRS method is presented. LSTRS was described in Rojas et al. [2000]. LSTRS is designed for large-scale quadratic problems with one norm constraint. The method is based on a reformulation of the trust-region subproblem as a parameterized eigenvalue problem, and consists of an iterative procedure that finds the optimal value for the parameter. The adjustment of the parameter requires the solution of a large-scale eigenvalue problem at each step. LSTRS relies on matrix-vector products only and has low and fixed storage requirements, features that make it suitable for large-scale computations. In the MATLAB implementation, the Hessian matrix of the quadratic objective function can be specified either explicitly, or in the form of a matrix-vector multiplication routine. Therefore, the implementation preserves the matrix-free nature of the method. A description of the LSTRS method and of the MATLAB software, version 1.2, is presented. Comparisons with other techniques and applications of the method are also included. A guide for using the software and examples are provided.

Improvement of substructuring reduction technique for large eigenproblems using an efficient dynamic condensation method

An accelerated substructuring reduction procedure for the iterated improved reduced system (IIRS) method is proposed. The iterated IIRS method can be combined with a substructuring scheme to provide an efficient methodology for large-scale eigenvalue problems. Not only can it reduce eigenvalue analysis errors through successive iterations, but the accuracy of the eigenanalysis is not sensitive to the selected master degrees of freedom. In practical structural eigenproblems, reducing the number of iterations can save a great deal of computation cost. The present substructuring technique modifies the iterative form of the transformation matrices in each substructure to achieve faster convergence. Applications of the present method to two numerical examples demonstrate that the proposed method can obtain lower eigensolutions of structures more accurately and efficiently, as compared with those of the current substructuring technique.

Parameter extraction of interdigital slow-wave coplanar waveguide circuits using finite difference frequency domain algorithm

The slow wave effect can be obtained by a capacitively loaded structure with a symmetrical interdigital line connected on both sides of the coplanar waveguide (CPW) central line. The ferroelectric thin film with high dielectric constant can reduce the size of circuit and make it possible to realize tunable devices such as filter by applying voltage on it. Actually, this kind of slow wave structure is a periodic guided-wave structure and can be analyzed by using classic finite difference frequency domain (FDFD) method for periodic guided-wave structures. However, the very compact slow-wave structures will usually result in simulation errors when the classic FDFD method is adopted, which will lead to a nonsymmetrical generalized eigenvalue problem. In this article, the shift-and-invert (SI) Arnoldi method is used to directly resolve this nonsymmetrical generalized eigenvalue problem. As a result, the accuracy of FDFD algorithm is improved. Especially for the large scale eigenvalue problem, SI method can also have a very fast speed of calculation. By means of its complex propagation constant obtained from simulation, one can extract circuit parameters of the interdigital capacitor. Consequently, one can analyze and design relevant resonators and filters in a quick and accurate manner, which are constructed with such interdigital slow wave structures. © 2008 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2008.

On Large-Scale Diagonalization Techniques for the Anderson Model of Localization

We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely, the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques in the implicitly restarted Lanczos method and in the Jacobi-Davidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete $LDL^T$ factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude.

Parallel solution of large-scale eigenvalue problem for master equation in protein folding dynamics

It is known that a master equation characterizes time evolution of trajectories and transition of states in protein folding dynamics. Solution of the master equation may require calculating eigenvalues for the corresponding eigenvalue problem. In this paper, we numerically study the folding rate for a dynamic problem of protein folding by solving a large-scale eigenvalue problem. Three methods, the implicitly restarted Arnoldi, Jacobi–Davidson, and QR methods are employed in solving the corresponding large-scale eigenvalue problem for the transition matrix of master equation. Comparison shows that the QR method demands tremendous computing resource when the length of sequence L > 10 due to extremely large size of matrix and CPU time limitation. The Jacobi–Davidson method may encounter convergence issue, for cases of L > 9 . The implicitly restarted Arnoldi method is suitable for solving problems among them. Parallelization of the implicitly restarted Arnoldi method is successfully implemented on a PC-based Linux cluster. The parallelization scheme mainly partitions the operation of matrix. For the Arnoldi factorization, we replicate the upper Hessenberg matrix H m for each processor, and distribute the set of Arnoldi vectors V m among processors. Each processor performs its own operation. The algorithm is implemented on a PC-based Linux cluster with message passing interface (MPI) libraries. Numerical experiment performing on our 32-nodes PC-based Linux cluster shows that the maximum difference among processors is within 10%. A 23-times speedup and 72% parallel efficiency are achieved when the matrix size is greater than 2 × 10 6 on the 32-nodes PC-based Linux cluster. This parallel approach enables us to explore large-scale dynamics of protein folding.

Error analysis of symplectic Lanczos method for Hamiltonian eigenvalue problem

A rounding error analysis for the symplectic Lanczos method is given to solve the large-scale sparse Hamiltonian eigenvalue problem. If no breakdown occurs in the method, then it can be shown that the Hamiltonian structure preserving requirement does not destroy the essential feature of the nonsymmetric Lanczos algorithm. The relationship between the loss of J-orthogonality among the symplectic Lanczos vectors and the convergence of the Ritz values in the symmetric Lanczos algorithm is discussed. It is demonstrated that under certain assumptions the computed J-orthogonal Lanczos vectors lose the J-orthogonality when some Ritz values begin to converge. Our analysis closely follows the recent works of Bai and Fabbender.

Analysis of vibration levels of large structural system with recursive component mode synthesis method: Theory and convergence

The component mode synthesis (CMS) method has been extensively used in industries. However, industry finite-element (FE) models need a more efficient CMS method for satisfactory performance since the size of FE models needs to be increased for a more accurate analysis. Recently, the recursive component mode synthesis (RCMS) method was introduced to solve large-scale eigenvalue problem efficiently. This article focuses on the convergence of the RCMS method with respect to different parameters, and evaluates the accuracy and performance compared with the Lanczos method.

Numerical solutions of a master equation for protein folding kinetics

The numerical solution of a master equation involves the calculation of eigenpairs for the corresponding transition matrix. In this paper, we computationally study the folding rate for a kinetics problem of protein folding by solving a large-scale eigenvalue problem. Three numerical methods, the implicitly restarted Arnoldi, the Jacobi-Davidson, and the QR methods are applied to solve the corresponding large-scale eigenvalue problem of the transition matrix of the master equation. Comparison among three methods is performed in terms of the computational efficiency. It is found that the QR method demands tremendous computing resource when the length of sequence L > 10 due to the extremely large size of matrix and CPU time limitation. The Jacobi-Davidson method may encounter convergence issues, for some testing cases with L > 9. Among the three solution methods the implicitly restarted Arnoldi method is suitable for solving the problem. Numerical examples with various residues are investigated.

On Large-Scale Diagonalization Techniques for the Anderson Model of Localization

We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques in the implicitly restarted Lanczos method and in the Jacobi--Davidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete LDLT factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude.

Gearbox vibratory analysis using carrying, coupling and slave substructures

Vibration analysis of complex structures is usually accomplished by analysing its finite element model. When the total number of degrees of freedom in the model is too large, we deal with large-scale eigenvalue problems. The classical approach for the solution of such problems consists in reducing the number of unknowns, allowing reducing the computational cost of eigensolver. Component mode synthesis or dynamic substructuring methods are appropriate tools for this reduction. In this paper, a novel approach in dynamic substructuring for numerical simulation of complex structures is presented and three type of substructure are considered. (1) Carrying substructure (plates and beams), (2) coupling substructure (organs joining carrying substructures), (3) slave substructure (elements connected to the carrying substructure). This approach is based on the use of the Craig-Bampton decomposition of the admissible displacement field. A numerical example is presented. A gearbox with elastic casing, two transmission shafts and one helical gear pair is analyzed and concepts for reduced noise and vibration are identified.

The second-order biorthogonalization procedure and its application to quadratic eigenvalue problems

Given a pair of matrices and starting vectors, we present a procedure to generate the biorthonormal basis of the second-order right and left Krylov subspaces. The application is to solve the large-scale quadratic eigenvalue problems via oblique projection technique. This method can take full advantage of the sparseness of large-scale system as well as the superior convergence behavior of Krylov subspace based methods by implicit linearization, which makes the solution acceptable in terms of both cost and time.

Solving large-scale eigenvalue problems in SciDAC applications

Large-scale eigenvalue problems arise in a number of DOE applications. This paper provides an overview of the recent development of eigenvalue computation in the context of two SciDAC applications. We emphasize the importance of Krylov subspace methods, and point out its limitations. We discuss the value of alternative approaches that are more amenable to the use of preconditioners, and report the progress on using the multi-level algebraic sub-structuring techniques to speed up eigenvalue calculation. In addition to methods for linear eigenvalue problems, we also examine new approaches to solving two types of non-linear eigenvalue problems arising from SciDAC applications.

9033 大規模固有値問題のグリッド環境向き並列解法(GS-D 一般セッション(行列計算))

9033 大規模固有値問題のグリッド環境向き並列解法(GS-D 一般セッション(行列計算))

An Algebraic Substructuring Method for Large-Scale Eigenvalue Calculation

This paper is concerned with solving large-scale eigenvalue problems by algebraic substructuring. Algebraic substructuring refers to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to form approximate solutions to the original problem. Through an algebraic analysis, we identify critical conditions under which a simple version of algebraic substructuring works well. This particular version of substructuring is identical to the component mode synthesis (CMS) method (see [R. R. Craig and M. C. C. Bampton, {\em Coupling of substructures for dynamic analysis}, AIAA J., 6 (1968), pp. 1313--1319] and [W. C. Hurty, {\em Vibrations of structure systems by component-mode synthesis}, J. Engrg. Mech., 86 (1960), pp. 51--69]) when the matrix reordering is based on a geometric partitioning of the computational domain. We observe an interesting connection between the accuracy of an approximate eigenpair obtained through substructuring and the distribution of the components of eigenvectors of a canonical matrix pencil congruent to the original problem. A priori error bounds for the smallest eigenpair approximation are developed. This development leads to a simple heuristic for choosing spectral components (modes) from each substructure. The effectiveness of such a heuristic is demonstrated with numerical examples. We show that algebraic substructuring can be effectively used to solve a generalized eigenvalue problem arising from the finite element analysis of an accelerator structure.