Arnoldi Algorithm Research Articles

Structure-Preserving Algorithms for Palindromic Quadratic Eigenvalue Problems Arising from Vibration of Fast Trains

In this paper, based on Patel's algorithm (1993), we propose a structure-preserving algorithm for solving palindromic quadratic eigenvalue problems (QEPs). We also show the relationship between the structure-preserving algorithm and the URV-based structure-preserving algorithm by Schröder (2007). For large sparse palindromic QEPs, we develop a generalized $\top$-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi algorithm for solving the resulting $\top$-skew-Hamiltonian pencils. Numerical experiments show that our proposed structure-preserving algorithms perform well on the palindromic QEP arising from a finite element model of high-speed trains and rails.

The Arnoldi Eigenvalue Iteration with Exact Shifts Can Fail

The restarted Arnoldi algorithm, implemented in the ARPACK software library and MATLAB's eigs command, is among the most common means of computing select eigenvalues and eigenvectors of a large, sparse matrix. To assist convergence, a starting vector is repeatedly refined via the application of automatically constructed polynomial filters whose roots are known as “exact shifts.” Though Sorensen proved the success of this procedure under mild hypotheses for Hermitian matrices, a convergence proof for the non-Hermitian case has remained elusive. The present note describes a class of examples for which the algorithm fails in the strongest possible sense; that is, the polynomial filter used to restart the iteration deflates the eigenspace one is attempting to compute.

The critical layer in pipe flow at high Reynolds number

We report the computation of a family of travelling wave solutions of pipe flow up to Re=75000. As in all lower branch solutions, streaks and rolls feature prominently in these solutions. For large Re, these solutions develop a critical layer away from the wall. Although the solutions are linearly unstable, the two unstable eigenvalues approach 0 as Re-->infinity at rates given by Re-0.41 and Re-0.87; surprisingly, the solutions become more stable as the flow becomes less viscous. The formation of the critical layer and other aspects of the Re-->infinity limit could be universal to lower branch solutions of shear flows. We give implementation details of the GMRES-hookstep and Arnoldi iterations used for computing these solutions and their spectra, while pointing out the new aspects of our method.

On orthogonal reduction to Hessenberg form with small bandwidth

Numerous algorithms in numerical linear algebra are based on the reduction of a given matrix A to a more convenient form. One of the most useful types of such reduction is the orthogonal reduction to (upper) Hessenberg form. This reduction can be computed by the Arnoldi algorithm. When A is Hermitian, the resulting upper Hessenberg matrix is tridiagonal, which is a significant computational advantage. In this paper we study necessary and sufficient conditions on A so that the orthogonal Hessenberg reduction yields a Hessenberg matrix with small bandwidth. This includes the orthogonal reduction to tridiagonal form as a special case. Orthogonality here is meant with respect to some given but unspecified inner product. While the main result is already implied by the Faber-Manteuffel theorem on short recurrences for orthogonalizing Krylov sequences (see Liesen and Strakos, SIAM Rev 50:485–503, 2008), we consider it useful to present a new, less technical proof. Our proof utilizes the idea of a “minimal counterexample”, which is standard in combinatorial optimization, but rarely used in the context of linear algebra.

Hybrid modal reduction for poroelastic materials

A modal-like projection method for poroelastic materials is proposed and implemented for finite element calculations. Non-physical Dirichlet conditions are imposed at the junction interface, involving constrained fluid displacements and free solid displacements. The ( u s , U f ) formulation is used. The resulting frequency-dependent eigenproblem is solved without simplification using the non-linear Arnoldi algorithm. The projection subspace is spanned by calculated dynamic modes and fluid static boundary functions. A convergence study is performed and results are compared to classical Craig and Bampton and MacNeal approaches. The hybrid basis proves to be efficient. To cite this article: C. Batifol et al., C. R. Mecanique 336 (2008).

Non-normal Lanczos methods for quantum scattering

This article presents a new complex absorbing potential (CAP) block Lanczos method for computing scattering eigenfunctions and reaction probabilities. The method reduces the problem of computing energy eigenfunctions to solving two energy dependent systems of equations. An energy independent block Lanczos factorization casts the system into a block tridiagonal form, which can be solved very efficiently for all energies. We show that CAP-Lanczos methods exhibit instability due to the non-normality of CAP Hamiltonians and may break down for some systems. The instability is not due to loss of orthogonality but to non-normality of the Hamiltonian matrix. While use of a Woods-Saxon exponential CAP-as opposed to a polynomial CAP-reduced non-normality, it did not always ensure convergence. Our results indicate that the Arnoldi algorithm is more robust for non-normal systems and less prone to break down. An Arnoldi version of our method is applied to a nonadiabatic tunneling Hamiltonian with excellent results, while the Lanczos algorithm breaks down for this system.

A new shift scheme for the harmonic Arnoldi method

The implicitly restarted harmonic Arnoldi algorithm by Morgan used those unwanted harmonic Ritz values as shifts—called Morgan’s harmonic shifts. In this paper, a new shift scheme is given for the harmonic Arnoldi algorithm. We first analyze the harmonic Ritz values w k + 1 , … , w m of A from the orthogonal complement of span of those wanted harmonic Ritz vectors with respect to K m ( A , v 1 ) , then present an implicitly restarted harmonic Arnoldi algorithm with w k + 1 , … , w m as shifts. Finally, through the numerical experiments, we mainly draw comparisons on our algorithm and Morgan’s algorithm, and show our algorithm often performed better than Morgan’s one.

Conjugate-normal matrices with conjugate-normal submatrices

In studying the reduction of a complex n × n matrix A to its Hessenberg form by the Arnoldi algorithm, T. Huckle discovered that an irreducible Hessenberg normal matrix with a normal leading principal m × m submatrix, where 1 < m < n, actually is tridiagonal. We prove a similar assertion for the conjugate-normal matrices, which play the same role in the theory of unitary congruences as the conventional normal matrices in the theory of unitary similarities. This fact is stated as a purely matrix-theoretic theorem, without any reference to Arnoldi-like algorithms. Bibliography: 2 titles.

Block Krylov–Schur method for large symmetric eigenvalue problems

Stewart’s Krylov–Schur algorithm offers two advantages over Sorensen’s implicitly restarted Arnoldi (IRA) algorithm. The first is ease of deflation of converged Ritz vectors, the second is the avoidance of the potential forward instability of the QR algorithm. In this paper we develop a block version of the Krylov–Schur algorithm for symmetric eigenproblems. Details of this block algorithm are discussed, including how to handle rank deficient cases and how to use varying block sizes. Numerical results on the efficiency of the block Krylov–Schur method are reported.

Skew-symmetric methods for nonsymmetric linear systems with multiple right-hand sides

By transforming nonsymmetric linear systems to the extended skew-symmetric ones, we present the skew-symmetric methods for solving nonsymmetric linear systems with multiple right-hand sides. These methods are based on the block and global Arnoldi algorithm which is formed by implementing orthogonal projections of the initial matrix residual onto a matrix Krylov subspace. The algorithms avoid the tediously long Arnoldi process and highly reduce expensive storage. Numerical experiments show that these algorithms are effective and give better practical performances than global GMRES for solving nonsymmetric linear systems with multiple right-hand sides.

Adaptive Rational Interpolation: Arnoldi and Lanczos-like Equations

The Arnoldi and Lanczos algorithms, which belong to the class of Krylov subspace methods, are increasingly used for model reduction of large-scale systems. The standard versions of the algorithms tend to create reduced order models that poorly approximate low frequency dynamics. Rational Arnoldi and Lanczos algorithms produce reduced models that approximate dynamics at various frequencies. This paper tackles the issue of developing simple Arnoldi and Lanczos equations for the rational case. This allows a simple error analysis to be carried out for both algorithms and permits the development of computationally efficient model reduction algorithms, where the frequencies at which the dynamics are to be matched can be updated adaptively.

An inexact Krylov–Schur algorithm for the unitary eigenvalue problem

We present an efficient inexact implicitly restarted Arnoldi algorithm to find a few eigenpairs of large unitary matrices. The approximating Krylov spaces are built using short-term recurrences derived from Gragg’s isometric Arnoldi process. The implicit restarts are done by the Krylov–Schur methodology of Stewart. All of the operations of the restart are done in terms of the Schur parameters generated by the isometric Arnoldi process. Numerical results confirm the effectiveness of the algorithm.

Model-order reductions for MIMO systems using global Krylov subspace methods

This paper presents theoretical foundations of global Krylov subspace methods for model order reductions. This method is an extension of the standard Krylov subspace method for multiple-inputs multiple-outputs (MIMO) systems. By employing the congruence transformation with global Krylov subspaces, both one-sided Arnoldi and two-sided Lanczos oblique projection methods are explored for both single expansion point and multiple expansion points. In order to further reduce the computational complexity for multiple expansion points, adaptive-order multiple points moment matching algorithms, or the so-called rational Krylov space method, are also studied. Two algorithms, including the adaptive-order rational global Arnoldi (AORGA) algorithm and the adaptive-order global Lanczos (AOGL) algorithm, are developed in detail. Simulations of practical dynamical systems will be conducted to illustrate the feasibility and the efficiency of proposed methods.

Dimensionally reduced Krylov subspace model reduction for large scale systems

This paper introduces a new mathematical approach that combines the concepts of dimensional reduction and Krylov subspace techniques for use in the model reduction problem for large-scale systems. Krylov subspace methods for model reduction uses the Arnoldi algorithm in order to construct the bases for controllability, observability, and oblique subspaces of state space realization. The newly developed algorithm uses principal component analysis along with Krylov oblique projection model reduction technique to provide computationally efficient and inexpensive model reduction method. To demonstrate the effectiveness of the proposed hybrid scheme the residual error, forward error and stability response analyses have been performed for various randomly generated large-scale systems.

The Ar 3 molecular system

We employ the three-dimensional Faddeev equations in the total angular momentum representation to investigate the contributions stemming from the various long-range three-body forces to the low-lying vibrational states of the Ar3 molecule. The resulting large matrices were handled using the Arnoldi's algorithm and a background potential which can eliminate the plethora of unphysical eigenvalues close to unity. We demonstrated that our method is suited for calculations in relatively heavy trimers without resorting to an explicit partial wave decomposition.

Parallel Arnoldi eigensolvers with enhanced scalability via global communications rearrangement

This paper presents several new variants of the single-vector Arnoldi algorithm for computing approximations to eigenvalues and eigenvectors of a non-symmetric matrix. The context of this work is the efficient implementation of industrial-strength, parallel, sparse eigensolvers, in which robustness is of paramount importance, as well as efficiency. For this reason, Arnoldi variants that employ Gram-Schmidt with iterative reorthogonalization are considered. The proposed algorithms aim at improving the scalability when running in massively parallel platforms with many processors. The main goal is to reduce the performance penalty induced by global communications required in vector inner products and norms. In the proposed algorithms, this is achieved by reorganizing the stages that involve these operations, particularly the orthogonalization and normalization of vectors, in such a way that several global communications are grouped together while guaranteeing that the numerical stability of the process is maintained. The numerical properties of the new algorithms are assessed by means of a large set of test matrices. Also, scalability analyses show a significant improvement in parallel performance.

A comparison between one-sided and two-sided Arnoldi-based model order reduction (MORe) techniques for fully coupled structural-acoustic analysis

Reduced order models are developed for fully coupled structural-acoustic unsymmetric matrix models, resulting from Cragg’s displacement/pressure formulation, using Krylov subspace techniques. The reduced order model is obtained by applying a Galerkin and Petrov-Galerkin projection of the coupled system matrices, from a higher dimensional subspace to a lower dimensional subspace, while matching the moments of the coupled higher dimensional system. Two such techniques, based on the Arnoldi algorithm, focusing on one-sided and two-sided moment matching, are presented. To validate the numerical techniques, an ABAQUS coupled structural-acoustic Benchmark problem is chosen and solved using the direct approach. First, the physical problem is modeled using ANSYS FE package and compared with closed form solutions. Next, ANSYS results are compared with nodal velocities obtained by generating reduced order models via moment matching. The results show that the reduced order models give a very significant reduction in computational time, while preserving the desired accuracy of the solution. It is also shown that the accuracy of the one-sided method could be further improved by using two-sided methods, where the Arnoldi vectors are optimized for chosen outputs. The accuracy of the approaches, convergence models, and computational times are compared.

Use of near-breakdowns in the block Arnoldi method for solving large Sylvester equations

In this paper we construct an approximate solution to large Sylvester equations of the form AX+XB=CD^T. The construction uses a new variant of the block Arnoldi algorithm which exploits the near-breakdowns, that is, the near singularities in the generated basis. As a consequence, the algorithm eliminates the directions which do not contribute to the approximate solution by keeping in the generated basis only the ''active'' vectors detected by a criterion based on the residual associated with the approximate solution. The effectiveness of the proposed algorithm is demonstrated on several examples, including the case where the matrix B has a small or a large size.

Weighted FOM-inverse vector iteration method for computing a few smallest (largest) eigenvalues of pair ( A, B)

The generalized eigenvalue problem AX = λ BX has special properties when ( A, B) is a symmetric and positive definite pair. We have recently developed a new method for computing a few smallest (largest) eigenvalues of a symmetric positive definite problem AX = λ BX [H. Saberi Najafi, A. Refahi, FOM-inverse vector iteration method for computing a few smallest (largest) eigenvalues of pair ( A, B), Appl. Math., in press] the method used three parameters, which are weighted Arnoldi, Inner and outer iterations parameters. In this article those parameters have been optimized to increase the speed of convergence. The method has been tested by some numerical examples.

A Power–Arnoldi algorithm for computing PageRank

AbstractThe PageRank algorithm plays an important role in modern search engine technology. It involves using the classical power method to compute the principle eigenvector of the Google matrix representing the web link graph. However, when the largest eigenvalue is not well separated from the second one, the power method may perform poorly. This happens when the damping factor is sufficiently close to 1. Therefore, it is worth developing new techniques that are more sophisticated than the power method. The approach presented here, called Power–Arnoldi, is based on a periodic combination of the power method with the thick restarted Arnoldi algorithm. The justification for this new approach is presented. Numerical tests illustrate the efficiency and convergence behaviour of the new algorithm. Copyright © 2007 John Wiley &amp; Sons, Ltd.