Nonsymmetric Eigenvalue Problem Research Articles

Definite Quadratic Eigenvalue Problems

Free vibrations of fluid-solids structures are governed by a nonsymmetric eigenvalue problem. This problem can be transformed into a definite quadratic eigenvalue problem. The present paper considers properties of definite problems and variational characterization for definite quadratic eigenvalue problem. We propose improvement of existing methods which determine whether a quadratic pencil Q(λ) is definite. The improved method is reflected in a better localization of start parameters μ and ξ where Q(μ) > 0 >Q(ξ) and μ >ξ.

Kronecker-Schur-Newton Method for the Matrix Inverse Roots

The computation of matrix inverse roots arises in solving nonlinear matrix equations, evaluating non-symmetric eigenvalue problems, computing some matrix functions, Control theory and several other areas of applications. It is possible to approximate the matrix inverse pth roots by exploiting iterative schemes. In this work, we propose an algorithm to compute principal matrix inverse pth root by employing Schur factorization and operators. Numerical experiments are performed to illustrate the properties of the modified method.

On the use of matrix-free shift-invert strategies for global flow instability analysis

A novel time-stepping shift-invert algorithm for linear stability analysis of laminar flows in complex geometries is presented. This method, based on a Krylov subspace iteration, enables the solution of complex non-symmetric eigenvalue problems in a matrix-free framework. Validations and comparisons to the classical exponential method have been performed in three different cases: (i) stenotic flow, (ii) backward-facing step and (iii) lid-driven swirling flow. Results show that this new approach speeds up the required Krylov subspace iterations and has the capability of converging to specific parts of the global spectrum. It is shown that, although the exponential method remains the method of choice if leading eigenvalues are sought, the performance of the present method could be dramatically improved with the use of a preconditioner. In addition, as opposed to other methods, this strategy can be directly applied to any time-stepper, regardless of the temporal or spatial discretization of the latter.

Fine-grained bulge-chasing kernels for strongly scalable parallel QR algorithms

The bulge-chasing kernel in the small-bulge multi-shift QR algorithm for the non-symmetric dense eigenvalue problem becomes a sequential bottleneck when the QR algorithm is run in parallel on a multicore platform with shared memory. The duration of each kernel invocation is short, but the critical path of the QR algorithm contains a long sequence of calls to the bulge-chasing kernel. We study the problem of parallelizing the bulge-chasing kernel itself across a handful of processor cores in order to reduce the execution time of the critical path. We propose and evaluate a sequence of four algorithms with varying degrees of complexity and verify that a pipelined algorithm with a slowly shifting block column distribution of the Hessenberg matrix is superior. The load-balancing problem is non-trivial and computational experiments show that the load-balancing scheme has a large impact on the overall performance. We propose two heuristics for the load-balancing problem and also an effective optimization method based on local search. Numerical experiments show that speed-ups are obtained for problems as small as 40x40 on two different multicore architectures.

Relative Perturbation Theory for Diagonally Dominant Matrices

In this paper, strong relative perturbation bounds are developed for a number of linear algebra problems involving diagonally dominant matrices. The key point is to parameterize diagonally dominant matrices using their off-diagonal entries and diagonally dominant parts and to consider small relative componentwise perturbations of these parameters. This allows us to obtain new relative perturbation bounds for the inverse, the solution to linear systems, the symmetric indefinite eigenvalue problem, the singular value problem, and the nonsymmetric eigenvalue problem. These bounds are much stronger than traditional perturbation results, since they are independent of either the standard condition number or the magnitude of eigenvalues/singular values. Together with previously derived perturbation bounds for the LDU factorization and the symmetric positive definite eigenvalue problem, this paper presents a complete and detailed account of relative structured perturbation theory for diagonally dominant matrices.

An iterative adaptive hp-FEM method for non-symmetric elliptic eigenvalue problems

We present a novel adaptive higher-order finite element (hp-FEM) algorithm to solve non-symmetric elliptic eigenvalue problems. This is an extension of our prior work on symmetric elliptic eigenvalue problems. The method only needs to make one call to a generalized eigensolver on the coarse mesh, and then it employs Newton’s or Picard’s methods to resolve adaptively a selected eigenvalue–eigenvector pair. The fact that the method does not need to make repeated calls to a generalized eigensolver not only makes it very efficient, but it also eliminates problems that pose great complications to adaptive algorithms, such as eigenvalue reordering or returning arbitrary linear combinations of eigenvectors associated with the same eigenvalue. New theoretical and numerical results for the non-symmetric case are presented.

Order 104 speedup in global linear instability analysis using matrix formation

A unified solution framework is presented for one-, two- or three-dimensional complex non-symmetric eigenvalue problems, respectively governing linear modal instability of incompressible fluid flows in rectangular domains having two, one or no homogeneous spatial directions. The solution algorithm is based on subspace iteration in which the spatial discretization matrix is formed, stored and inverted serially. Results delivered by spectral collocation based on the Chebyshev-Gauss–Lobatto (CGL) points and a suite of high-order finite-difference methods comprising the previously employed for this type of work Dispersion-Relation-Preserving (DRP) and Padé finite-difference schemes, as well as the Summation-by-parts (SBP) and the new high-order finite-difference scheme of order q (FD-q) have been compared from the point of view of accuracy and efficiency in standard validation cases of temporal local and BiGlobal linear instability. The FD-q method has been found to significantly outperform all other finite difference schemes in solving classic linear local, BiGlobal, and TriGlobal eigenvalue problems, as regards both memory and CPU time requirements. Results shown in the present study disprove the paradigm that spectral methods are superior to finite difference methods in terms of computational cost, at equal accuracy, FD-q spatial discretization delivering a speedup of O(104). Consequently, accurate solutions of the three-dimensional (TriGlobal) eigenvalue problems may be solved on typical desktop computers with modest computational effort.

Minimization Principles for the Linear Response Eigenvalue Problem I: Theory

We present two theoretical results for the linear response eigenvalue problem. The first result is a minimization principle for the sum of the smallest eigenvalues with the positive sign. The second result is Cauchy-like interlacing inequalities. Although the linear response eigenvalue problem is a nonsymmetric eigenvalue problem, these results mirror the well-known trace minimization principle and Cauchy's interlacing inequalities for the symmetric eigenvalue problem.

Function Value Recovery and Its Application in Eigenvalue Problems

Function value recovery techniques for linear finite elements are discussed. Using the recovered function and its gradient, we are able to enhance the eigenvalue approximation and increase its convergence rate to $h^{2\alpha}$, where $\alpha > 1$ is the superconvergence rate of the recovered gradient. This is true in both symmetric and nonsymmetric eigenvalue problems.

Acceleration of Hessenberg Reduction for Nonsymmetric Eigenvalue Problems in a Hybrid CPU-GPU Computing Environment

Solution of large-scale dense nonsymmetric eigenvalue problem is required in many areas of scientific and engineering computing, such as vibration analysis of automobiles and analysis of electronic diffraction patterns. In this study, we focus on the Hessenberg reduction step and consider accelerating it in a hybrid CPU-GPU computing environment. Considering that the Hessenberg reduction algorithm consists almost entirely of BLAS (Basic Linear Algebra Subprograms) operations, we propose three approaches for distributing the BLAS operations between CPU and GPU. Among them, the third approach, which assigns small-size BLAS operations to CPU and distributes large-size BLAS operations between CPU and GPU in some optimal manner, was found to be consistently faster than the other two approaches. On a machine with an Intel Core i7 processor and an NVIDIA Tesla C1060 GPU, this approach achieved 3.2 times speedup over the CPU-only case when computing the Hessenberg form of a 8,192 × 8,192 real matrix.

Symmetrization Procedure for Structure-Fluid Cavity System Involving Pseudostatic Correction

This paper deals with a pressure-displacement formulation based on the finite-element method of an elastoacoustic coupling problem. Since directly solving the resulting system isCPU intensive for large models, the solution is usually based on only a few uncoupled structural modes in vacuum and rigid cavity modes. However this classical pressuredisplacement formulation leads to nonsymmetric eigenvalue problems. Furthermore, if this method is accurate for weak coupling models, significant errors occur when the truncated modes are coupled to the ones used in the analysis. The present paper suggests a method combining the pseudostatic correction of the truncated modes in modal analysis with a symmetrization technique of the eigenvalue problems. It allows the calculation of the real coupled modes and frequencies, as well as the computation of an accurate elastoacoustic response, even for the strong coupling of a structure with a heavy fluid (liquid). It is less CPU expensive than the classical method, because it leads to a diagonalized system.

Accelerating the reduction to upper Hessenberg, tridiagonal, and bidiagonal forms through hybrid GPU-based computing

We present a Hessenberg reduction (HR) algorithm for hybrid systems of homogeneous multicore with GPU accelerators that can exceed 25× the performance of the corresponding LAPACK algorithm running on current homogeneous multicores. This enormous acceleration is due to proper matching of algorithmic requirements to architectural strengths of the system’s hybrid components. The results described in this paper are significant because the HR has not been properly accelerated before on homogeneous multicore architectures, and it plays a significant role in solving non-symmetric eigenvalue problems. Moreover, the ideas from the hybrid HR are used to develop a hybrid tridiagonal reduction algorithm (for symmetric eigenvalue problems) and a bidiagonal reduction algorithm (for singular value decomposition problems). Our approach demonstrates a methodology that streamlines the development of a large and important class of algorithms on modern computer architectures of multicore and GPUs. The new algorithms can be directly used in the software stack that relies on LAPACK.

A-posteriori residual bounds for Arnoldi’s methods for nonsymmetric eigenvalue problems

Convergence of the implicitly restarted Arnoldi (IRA) method for nonsymmetric eigenvalue problems has often been studied by deriving bounds for the angle between a desired eigenvector and the Krylov projection subspace. Bounds for residual norms of approximate eigenvectors have been less studied and this paper derives a new a-posteriori residual bound for nonsymmetric matrices with simple eigenvalues. The residual vector is shown to be a linear combination of exact eigenvectors and a residual bound is obtained as the sum of the magnitudes of the coefficients of the eigenvectors. We numerically illustrate that the convergence of the residual norm to zero is governed by a scalar term, namely the last element of the wanted eigenvector of the projected matrix. Both cases of convergence and non-convergence are illustrated and this validates our theoretical results. We derive an analogous result for implicitly restarted refined Arnoldi (IRRA) and for this algorithm, we numerically illustrate that convergence is governed by two scalar terms appearing in the linear combination which drives the residual norm to zero. We provide a set of numerical results that validate the residual bounds for both variants of Arnoldi methods.

A Novel Parallel QR Algorithm for Hybrid Distributed Memory HPC Systems

A novel variant of the parallel QR algorithm for solving dense nonsymmetric eigenvalue problems on hybrid distributed high performance computing systems is presented. For this purpose, we introduce the concept of multiwindow bulge chain chasing and parallelize aggressive early deflation. The multiwindow approach ensures that most computations when chasing chains of bulges are performed in level 3 BLAS operations, while the aim of aggressive early deflation is to speed up the convergence of the QR algorithm. Mixed MPI-OpenMP coding techniques are utilized for porting the codes to distributed memory platforms with multithreaded nodes, such as multicore processors. Numerous numerical experiments confirm the superior performance of our parallel QR algorithm in comparison with the existing ScaLAPACK code, leading to an implementation that is one to two orders of magnitude faster for sufficiently large problems, including a number of examples from applications.

A new method for accelerating Arnoldi algorithms for large scale eigenproblems

We propose a new method for accelerating the convergence of the implicitly restarted Arnoldi (IRA) algorithm for the solution of large sparse nonsymmetric eigenvalue problems. A new relationship between the residual of the current step and the residual in the previous step is derived and we use this relationship to develop a technique for dynamically switching the Krylov subspace dimension at successive cycles. We give numerical results for various difficult nonsymmetric eigenvalue problems that demonstrate the capability of the dynamic switching strategy for significantly accelerating the convergence of Arnoldi algorithms. For some large scale difficult eigenvalue problems that arise in the fields of computational fluid dynamics, electrical engineering and materials science, our strategy leads to significant reductions in the number of matrix–vector products, orthogonalization costs and computational time.

Extracting the left and right critical eigenvectors from the LDU-decomposed non-symmetric Jacobian matrix in stability problems

In the previous publications of the authors, an eigenanalysis-free computational procedure has been proposed to extract the bifurcation buckling mode(s) from the LDL T -decomposed symmetric stiffness matrix in the vicinity of a stability point. Any eigensolver, for instance, inverse iteration or subspace method, is not necessary. The procedure has been verified in numerical examples and well works in multiple and clustered bifurcation problems too. This present paper will extend the eigenanalysis-free procedure to the LDU-decomposed non-symmetric Jacobian matrix, from which both left and right critical eigenvectors relevant to the stability point may be extracted in a similar way. The idea is mathematical and totally independent of the physical problem to be solved, so that it is applicable to any non-symmetric square matrix in stability problems including plasticity with non-associated flow rules, contact and fluid-structure interaction. The linear-algebraic background of non-symmetric eigenvalue problems is firstly described. The present paper will then mention the role play of the left and right critical eigenvectors in stability analysis and the eigenanalysis-free LDU-procedure is proposed. Numerical examples of elastoplastic bifurcation are illustrated for verification and discussion. In APPENDICES, a bench model visualizes the mechanical meaning of the left and right critical singular vectors of a rectangular matrix.

Preconditioning constrained eigenvalue problems

The purpose of our paper is to introduce a robust preconditioning scheme for the numerical solution of the leftmost eigenvalues and corresponding eigenvectors of a constrained eigenvalue problem. This constrained eigenvalue problem is congruent to a nonsymmetric eigenvalue problem with nontrivial Jordan blocks associated with infinite eigenvalues. The proposed preconditioning scheme is relevant to the application of Krylov subspace methods and preconditioned eigensolvers. The two key results are a semi-orthogonal decomposition and a transformation process that implicitly combines a preconditioning step followed by abstract projection onto the subspace associated with the finite eigenvalues. Numerical results demonstrate the effectiveness of the preconditioning scheme.

A nonsymmetric state-variable decomposition for modal analysis

A modal decomposition strategy based on state-variable ensembles is formulated. A nonsymmetric, generalized eigenvalue problem is constructed. The data-based eigenvalue problem is related to the generalized eigenvalue problem associated with free-vibration solutions of the state-variable formulation of linear multi-degree-of-freedom systems. For linear free-response data, the inverse-transpose of the eigenvector matrix converges to the state-variable modal eigenvectors, and the eigenvalues of the nonsymmetric eigenvalue problem approximate those of the state-variable model. As such, the eigenvalues lead to estimates of frequencies and modal damping. The interpretation holds for linear systems with multi-modal free responses, whether damping is small or somewhat large, modal or nonmodal, and without the need of input data.

A Parallel Implementation of the Nonsymmetric QR Algorithm for Distributed Memory Architectures

One approach to solving the nonsymmetric eigenvalue problem in parallel is to parallelize the QR algorithm. Not long ago, this was widely considered to be a hopeless task. Recent efforts have led to significant advances, although the methods proposed up to now have suffered from scalability problems. This paper discusses an approach to parallelizing the QR algorithm that greatly improves scalability. A theoretical analysis indicates that the algorithm is ultimately not scalable, but the nonscalability does not become evident until the matrix dimension is enormous. Experiments on the Intel Paragon system, the IBM SP2 supercomputer, the SGI Origin 2000, and the Intel ASCI Option Red supercomputer are reported.

A posteriori error control for finite element approximations of elliptic eigenvalue problems

We develop a new approach to a posteriori error estimation for Galerkin finite element approximations of symmetric and nonsymmetric elliptic eigenvalue problems. The idea is to embed the eigenvalue approximation into the general framework of Galerkin methods for nonlinear variational equations. In this context residual-based a posteriori error representations are available with explicitly given remainder terms. The careful evaluation of these error representations for the concrete situation of an eigenvalue problem results in a posteriori error estimates for the approximations of eigenvalues as well as eigenfunctions. These suggest local error indicators that are used in the mesh refinement process.