Nonsymmetric Eigenvalue Problem Research Articles

Reduced basis method for non-symmetric eigenvalue problems: application to the multigroup neutron diffusion equations

In this article, we propose a reduced basis method for parametrized non-symmetric eigenvalue problems arising in the loading pattern optimization of a nuclear core in neutronics. To this end, we derive a posteriori error estimates for the smallest eigenvalue which is assumed to be simple and the associated left and right eigenvectors. The practical computation of these estimators requires the estimation of a constant called prefactor, which we can express as the spectral norm of some operator. We provide some elements of theoretical analysis which illustrate the link between the expression of the prefactor we obtain here and its well-known expression in the case of symmetric eigenvalue problems, either using the notion of numerical range of the operator, or via a perturbative analysis. Lastly, we propose a practical method in order to estimate this prefactor which yields interesting numerical results on actual test cases. We provide detailed numerical simulations on two-dimensional examples including a multigroup neutron diffusion equation.

A New Quotient and Iterative Detection Method in an Affine Krylov Subspace for Solving Eigenvalue Problems

The paper solves the eigenvalues of a symmetric matrix by using three novel algorithms developed in the m-dimensional affine Krylov subspace. The n-dimensional eigenvector is superposed by a constant shifting vector and an m-vector. In the first algorithm, the m-vector is derived as a function of eigenvalue by maximizing the Rayleigh quotient to generate the first characteristic equation, which, however, is not easy to determine the eigenvalues since its roots are not of simple ones, exhibiting turning points, spikes, and even no intersecting point to the zero line. To overcome that difficulty by the first algorithm, we propose the second characteristic equation through a new quotient with the inner product of the shifting vector to the eigen-equation. The Newton method and the fictitious time integration method are convergent very fast due to the simple roots of the second characteristic equation. For both symmetric and nonsymmetric eigenvalue problems solved by the third algorithm, we develop a simple iterative detection method to maximize the Euclidean norm of the eigenvector in terms of the eigen-parameter, of which the peaks of the response curve correspond to the eigenvalues. Through a few finer tunings to the smaller intervals sequentially, a very accurate eigenvalue and eigenvector can be obtained. The efficiency and accuracy of the proposed iterative algorithms are verified and compared to the Lanczos algorithm and the Rayleigh quotient iteration method.

Inexact rational Krylov subspace method for eigenvalue problems

AbstractAn inexact rational Krylov subspace method is studied to solve large‐scale nonsymmetric eigenvalue problems. Each iteration (outer step) of the rational Krylov subspace method requires solution to a shifted linear system to enlarge the subspace, performed by an iterative linear solver for large‐scale problems. Errors are introduced at each outer step if these linear systems are solved approximately by iterative methods (inner step), and they accumulate in the rational Krylov subspace. In this article, we derive an upper bound on the errors introduced at each outer step to maintain the same convergence as exact rational Krylov subspace method for approximating an invariant subspace. Since this bound is inversely proportional to the current eigenresidual norm of the target invariant subspace, the tolerance of iterative linear solves at each outer step can be relaxed with the outer iteration progress. A restarted variant of the inexact rational Krylov subspace method is also proposed. Numerical experiments show the effectiveness of relaxing the inner tolerance to save computational cost.

On relaxed filtered Krylov subspace method for non-symmetric eigenvalue problems

In this paper, by introducing a class of relaxed filtered Krylov subspaces, we propose the relaxed filtered Krylov subspace method for computing the eigenvalues with the largest real parts and the corresponding eigenvectors of non-symmetric matrices. As by-products, the generalizations of the filtered Krylov subspace method and the Chebyshev–Davidson method for solving non-symmetric eigenvalue problems are also presented. We give the convergence analysis of the complex Chebyshev polynomial, which plays a significant role in the polynomial acceleration technique. In addition, numerical experiments are carried out to show the robustness of the relaxed filtered Krylov subspace method and its great superiority over some state-of-the-art iteration methods.

Multilevel correction adaptive finite element method for solving nonsymmetric eigenvalue problems

Large-scale nonsymmetric eigenvalue problems are common in various fields of science and engineering computing. However, their efficient handling is challenging, and research on their solution algorithms is limited. In this study, a new multilevel correction adaptive finite element method is designed for solving nonsymmetric eigenvalue problems based on the adaptive refinement technique and multilevel correction scheme. Different from the classical adaptive finite element method, which requires solving a nonsymmetric eigenvalue problem in each adaptive refinement space, our approach requires solving a symmetric linear boundary value problem in the current refined space and a small-scale nonsymmetric eigenvalue problem in an enriched correction space. Since it is time-consuming to solve a large-scale nonsymmetric eigenvalue problem directly in adaptive spaces, the proposed method can achieve nearly the same efficiency as the classical adaptive algorithm when solving the symmetric linear boundary value problem. In addition, the corresponding convergence and optimal complexity are verified theoretically and demonstrated numerically.

Preconditioned inexact Newton-like method for large nonsymmetric eigenvalue problems

<p style='text-indent:20px;'>An efficiently preconditioned Newton-like method for the computation of the eigenpairs of large and sparse nonsymmetric matrices is proposed. A sequence of preconditioners based on the Broyden-type rank-one update formula are constructed for the solution of the linearized Newton system. The properties of the preconditioned matrix are investigated. Numerical results are given which reveal that the new proposed algorithms are efficient.</p>

A new constrained optimization model for solving the nonsymmetric stochastic inverse eigenvalue problem

ABSTRACT The stochastic inverse eigenvalue problem aims to reconstruct a stochastic matrix from its spectrum. Recently, Zhao et al. [A geometric nonlinear conjugate gradient method for stochastic inverse eigenvalue problems. SIAM J Numer Anal. 2016;54(4):2015–2035] proposed a constrained optimization model on the manifold of so-called isospectral matrices and adapted a modified. Polak-Ribière-Polyak conjugate gradient method to the geometry of this manifold. However, not every stochastic matrix is an isospectral one and the model in Zhao et al. is based on the assumption that for each stochastic matrix there exists a (possibly different) isospectral, stochastic matrix with the same spectrum. We are not aware of such a result in the literature, but prove the claim at least for 3 × 3 matrices. In this paper, we suggest to extend the above model by considering matrices which differ from isospectral ones only by multiplication with a block diagonal matrix with 2 × 2 blocks from the special linear group. First, we show that each stochastic matrix can be written in such a form. We prove that our model has a minimizer and show how the Polak–Ribiére–Polyak conjugate gradient method works on the corresponding more general manifold. We demonstrate by numerical examples that the new, more general method performs similarly as those in Zhao et al.

Task‐based, GPU‐accelerated and robust library for solving dense nonsymmetric eigenvalue problems

SummaryIn this paper, we present the StarNEig library for solving dense nonsymmetric standard and generalized eigenvalue problems. The library is built on top of the StarPU runtime system and targets both shared and distributed memory machines. Some components of the library have support for GPU acceleration. The library currently applies to real matrices with real and complex eigenvalues and all calculations are done using real arithmetic. Support for complex matrices is planned for a future release. This paper is aimed at potential users of the library. We describe the design choices and capabilities of the library, and contrast them to existing software such as LAPACK and ScaLAPACK. StarNEig implements a ScaLAPACK compatibility layer which should assist new users in the transition to StarNEig. We demonstrate the performance of the library with a sample of computational experiments.

EIGENVALUES EVALUATION OF GENERALLY DAMPED ELASTIC DISC BRAKE MODEL LOADED WITH NON-CONSERVATIVE FRICTION FORCE

This paper deals with the evaluation of eigenvalues of a linear damped elastic two-degrees-of-freedom system under a non- onservative loading. As a physical interpretation of a proposed mathematical model, a simplified disk brake model is considered. A spectral analysis is performed to predict an eigenvalues bifurcation, known as the Krein collision, leading to double eigenvalues, one of them having a positive real part causing a vibration instability of the mechanical systems. This defective behaviour of eigenvalues is studied with respect to a magnitude of non-conservative Coulomb friction force, through the variation of the friction coefficient. The influence of a proportional versus general damping on the system stability is further analysed. The generalized non-symmetric eigenvalue problem calculation is employed for spectral analyses, while a modal decomposition is performed to obtain a time-domain response of the system. The analyses are compared with an experiment.

Accurate Eigenvalues of Some Generalized Sign Regular Matrices via Relatively Robust Representations

In this paper, we consider how to accurately solve the nonsymmetric eigenvalue problem for a class of generalized sign regular matrices including extremely ill-conditioned quasi-Cauchy and quasi-Vandermonde matrices. The problem of performing accurate computations with structured matrices is very much a representation problem. We first develop a relatively robust representation (RRR) for this class of matrices by introducing a free parameter, which exceeds an essential threshold, into an indefinite factorization. We then design a new $$O(n^{3})$$ algorithm to compute all the eigenvalues of such matrices with high relative accuracy, as warranted by the RRR. Error analysis and numerical experiments are performed to illustrate the high relative accuracy.

A cascadic multigrid method for nonsymmetric eigenvalue problem

In this paper, a cascadic multigrid method is proposed to solve nonsymmetric eigenvalue problems. Based on the multilevel correction method, the proposed method transforms a nonsymmetric eigenvalue problem solving on the finest finite element space to linear smoothing steps on a sequence of multilevel finite element spaces and some nonsymmetric eigenvalue problems solving on a very low dimensional space. Choosing the sequence of finite element spaces and the number of smoothing steps appropriately, we obtain the optimal convergence rate with the optimal computing complexity. Some numerical examples are provided to validate the theoretical results and the efficiency of this proposed scheme.

Numerical comparison of eigenvalue algorithms for vibroacoustic problems

This paper concerns the study of vibroacoustic problems. By considering a displacement-pressure formulation, a non-symmetric eigenvalue problem is obtained. In order to solve it, three numerical schemes are compared: the classical ARPACK solver, an indicator method (initially proposed in [4]) which has the property to be null at the eigenvalues, and an original method based on the analysis of Taylor series expansions near a singularity. Numerical results show all the evaluated numerical methods give accurate results but the indicator method requires the lowest computational times. Nevertheless, the original method based on the behavior of the perturbation method close to eigenvalues seems to be a very promising technique.

Two-sided orthogonal reductions to condensed forms on asymmetric multicore processors

Abstract We investigate how to leverage the heterogeneous resources of an Asymmetric Multicore Processor (AMP) in order to deliver high performance in the reduction to condensed forms for the solution of dense eigenvalue and singular-value problems. The routines that realize this type of two-sided orthogonal reductions (TSOR) in LAPACK are especially challenging, since a significant fraction of their floating-point operations are cast in terms of memory-bound kernels while the remaining part corresponds to efficient compute-bound kernels. To deal with this scenario: (1) we leverage implementations of memory-bound and compute-bound kernels specifically tuned for AMPs; (2) we select the algorithmic block size for the TSOR routines via a practical model; and (3) we adjust the type and number of cores to use at each step of the reduction. Our experiments validate the model and assess the performance of our asymmetry-aware TSOR routines, using an ARMv7 big.LITTLE AMP, for three key operations: the reduction to tridiagonal form for symmetric eigenvalue problems, the reduction to Hessenberg form for non-symmetric eigenvalue problems, and the reduction to bidiagonal form for singular-value problems.

Few-body integrodifferential equation on Lagrange mesh

The two-variable integrodifferential equation for few-body systems is solved using the Lagrange-mesh method. The method transforms the equation into a system of algebraic equations that are solved as a non-symmetric matrix eigenvalue problem. Convergence properties of the solution to the integrodifferential equation in relation to the problem parameters is investigated. The accuracy of the converged solution is tested by calculating the binding energies and root-mean-square radii of selected few-body systems. The results are compared to those generated by other methods.

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

We present two efficient iterative algorithms for solving the linear response eigenvalue problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into an eigenvalue problem that involves the product of two matrices M and K. We show that, because MK is self-adjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by a modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. We show that the other component of the eigenvector can be easily recovered in an inexpensive postprocessing procedure. As a result, the algorithms we present here become more efficient than existing methods that try to approximate both components of the eigenvectors simultaneously. In particular, our numerical experiments demonstrate that the new algorithms presented here consistently outperform the existing state-of-the-art Davidson type solvers by a factor of two in both solution time and storage.

Computable Error Estimates for a Nonsymmetric Eigenvalue Problem

AbstractWe provide some computable error estimates in solving a nonsymmetric eigenvalue problem by general conforming finite element methods on general meshes. Based on the complementary method, we first give computable error estimates for both the original eigenfunctions and the corresponding adjoint eigenfunctions, and then we introduce a generalised Rayleigh quotient to deduce a computable error estimate for the eigenvalue approximations. Some numerical examples are presented to illustrate our theoretical results.

Eigensolutions to a vibroacoustic interior coupled problem with a perturbation method

In this paper, an efficient and robust numerical method is proposed to solve non-symmetric eigenvalue problems resulting from the spatial discretization with the finite element method of a vibroacoustic interior problem. The proposed method relies on a perturbation method. Finding the eigenvalues consists in determining zero values of a scalar that depends on angular frequency. Numerical tests show that the proposed method is not sensitive to poorly conditioned matrices resulting from the displacement–pressure formulation. Moreover, the computational times required with this method are lower than those needed with a classical technique such as, for example, the Arnoldi method.

On a Non-Symmetric Eigenvalue Problem Governing Interior Structural–Acoustic Vibrations

Small amplitude vibrations of a structure completely filled with a fluid are considered. Describing the structure by displacements and the fluid by its pressure field, the free vibrations are governed by a non-self-adjoint eigenvalue problem. This survey reports on a framework for taking advantage of the structure of the non-symmetric eigenvalue problem allowing for a variational characterization of its eigenvalues. Structure-preserving iterative projection methods of the the Arnoldi and of the Jacobi–Davidson type and an automated multi-level sub-structuring method are reviewed. The reliability and efficiency of the methods are demonstrated by a numerical example.

An Adaptive Finite Element Method for the Transmission Eigenvalue Problem

The classical weak formulation of the Helmholtz transmission eigenvalue problem can be linearized into an equivalent nonsymmetric eigenvalue problem. Based on this nonsymmetric eigenvalue problem, we first discuss the a posteriori error estimates and adaptive algorithm of conforming finite elements for the Helmholtz transmission eigenvalue problem. We give the a posteriori error indicators for primal eigenfunction, dual eigenfunction and eigenvalue. Theoretical analysis shows that the indicators for both primal eigenfunction and dual eigenfunction are reliable and efficient and that the indicator for eigenvalue is reliable. Numerical experiments confirm our theoretical analysis.

Multilevel finite element discretizations based on local defect correction for nonsymmetric eigenvalue problems

Based on the work of Xu and Zhou (2000), we establish new three-level and multilevel finite element discretizations by local defect correction techniques. Theoretical analysis and numerical experiments show that the discretizations are simple and easy to implement, and can be used to solve nonsymmetric eigenvalue problems with non smooth eigenfunctions efficiently. We also discuss the local error estimates of finite element approximations; it is a new feature here that the estimates apply to the local domains containing corner points.