Arnoldi Method Research Articles

Floquet analysis of spatially periodic thermocapillary convection in a low-Prandtl-number liquid bridge

Secondary instability of thermocapillary convection was investigated for a half-zone liquid bridge with low Prandtl number fluids. The liquid bridge was suspended between two cylindrical flat disks, which were maintained at different temperatures. The thermocapillary-driven flow formed an axisymmetric steady toroidal vortex. If the temperature difference between the two disks exceeded a certain threshold, the axisymmetric flow transitioned to an azimuthal periodic steady flow. The objective in the analysis of secondary instability is to examine the stability limit of this steady flow. Employing Floquet analysis, we obtained the neutral stability curves as a function of the aspect ratio (radius/height) of the liquid bridge and identified the critical modes. The critical modes of the Floquet analysis are referred to as critical Floquet modes and are classified by Floquet parameter β, which characterizes a set of azimuthal wave numbers of the Floquet mode. The Newton–Krylov method and Arnoldi method are implemented to solve the large-scale nonlinear equations and generalized eigenvalue problems, respectively. Two critical Floquet modes were observed having different Floquet parameters, β = 0 and β = 1, which appear preferential in the liquid bridge for high and low aspect ratios, respectively. The mode with β = 1 was steady for low aspect ratios and was able to change to an oscillatory mode with a frequency of oscillation that rapidly increased with increasing aspect ratio. We visualized the temperature and azimuthal velocity distributions of the critical Floquet modes and its Fourier components. We concluded that the mode with β = 1 leads to a pulsating oscillation, denoted as “P-type,” whereas that with β = 0 leads to a twisting oscillation, denoted as “T-type.” Both types of oscillations were reported in previous studies. The present neutral stability curves are for the most part in good agreement with critical Reynolds numbers previously obtained in numerical simulations.

An Improved IRA Algorithm and Its Application in Critical Eigenvalues Searching for Low Frequency Oscillation Analysis

On the basis of implicitly restarted Arnoldi (IRA) method, an improved algorithm is proposed, in which the dimension of Krylov subspace is dynamically increased to compute eigenvalues in specified circle. First, the radius of searching circles is dynamically expanded through automatically increasing the number of eigenvalues and the dimension of Krylov subspace, based on the locking mechanism. Second, the region where the low frequency oscillation modes located is divided into small independent computing units, which are covered by specified searching circles. Third, the independent computing units can be calculated simultaneously with no effects on each other. The proposed method can avoid eigenvalues missing caused by inappropriate resetting of search number in equidistant-searching IRA method. Furthermore, no manual intervention is needed in the proposed method. Two systems with 570 and 5272 state variables are tested in this paper, and the results indicate that the proposed method is efficient, reliable, and practical.

A global rational Arnoldi method for model reduction

In this paper, we propose two new approaches for model order reduction of large-scale multi-input multi-output (MIMO) linear time invariant dynamical systems (LTI). These methods are based on a generalization of the global Arnoldi algorithm which is used to generate projection subspaces. An adaptive procedure for the selection of shift parameters is proposed and some simple Arnoldi relations are established in order to compute error bounds on the transfer function. Numerical experiments showing the effectiveness of these approaches are provided.

On the structured backward error of inexact Arnoldi methods for (skew)-Hermitian and (skew)-symmetric eigenvalue problems

In this paper, we present the inexact structure preserving Arnoldi methods for structured eigenvalue problems. They are called structure preserving because the computed eigenvalues and eigenvectors can preserve the desirable properties of the structures of the original matrices, even with large errors involved in the computation of matrix-vector products. A backward error matrix is called structured if it has the same structure as the original matrix. We derive a common form for the structured backward errors that can be used for different structure preserving processes, and prove the derived form has the minimum Frobenius norm among all possible backward errors. Furthermore, we employ the derived backward errors for some specific structure preserving processes to estimate the accuracy of the solutions obtained by inexact Arnoldi methods for eigenvalue problems. We aim to give, wherever possible, formulae that are inexpensive to compute so that they can be used routinely in practice. Numerical experiments are provided to support the theoretical results.

Pade近似を用いた数値等角写像計算のArnoldi法による精度改善(行列・固有値問題の解法とその応用, 平成17年研究部会連合発表会)

Pade近似を用いた数値等角写像計算のArnoldi法による精度改善(行列・固有値問題の解法とその応用, 平成17年研究部会連合発表会)

Conical quantum dot: Electronic states and dipole moment

The electronic states in a conical quantum dot in the framework of the adiabatic approximation as well as the combined approach with the perturbation theory have been considered. The obtained results have been compared with the results of numerical methods–the finite element and the Arnoldi methods. The interpolation formula for the energy correction and its dependence on the geometric parameters of the conical quantum dot have been obtained. The comparing of the wave functions obtained by different methods and the range of applicability of different methods have been defined. The dependence of the z-component of the dipole moment on the geometric parameters of the structure has been considered.

The Waveguide Eigenvalue Problem and the Tensor Infinite Arnoldi Method

We present a new computational approach for a class of large-scale nonlinear eigenvalue problems (NEPs) that are nonlinear in the eigenvalue. The contribution of this paper is twofold. We derive a new iterative algorithm for NEPs, the tensor infinite Arnoldi method (TIAR), which is applicable to a general class of NEPs, and we show how to specialize the algorithm to a specific NEP: the waveguide eigenvalue problem. The waveguide eigenvalue problem arises from a finite-element discretization of a partial differential equation used in the study waves propagating in a periodic medium. The algorithm is successfully applied to accurately solve benchmark problems as well as complicated waveguides. We study the complexity of the specialized algorithm with respect to the number of iterations $m$ and the size of the problem $n$, both from a theoretical perspective and in practice. For the waveguide eigenvalue problem, we establish that the computationally dominating part of the algorithm has complexity $\mathcal{O}(nm^2+\sqrt{n}m^3)$. Hence, the asymptotic complexity of TIAR applied to the waveguide eigenvalue problem, for $n\rightarrow\infty$, is the same as for Arnoldi's method for standard eigenvalue problems.

Krylov--Schur-Type Restarts for the Two-Sided Arnoldi Method

We consider the two-sided Arnoldi method and propose a two-sided Krylov--Schur-type restarting method. We discuss the restart for standard Rayleigh--Ritz extraction as well as harmonic Rayleigh--Ritz extraction. Additionally, we provide error bounds for Ritz values and Ritz vectors in the context of oblique projections and present generalizations of, e.g., the Bauer--Fike theorem and Saad's theorem. Applications of the two-sided Krylov--Schur method include the simultaneous computation of left and right eigenvectors and the computation of eigenvalue condition numbers. We demonstrate how the method can be used to find the least sensitive eigenvalues of a nonnormal matrix and how to approximate pseudospectra by using left and right shift-invariant subspaces. The results demonstrate that significant improvements in quality can be obtained over approximations with the (one-sided) Krylov--Schur method.

Research on vibration characteristics and its key influencing factors of new mechanical elastic wheel

This paper presents the vibration characteristics and its key influencing factors of a new mechanical elastic wheel (MEW). The MEW was modeled as a ring on elastic foundations (REF) with distributed spring stiffness in the radial and tangential directions. The general forced solutions of inextensible vibration were derived by the use of a modal expansion technique and Arnoldi method, and the accuracy of the solutions had been validated by FEM simulation and modal test under free suspension and various loading situations. The natural frequencies and mode shapes of the rotating MEW could be obtained under free suspension and various loading situations. Moreover, the effects of various rotational speed, loading and different number of hinges on natural frequencies were investigated. Finally, the effect of different number of hinges on the damping ratio of the MEW radial modes was also analyzed. The analysis results reflect the objective law of the actual vibration characteristics of the MEW, and provide a reference for the MEW structure optimization and the vibration characteristics of the whole vehicle.

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.

A modified second-order Arnoldi method for solving the quadratic eigenvalue problems

Based on a pair of square matrices A and B and a vector u, a modified second-order Krylov subspace Rn(A,B;u) is first defined, which generalizes the standard Krylov subspace and the second-order Krylov subspace proposed by Bai and Su (2005). Then a modified second-order Arnoldi procedure for generating an orthonormal basis of Rn(A,B;u) has been presented. By applying the standard Rayleigh–Ritz orthogonal projection technique, a modified second-order Arnoldi method (MSOAR) for solving a large-scale quadratic eigenvalue problems (QEP) has been proposed. Finally, numerical experiments are given to show the efficiency of the new method.

A new shift strategy for the implicitly restarted generalized\\ second-order Arnoldi method

In the implicitly restarted generalized second-order Arnoldi (GSOAR) method, we can get $2m$ approximate eigenpairs by projecting the quadratic eigenvalue problem (QEP) onto an $m$-dimensional subspace. During implicitly restarted processes, the problem is the mismatch between the number of shifts and the dimension of the subspace. In order to cure the problem, a new shift strategy for GSOAR method is proposed in this paper. We proof that the novel method can use all $2l$ shift candidates and preserve the special structure. Numerical experiments illustrate that the new method enhances the overall efficiency of the algorithm by increasing the efficiency of every restart process.

On some properties of the extended block and global Arnoldi methods with applications to model reduction

The aim of the present paper is to give some new algebraic properties of the extended block and the extended global Arnoldi algorithms. These results are then applied on moment matching methods for model reductions in large-scale dynamical systems to get low-order models that approximate the original models by matching moments and Markov parameters at the same time. Some numerical examples are given to show the effectiveness of the methods on some benchmark tests.

An Arnoldi-Frontal Approach for the Stability Analysis of Flows in a Collapsible Channel

In this paper, we present a new approach based on a combination of the Arnoldi and frontal methods for solving large sparse asymmetric and generalized complex eigenvalue problems. The new eigensolver seeks the most unstable eigensolution in the Krylov subspace and makes use of the efficiency of the frontal solver developed for the finite element methods. The approach is used for a stability analysis of flows in a collapsible channel and is found to significantly improve the computational efficiency compared to the traditionally used QZ solver or a standard Arnoldi method. With the new approach, we are able to validate the previous results obtained either on a much coarser mesh or estimated from unsteady simulations. New neutral stability solutions of the system have been obtained which are beyond the limits of previously used methods.

Comparison of non-linear eigensolvers for modal analysis of frequency dependent laminated visco-elastic sandwich plates

Modal analysis of frequency dependent visco-elastic sandwich structures with finite element models leads to a non-linear eigenvalue problem. Many methods have been proposed so far to solve this problem. The present paper compares the asymptotic numerical method to some representative algorithms such as non-linear Arnoldi method, non-linear Jacobi–Davidson method, inverse iteration and iterative shift-invert method. Three mesh refinement levels 1 (~5000 dofs), 2 (~20000 dofs) and 3 (~45000 dofs) have been used. Computational times and accuracy are compared on relevant use cases with frequency dependent visco-elastic materials. Three layered laminated sandwich plates with visco-elastic layers are considered as test cases and two different frequency and temperature dependent materials (ISD112 and PVB) are employed. It is shown that the non-linear Arnoldi method provides the best results in terms of accuracy and CPU time, inverse iteration being the most efficient method for small size problems. Moreover, it is shown that fractional derivative models commonly used to model frequency dependent viscoelastic materials are more CPU intensive than generalized Maxwell models when performing modal analysis of frequency dependent viscoelastic sandwich structures.

Large-Scale Characteristic Mode Analysis With Fast Multipole Algorithms

Large-scale characteristic mode analysis (CMA) poses challenges in computational electromagnetics as it calls for efficient solutions of large dense generalized eigenvalue problems. In this paper, we consider two applications that involve large-scale CMA, and demonstrate that fast multipole algorithms (FMAs) can be easily incorporated into the implicitly restarted Arnoldi method (IRAM) for eigenanalysis after simple modifications. The first application performs CMA for large platforms made by closed perfectly conducting surfaces. Multilevel FMA (MLFMA) is embedded into a combined field integral equation-based theory of characteristic mode (TCM). The second application addresses multiscale modeling of small but geometrically complicated objects, which possess fine subwavelength structures. An augmented electric field integral equation-based TCM is formulated, and low-frequency (LF-)FMA is adopted to accelerate the required matrix–vector products.

An Arnoldi-Inout algorithm for computing PageRank problems

The PageRank algorithm plays an important role in the web search engines. To speed up the convergence behavior for computing PageRank, we propose a new method, called as Arnoldi-Inout, which is the inner–outer iteration method modified with the thick restarted Arnoldi method. The description and convergence of the new algorithm are discussed in detail. Numerical results are given to illustrate the efficiency of the new algorithm.

Greedy Multipoint Model-Order Reduction Technique for Fast Computation of Scattering Parameters of Electromagnetic Systems

This paper attempts to develop a new automated multipoint model-order reduction (MOR) technique, based on matching moments of the system input–output function, which would be suited for fast and accurate computation of scattering parameters for electromagnetic (EM) systems over a wide frequency band. To this end, two questions are addressed. Firstly, the cost of the wideband reduced model generation is optimized by automating a greedy multipoint MOR scheme. This is achieved by introducing a new dual local-global model convergence scheme, which applies fast and reliable a posteriori error estimates to check both local model convergence, used to select the number of moments at a single expansion point, and global model convergence, used to optimally select the expansion points. Secondly, the question of optimal convergence measure is addressed by proposing an enhanced a posteriori error estimator particularly suited for scattering parameter computations for lossy EM systems. The effectiveness and efficiency of the proposed automated scheme is verified through numerical simulations using reduced-order models for examples of a bandstop dielectric resonator filter and a dielectric resonator antenna for a wide frequency band, and compared against the results obtained using the full-order model, a reduced model generated with the optimal greedy point selection algorithm, as well as the reduced-order models obtained using the reduced basis method (RBM) and the single-point second-order Arnoldi method for passive order reduction (SAPOR) method.

Model order reduction of MIMO bilinear systems by multi-order Arnoldi method

In this paper, we present a time domain model order reduction method for multi-input multi-output (MIMO) bilinear systems by general orthogonal polynomials. The proposed method is based on a multi-order Arnoldi algorithm applied to construct the projection matrix. The resulting reduced model can match a desired number of expansion coefficient terms of the original system. The approximate error estimate of the reduced model is given. And we also briefly discuss the stability preservation of the reduced model in some cases. Additionally, in combination with Krylov subspace methods, we propose a two-sided projection method to generate reduced models which capture properties of the original system in the time and frequency domain simultaneously. The effectiveness of the proposed methods is demonstrated by two numerical examples.

Restarting iterative projection methods for Hermitian nonlinear eigenvalue problems with minmax property

In this work we present a new restart technique for iterative projection methods for nonlinear eigenvalue problems admitting minmax characterization of their eigenvalues. Our technique makes use of the minmax induced local enumeration of the eigenvalues in the inner iteration. In contrast to global numbering which requires including all the previously computed eigenvectors in the search subspace, the proposed local numbering only requires a presence of one eigenvector in the search subspace. This effectively eliminates the search subspace growth and therewith the super-linear increase of the computational costs if a large number of eigenvalues or eigenvalues in the interior of the spectrum are to be computed. The new restart technique is integrated into nonlinear iterative projection methods like the Nonlinear Arnoldi and Jacobi-Davidson methods. The efficiency of our new restart framework is demonstrated on a range of nonlinear eigenvalue problems: quadratic, rational and exponential including an industrial real-life conservative gyroscopic eigenvalue problem modeling free vibrations of a rolling tire. We also present an extension of the method to problems without minmax property but with eigenvalues which have a dominant either real or imaginary part and test it on two quadratic eigenvalue problems.