A meshless Chebyshev collocation method for eigenvalue problems of the Helmholtz equation

Dielectric waveguide formulations usually result in generalized eigenvalue problems which are non-Hermitian in nature. We solve this class of problems by a two stage process. In the first stage, a standard lumped eigenvalue problem is solved for a few eigenpairs of interest (usually the guided modes) by Arnoldi's method accelerated by Chebyshev polynomials. The solution is then refined for the actual unlumped generalised eigenvalue problem by using inflated inverse iteration. After the eigenvalues and eigenvectors have been calculated at the first point (preferably the highest frequency), the eigenpairs for the rest of the frequencies of interest are calculated for progressively smaller frequencies by inflated inverse iteration alone, using the eigenvectors of the last step as the initial guess. To demonstrate the efficacy of this method, we use a variational formulation for anisotropic, dielectric waveguides based only on the E/sub s/ components or only on the H/sub s/ components of the electromagnetic fields that was presented by Chew and Nasir (1989). In this formulation it was shown that due of the imposition of the divergence condition the spurious waveguide modes were eliminated.

The hyperbolic quadratic eigenvalue problem (HQEP) was shown to admit Courant–Fischer type min–max principles in 1955 by Duffin and Cauchy type interlacing inequalities in 2010 by Veselić. It can be regarded as the closest analog (among all kinds of quadratic eigenvalue problem) to the standard Hermitian eigenvalue problem (among all kinds of standard eigenvalue problem). In this paper, we conduct a systematic study on the HQEP both theoretically and numerically. On the theoretical front, we generalize Wielandt–Lidskii type min–max principles and, as a special case, Fan type trace min/max principles and establish Weyl type and Wielandt–Lidskii–Mirsky type perturbation results when an HQEP is perturbed to another HQEP. On the numerical front, we justify the natural generalization of the Rayleigh–Ritz procedure with existing principles and our new optimization principles, and, as consequences of these principles, we extend various current optimization approaches—steepest descent/ascent and nonlinear conjugate gradient type methods for the Hermitian eigenvalue problem—to calculate a few extreme eigenvalues (of both positive and negative type). A detailed convergence analysis is given for the steepest descent/ascent methods. The analysis reveals the intrinsic quantities that control convergence rates and consequently yields ways of constructing effective preconditioners. Numerical examples are presented to demonstrate the proposed theory and algorithms.

DUAL FORMULATION OF MULTIPLE RECIPROCITY METHOD FOR THE ACOUSTIC MODE OF A CAVITY WITH A THIN PARTITION

Eigenvalue problem derivatives computation for a complex matrix using the adjoint method

The dynamic analysis of a structural system with different material and geometric properties leads to linear eigenvalue problem such as generalized (or standard) eigenvalue problem. In general, the material and geometric properties are assumed to be in the form of crisp (or exact). However, due to several errors and insufficient or incomplete information of data, the uncertainties are assumed to be present in the material and geometric properties. These uncertain material and geometric properties may be modeled through convex normalized fuzzy sets. In standard fuzzy arithmetic, all the operands are assumed to be independent of each other. But when they are partially or completely dependent on each other, the standard fuzzy arithmetic results in a wider range. This situation is known as the “dependency problem” or “overestimation problem”. In this regard, a fuzzy-affine approach is developed to overcome the dependency problem. This proposed approach may improve the outer enclosures and give tighter bounds to the fuzzy solution set. This chapter deals with the dynamic analysis of various structural systems viz. multi degrees-of-freedom spring-mass structural system, multi-storey frame structural system, etc. by adopting the proposed approach. Several numerical examples related to the dynamic analysis of these structural systems have been worked out to illustrate the reliability and efficiency of the present approach.KeywordsAffine arithmeticFuzzy-affine arithmeticDynamic analysis of structural systemFuzzy generalized eigenvalue problemFuzzy standard eigenvalue problem

A matrix-free isogeometric Galerkin method for Karhunen–Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature

The generalized eigenvalue problem provides a suitable framework for reliable solutions of many system theoretic, control, and estimation problems. A general algorithm for solving the matrix algebraic Riccati equation (ARE) which utilizes a pencil structure is described here. This algorithm avoids unnecessary inversion of cost or transition matrices, making it a numerically sound way to solve for the gains and/or ARE with singular quadratic costs, for cases satisfying detectability and stabilizability conditions. Examples are solution with discrete dead-beat control, noiseless measurements in Kalman filters and time-delays in discrete-time systems, which cause difficulties in the Hamiltonian standard eigenvalue problem formulation. The ARE algorithm implementation and numerical examples are shown.

In this article, we propose the Fast Algorithms for Maxwell’s Equations (FAME) package for solving Maxwell’s equations for modeling three-dimensional photonic crystals. FAME combines the null-space free method with fast Fourier transform (FFT)-based matrix-vector multiplications to solve the generalized eigenvalue problems (GEPs) arising from Yee’s discretization. The GEPs are transformed into a null-space free standard eigenvalue problem with a Hermitian positive-definite coefficient matrix. The computation times for FFT-based matrix-vector multiplications with matrices of dimension 7 million are only 0.33 and 3.6 × 10 − 3 seconds using MATLAB with an Intel Xeon CPU and CUDA C++ programming with a single NVIDIA Tesla P100 GPU, respectively. Such multiplications significantly reduce the computational costs of the conjugate gradient method for solving linear systems. We successfully use FAME on a single P100 GPU to solve a set of GEPs with matrices of dimension more than 19 million, in 127 to 191 seconds per problem. These results demonstrate the potential of our proposed package to enable large-scale numerical simulations for novel physical discoveries and engineering applications of photonic crystals.

Time-domain simulation of acoustic wave propagation and interaction with flexible structures using Chebyshev collocation method

Capacitance matrix technique for avoiding spurious eigenmodes in the solution of hydrodynamic stability problems by Chebyshev collocation method

In several applications in science and engineering, different types of matrix problems emerge from the discretization of partial differential equations. This thesis is devoted to the development of new algorithms to solve this kind of problems. In particular, when the matrices involved are sparse and non-symmetric. The new algorithms are based on the Induced Dimension Reduction method [IDR(s)]. IDR(s) is a Krylov subspace method originally proposed in 2008 to solve systems of linear equations. IDR(s) has received considerable attention due to its stable and fast convergence. It is, therefore, natural to ask if it is possible to extend IDR(s) to solve other matrix problems, and if so, to compare those extensions with other well-established methods. This work aims to answer these questions. The main matrix problems considered in this dissertation are: the standard eigenvalue problem, the quadratic eigenvalue problem, the solution of systems of linear equations, the solution of sequences of systems of linear equations, and linear matrix equations. We focus on examples that arise from the discretization of partial differential equations.

The finite difference frequency domain (FDFD) method can be used to analyze the properties of arbitrary complex close/open periodic guided-wave structures. For the open problem, it probably results in a large scale nonsymmetrical generalized eigenvalue problem. In the analysis of the leaky-wave antennas, because the leakage constant (attenuation constant) increases a lot in the working frequency range, the extraction process of the eigenvalues (complex propagation constants) is difficult to converge when the shift-and-invert Arnoldi technique is used to resolve this large scale nonsymmetrical generalized eigenvalue problem. In this paper, a refined shift-and-invert Arnoldi algorithm is adopted to speed the convergence rate by using refined Ritz vectors to approximate the desired eigenvectors. Since the refined method always converges, a simple FDFD method with six field components can be used. The difference equations of this method are much simpler than those of the previous method because the longitudinal field components don't have to be eliminated in order to approximately transform the generalized eigenvalue problem to a standard eigenvalue problem. Consequently, the accuracy of the method can be improved and the boundary conditions can be implemented easily.

To solve quadratic eigenvalue problems (QEPs), especially the gyroscopic systems, two methods are proposed: an iterative direct detection method (DDM) of the complex eigenvalues of the original QEP, and a split-linearization method (SLM) for finding the solvent matrix, which results to a standard linear eigenvalue problem (LEP) being solved to compute all eigenvalues by the symmetry extension. Reducing the dimension to one-half, the LEP is recast in a simpler QEP involving the square of the solvent. We set up two new merit functions which are minimized to detect the complex eigenvalues from the original QEP and a simpler QEP. For each eigen-parameter the merit function consists of the Euclidean norm of each derived eigen-equation, whose vector variable is solved from a derived inhomogeneous linear system. Then, the golden section search algorithm is employed to minimize the merit functions and locate the complex eigenvalue as a local minimal point. The results are compared with that computed by the cyclic-reduction-based solvent (CRS) method.

本文中我们对Helmholtz方程透射特征值问题提出一种带位移求逆的算法，此算法可以快速有效地求出任意给定的 σ 附近的几个实特征值及对应的特征向量。首先，我们用连续有限元方法对Helmholtz方程透射特征值问题进行离散，并将离散后的广义特征值问题化为一个二次特征值问题，进而对其进行线性化得到一个新的广义特征值问题。这个新的广义特征值问题排除了没有物理意义的零特征值的干扰，保留了所有的非零特征值。我们还利用位移求逆的技术，求得给定的 σ 附近的几个实特征对。我们所提出的算法对透射特征值问题的折射率没有特别的限制，即折射率可为正或负亦或是任意的实函数。最后的数值算例验证了该算法的有效性。 In this paper, we put forward a shift-and-invert algorithm to solve the transmission eigenvalue problem of Helmholtz equation, which can quickly and efficiently find out the several eigenvalues and the corresponding eigenvectors near arbitrarily given σ. First, we use the continuous finite element method to discrete the transmission eigenvalue problem of Helmholtz equation, and dis-crete the generalized eigenvalue problem into a quadratic eigenvalue problem, and then a new generalized eigenvalue problem is obtained by linearization. The new generalized eigenvalue problem eliminates the distraction of nonphysical zero eigenvalues, and preserves all the nonzero eigenvalues. Further through the use of shift-and-invert technology, we can quickly and efficiently get several real eigenpairs near given σ. The proposed algorithm has no special restrictions to the refractive index of the transmission eigenvalue problems, that is to say, the refractive index can be positive or negative or positive and negative real numbers. The final numerical example verifies the effectiveness of our algorithm.

Thermal buckling response of laminated and sandwich plates using refined 2-D models