Particle swarm optimization method for solving quadratic eigenvalues

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.

A multiscale analysis and computational method based on the Second-Order Two-Scale (SOTS) approach are proposed for the elastic quadratic eigenvalue problems in the periodic composite domain. Two typical quadratic eigenvalue problems with different damping effects are considered, and by the asymptotic expansions of both the eigenfunctions and eigenvalues, the first- and second-order cell functions, the microscale features of this heterogeneous materials are defined successively. Then, the homogenized quadratic eigenvalue problems are derived and the second-order expansions of the eigenfunctions are formed. The eigenvalues are also broadened to the second-order terms by introducing proper auxiliary elastic functions defined in the composite structure, and the nonlinear expressions of the correctors of the eigenvalues are derived. The finite element procedures are established, linearized methods are discussed for solving the quadratic eigenvalue problems and the second-order asymptotic computations are performed. Effectiveness of the asymptotic model is demonstrated by both the qualitative and quantitative comparisons between the computed SOTS approximations and the reference solutions, and the converging behavior of the eigenfunctions are numerically verified. It is also indicated that the second-order correctors are of importance to reconstruct the detailed information of the original eigenfunctions within the micro cells.

Numerical solution of a quadratic eigenvalue problem*1

SummaryHeavily damped quadratic eigenvalue problem (QEP) is a special type of QEPs. It has a large gap between small and large eigenvalues in absolute value. One common way for solving QEP is to linearize the original problem via linearizations. Previous work on the accuracy of eigenpairs of not heavily damped QEP focuses on analyzing the backward error of eigenpairs relative to linearizations. The objective of this paper is to explain why different linearizations lead to different errors when computing small and large eigenpairs. To obtain this goal, we bound the backward error of eigenpairs relative to the linearization methods. Using these bounds, we build upper bounds of growth factors for the backward error. We present results of numerical experiments that support the predictions of the proposed methods.

Model updating for quadratic eigenvalue problems (QEPs) is pro- posed by Friswell, Inman and Pilkey (1998), to incorporate the measured model data into the model which only produces the mass and stifiness matrices, that closely match the experimental model data. In this paper, we consider a numer- ical model for updating QEPs which produces not only the mass and stifiness matrices, but also the damping matrix will be updated. To this end the com- plete set of eigenpairs will be employed.

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.

We propose a novel formulation of minimal case solutions for determining the relative pose of perspective and generalized cameras given a partially known rotation, namely, a known axis of rotation. An axis of rotation may be easily obtained by detecting vertical vanishing points with computer vision techniques, or with the aid of sensor measurements from a smart phone. Given a known axis of rotation, our algorithms solve for the angle of rotation around the known axis along with the unknown translation. We formulate these relative pose problems as Quadratic Eigen value Problems which are very simple to construct. We run several experiments on synthetic and real data to compare our methods to the current state-of-the-art algorithms. Our methods provide several advantages over alternatives methods, including efficiency and accuracy, particularly in the presence of image and sensor noise as is often the case for mobile devices.

Structured eigenvalue methods for the computation of corner singularities in 3D anisotropic elastic structures

We consider quadratic eigenvalue problems with large and sparse matrices depending on a parameter. Problems of this type occur, for example, in the stability analysis of spatially discretized and p...

Retraction Note: UPSO-FSVRNET: Fuzzy Identification Approach in a VANET Environment Based on Fuzzy Support Vector Regression and Unified Particle Swarm Optimization

Passive control of damaged composite laminates with optimized location of piezoelectric fiber composite patches

Given a quadratic two-parameter matrix polynomial Q(λ, μ), we develop a systematic approach to generating a vector space of linear two-parameter matrix polynomials. The purpose for constructing this vector space is that potential linearizations of Q(λ, μ) lie in it. Then, we identify a set of linearizations and describe their constructions. Finally, we determine a class of linearizations for a quadratic two-parameter eigenvalue problem.

Chapter 6 - Quadratic forms and Eigenvalue problems

In the past decade or so, semi-definite programming (SDP) has emerged as a powerful tool capable of handling a remarkably wide range of problems. This article describes an innovative application of SDP techniques to quadratic inverse eigenvalue problems (QIEPs). The notion of QIEPs is of fundamental importance because its ultimate goal of constructing or updating a vibration system from some observed or desirable dynamical behaviors while respecting some inherent feasibility constraints well suits many engineering applications. Thus far, however, QIEPs have remained challenging both theoretically and computationally due to the great variations of structural constraints that must be addressed. Of notable interest and significance are the uniformity and the simplicity in the SDP formulation that solves effectively many otherwise very difficult QIEPs.

We consider even grade polynomial eigenvalue problems (PEP) to study their eigenvalue location and condition number bounds. We converted the even grade PEP into a Quadratic eigenvalue problem (QEP) of high dimension via quadratification techniques. We study an eigenvalue localization method that applies block Gersgorin sets to certain linearization classes of quadratic matrix polynomials originating from well-established vector spaces. We presented a few results on the bounding ratios of condition number of certain linearization classes with respect to that of original PEP and QEP induced due to quadratification. Relevant computational works are performed using a model numerical example to show the effectiveness of the results in terms of calculating eigenvalue and bounds of condition number ratios.