Jacobi-Davidson methods for polynomial two-parameter eigenvalue problems

We propose Jacobi-Davidson type methods for polynomial two-parameter eigenvalue problems (PMEP). Such problems can be linearized as singular two-parameter eigenvalue problems, whose matrices are of dimension k(k+1)n/2, where k is the degree of the polynomial and n is the size of the matrix coefficients in the PMEP. When k^2n is relatively small, the problem can be solved numerically by computing the common regular part of the related pair of singular pencils. For large k^2n, computing all solutions is not feasible and iterative methods are required. When k is large, we propose to linearize the problem first and then apply Jacobi-Davidson to the obtained singular two-parameter eigenvalue problem. The resulting method may for instance be used for computing zeros of a system of scalar bivariate polynomials close to a given target. On the other hand, when k is small, we can apply a Jacobi-Davidson type approach directly to the original matrices. The original matrices are projected onto a low-dimensional subspace, and the projected polynomial two-parameter eigenvalue problems are solved by a linearization. Keywords: Polynomial two-parameter eigenvalue problem (PMEP), quadratic two-parameter eigenvalue problem (QMEP), Jacobi-Davidson, correction equation, singular generalized eigenvalue problem, bivariate polynomial equations, determinantal representation, delay differential equations (DDEs), critical delays

Multiparameter spectral analysis for aeroelastic instability problems

This paper presents a novel application of multiparameter spectral theory to the study of structural stability, with particular emphasis on aeroelastic flutter. Methods of multiparameter analysis allow the development of new solution algorithms for aeroelastic flutter problems; most significantly, a direct solver for polynomial problems of arbitrary order and size, something which has not before been achieved. Two major variants of this direct solver are presented, and their computational characteristics are compared. Both are effective for smaller problems arising in reduced-order modelling and preliminary design optimization. Extensions and improvements to this new conceptual framework and solution method are then discussed.Comment: 20 pages, 8 figure

Fixing Nonconvergence of Algebraic Iterative Reconstruction with an Unmatched Backprojector

We consider algebraic iterative reconstruction methods with applications in image reconstruction. In particular, we are concerned with methods based on an unmatched projector/backprojector pair; i.e., the backprojector is not the exact adjoint or transpose of the forward projector. Such situations are common in large-scale computed tomography, and we consider the common situation where the method does not converge due to the nonsymmetry of the iteration matrix. We propose a modified algorithm that incorporates a small shift parameter, and we give the conditions that guarantee convergence of this method to a fixed point of a slightly perturbed problem. We also give perturbation bounds for this fixed point. Moreover, we discuss how to use Krylov subspace methods to efficiently estimate the leftmost eigenvalue of a certain matrix to select a proper shift parameter. The modified algorithm is illustrated with test problems from computed tomography

An SVD-approach to Jacobi-Davidson solution of nonlinear Helmholtz eigenvalue problems

Numerical solution of the Helmholtz equation in an infinite domain often involves restriction of the domain to a bounded computational window where a numerical solution method is applied. On the boundary of the computational window artificial transparent boundary conditions are posed, for example, widely used perfectly matched layers (PMLs) or absorbing boundary conditions (ABCs). Recently proposed transparent-influx boundary conditions (TIBCs) resolve a number of drawbacks typically attributed to PMLs and ABCs, such as introduction of spurious solutions and the inability to have a tight computational window. Unlike the PMLs or ABCs, the TIBCs lead to a nonlinear dependence of the boundary integral operator on the frequency. Thus, a nonlinear Helmholtz eigenvalue problem arises. \ud This paper presents an approach for solving such nonlinear eigenproblems which is based on a truncated singular value decomposition (SVD) polynomial approximation of the nonlinearity and subsequent solution of the obtained approximate polynomial eigenproblem with the Jacobi-Davidson method