Convergence Analysis of Extended LOBPCG for Computing Extreme Eigenvalues

This paper is concerned with the convergence analysis of an extended variation of the locally optimal preconditioned conjugate gradient method (LOBPCG) for the extreme eigenvalue of a Hermitian matrix polynomial which admits some extended form of Rayleigh quotient. This work is a generalization of the analysis by Ovtchinnikov (SIAM J. Numer. Anal., 46(5):2567-2592, 2008). As instances, the algorithms for definite matrix pairs and hyperbolic quadratic matrix polynomials are shown to be globally convergent and to have an asymptotically local convergence rate. Also, numerical examples are given to illustrate the convergence.Comment: 21 pages, 2 figure

The Generalized Graetz Problem in Finite Domains

We consider the generalized Graetz problem associated with stationary convection-diffusion inside a domain having any regular three-dimensional translationally invariant section and finite or semi-infinite extent. Our framework encompasses any previous “extended” and “conjugated” Graetz generalizations and provides theoretical bases for computing the orthogonal set of generalized two-dimensional Graetz modes. The theoretical framework includes both heterogeneous and possibly anisotropic diffusion tensors. In the case of semi-infinite domains, the existence of a bounded solution is shown from the analysis of two-dimensional operator eigenvectors which form a basis of L2 . In the case of finite domains a similar basis can be exhibited, and the mode’s amplitudes can be obtained from the inversion of newly defined finite domain operator. Our analysis includes both the theoretical and practical issues associated with this finite domain operator inversion as well as its interpretation as a multireflection image method. Error estimates are provided when numerically truncating the spectrum to a finite number of modes. Numerical examples are validated for reference configurations and provided in nontrivial cases. Our methodology shows how to map the solution of stationary convection-diffusion problems in finite three-dimensional domains into a two-dimensional operator spectrum, which leads to a drastic reduction in computational cost

A simple criterion of transverse linear instability for solitary waves

We prove an abstract instability result for an eigenvalue problem with parameter. We apply this criterion to show the transverse linear instability of solitary waves on various examples from mathematical physics.Comment: The main result has been improved and its proof simplifie