Localization theorems for nonlinear eigenvalue problems

Let T : \Omega \rightarrow \bbC^{n \times n} be a matrix-valued function that is analytic on some simply-connected domain \Omega \subset \bbC. A point $\lambda \in \Omega$ is an eigenvalue if the matrix $T(\lambda)$ is singular. In this paper, we describe new localization results for nonlinear eigenvalue problems that generalize Gershgorin's theorem, pseudospectral inclusion theorems, and the Bauer-Fike theorem. We use our results to analyze three nonlinear eigenvalue problems: an example from delay differential equations, a problem due to Hadeler, and a quantum resonance computation.Comment: Submitted to SIMAX. 22 pages, 11 figure

Localization for MCMC: sampling high-dimensional posterior distributions with local structure

We investigate how ideas from covariance localization in numerical weather prediction can be used in Markov chain Monte Carlo (MCMC) sampling of high-dimensional posterior distributions arising in Bayesian inverse problems. To localize an inverse problem is to enforce an anticipated "local" structure by (i) neglecting small off-diagonal elements of the prior precision and covariance matrices; and (ii) restricting the influence of observations to their neighborhood. For linear problems we can specify the conditions under which posterior moments of the localized problem are close to those of the original problem. We explain physical interpretations of our assumptions about local structure and discuss the notion of high dimensionality in local problems, which is different from the usual notion of high dimensionality in function space MCMC. The Gibbs sampler is a natural choice of MCMC algorithm for localized inverse problems and we demonstrate that its convergence rate is independent of dimension for localized linear problems. Nonlinear problems can also be tackled efficiently by localization and, as a simple illustration of these ideas, we present a localized Metropolis-within-Gibbs sampler. Several linear and nonlinear numerical examples illustrate localization in the context of MCMC samplers for inverse problems.Comment: 33 pages, 5 figure

Two-Level discretization techniques for ground state computations of Bose-Einstein condensates

This work presents a new methodology for computing ground states of Bose-Einstein condensates based on finite element discretizations on two different scales of numerical resolution. In a pre-processing step, a low-dimensional (coarse) generalized finite element space is constructed. It is based on a local orthogonal decomposition and exhibits high approximation properties. The non-linear eigenvalue problem that characterizes the ground state is solved by some suitable iterative solver exclusively in this low-dimensional space, without loss of accuracy when compared with the solution of the full fine scale problem. The pre-processing step is independent of the types and numbers of bosons. A post-processing step further improves the accuracy of the method. We present rigorous a priori error estimates that predict convergence rates H^3 for the ground state eigenfunction and H^4 for the corresponding eigenvalue without pre-asymptotic effects; H being the coarse scale discretization parameter. Numerical experiments indicate that these high rates may still be pessimistic.Comment: Accepted for publication in SIAM J. Numer. Anal., 201

The spectrum of kernel random matrices

We place ourselves in the setting of high-dimensional statistical inference where the number of variables $p$ in a dataset of interest is of the same order of magnitude as the number of observations $n$. We consider the spectrum of certain kernel random matrices, in particular $n\times n$ matrices whose $(i,j)$th entry is $f(X_i'X_j/p)$ or $f(\Vert X_i-X_j\Vert^2/p)$ where $p$ is the dimension of the data, and $X_i$ are independent data vectors. Here $f$ is assumed to be a locally smooth function. The study is motivated by questions arising in statistics and computer science where these matrices are used to perform, among other things, nonlinear versions of principal component analysis. Surprisingly, we show that in high-dimensions, and for the models we analyze, the problem becomes essentially linear--which is at odds with heuristics sometimes used to justify the usage of these methods. The analysis also highlights certain peculiarities of models widely studied in random matrix theory and raises some questions about their relevance as tools to model high-dimensional data encountered in practice.Comment: Published in at http://dx.doi.org/10.1214/08-AOS648 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Integral Equation Methods in Optical Waveguide Theory

Optical waveguides are regular dielectric rods having various cross-sectional shapes where generally the permittivity may vary in the waveguide's cross section. The permittivity of the surrounding medium may be a step-index function of coordinates. The eigenvalue problems for natural modes (surface and leaky eigenmodes) of inhomogeneous optical waveguides in the weakly guiding approximation formulated as problems for Helmholtz equations with partial radiation conditions at infinity in the cross-sectional plane. The original problems are reduced with the aid of the integral equation method (using appropriate Green functions) to nonlinear spectral problems with Fredholm integral operators. Theorems on the spectrum localization are proved. It is shown that the sets of all eigenvalues of the original problems may consist of isolated points on the Riemann surface and each eigenvalue depends continuously on the frequency and permittivity and can appear or disappear only at the boundary of the Riemann surface. The original problems for surface waves are reduced to linear eigenvalue problems for integral operators with real-valued symmetric polar kernels. The existence, localization, and dependence on parameters of the spectrum are investigated. The collocation method for numerical calculations of the natural modes is proposed, the convergence of the method is proved, and some results of numerical experiments are discussed. © Springer International Publishing Switzerland 2013

Long time dynamics and coherent states in nonlinear wave equations

We discuss recent progress in finding all coherent states supported by nonlinear wave equations, their stability and the long time behavior of nearby solutions.Comment: bases on the authors presentation at 2015 AMMCS-CAIMS Congress, to appear in Fields Institute Communications: Advances in Applied Mathematics, Modeling, and Computational Science 201