Linear Algebra with Disordered Sparse Matrices that have Spatial Structure: Theory and Computation

TL;DR

This thesis advances the understanding of disordered sparse matrices with spatial structure by developing new algorithms, analyzing errors, and exploring localization phenomena in matrix functions within scientific computing.

Contribution

It introduces novel O(N) and O(N log N) algorithms, a new sigma model for disordered systems, and provides theoretical and numerical insights into localization and eigenfunction properties.

Findings

Errors in O(N) algorithms for matrix functions are characterized.

Localization and multifractality of eigenfunctions are numerically demonstrated.

A new sigma model reproduces results of the supersymmetric sigma model.

Abstract

This Ph.D. thesis contains original contributions to several areas within the disciplines of disordered systems, numerical linear algebra, and scientific computing: (1) Theoretical and numerical study of the errors caused by using certain O(N) algorithms for evaluating matrix functions. (2) Numerical results on the localization of matrix functions, and on the length scales of eigenfunctions. (3) A simple model which generalizes Berry's model of wavefunctions in chaotic systems to include both localization and multifractality. (4) A proposal of a new sigma model for disordered systems which does not involve graded matrixes but should reproduce the same results as the supersymmetric sigma model. A detailed derivation of the new sigma model is provided. (5) Proposals for many new O(N) algorithms, and for many O(N log N) algorithms suitable to systems with many length scales. (6) A review of the problems which computing can cause for the physics community, and of the physics community's current efforts to handle those problems. Following this review is a set of recommendations for better managing these problems.