Lectures on Randomized Numerical Linear Algebra

This chapter is based on lectures on Randomized Numerical Linear Algebra from the 2016 Park City Mathematics Institute summer school on The Mathematics of Data.Comment: To appear in the edited volume of lectures from the 2016 PCMI summer schoo

Fast randomized iteration: diffusion Monte Carlo through the lens of numerical linear algebra

We review the basic outline of the highly successful diffusion Monte Carlo technique commonly used in contexts ranging from electronic structure calculations to rare event simulation and data assimilation, and propose a new class of randomized iterative algorithms based on similar principles to address a variety of common tasks in numerical linear algebra. From the point of view of numerical linear algebra, the main novelty of the Fast Randomized Iteration schemes described in this article is that they work in either linear or constant cost per iteration (and in total, under appropriate conditions) and are rather versatile: we will show how they apply to solution of linear systems, eigenvalue problems, and matrix exponentiation, in dimensions far beyond the present limits of numerical linear algebra. While traditional iterative methods in numerical linear algebra were created in part to deal with instances where a matrix (of size $\mathcal{O}(n^2)$) is too big to store, the algorithms that we propose are effective even in instances where the solution vector itself (of size $\mathcal{O}(n)$) may be too big to store or manipulate. In fact, our work is motivated by recent DMC based quantum Monte Carlo schemes that have been applied to matrices as large as $10^{108} \times 10^{108}$. We provide basic convergence results, discuss the dependence of these results on the dimension of the system, and demonstrate dramatic cost savings on a range of test problems.Comment: 44 pages, 7 figure

Two-dimensional Banach spaces with Polynomial numerical index zero

We study two-dimensional Banach spaces with polynomial numerical indices equal to zero.Comment: 12 pages, to appear in Linear Algebra App

Gaussian elimination as an iterative algorithm

Gaussian elimination (GE) for solving an $n \times n$ linear system of equations $Ax=b$ is the archetypical direct method of numerical linear algebra, as opposed to iterative. In this note we want to point out that GE has an iterative side too