Fast Direct Solvers

TL;DR

This survey reviews fast direct solvers for linear systems from PDEs and integral equations, highlighting recent algorithms that achieve near-linear time complexity by exploiting data-sparsity.

Contribution

It unifies sparse and dense fast direct solver methods, introduces key concepts, and guides users in selecting appropriate algorithms for different applications.

Findings

Algorithms achieve near-linear time complexity.

Matrix inverse approximations exploit low-rank structures.

Provides a unified framework for sparse and dense solvers.

Abstract

This survey describes a class of methods known as "fast direct solvers". These algorithms address the problem of solving a system of linear equations $\boldsymbol{Ax}=\boldsymbol{b}$ arising from the discretization of either an elliptic PDE or of an associated integral equation. The matrix $\boldsymbol{A}$ will be sparse when the PDE is discretized directly, and dense when an integral equation formulation is used. In either case, industry practice for large scale problems has for decades been to use iterative solvers such as multigrid, GMRES, or conjugate gradients. A direct solver, in contrast, builds an approximation to the inverse of $\boldsymbol{A}$, or alternatively, an easily invertible factorization (e.g. LU or Cholesky). A major development in numerical analysis in the last couple of decades has been the emergence of algorithms for constructing such factorizations or performing such inversions in linear or close to linear time. Such methods must necessarily exploit that the matrix $\boldsymbol{A}^{-1}$ is "data-sparse", typically in the sense that it can be tessellated into blocks that have low numerical rank. This survey provides a unifying context to both sparse and dense fast direct solvers, introduces key concepts with a minimum of notational overhead, and provides guidance to help a user determine the best method to use for a given application.