A practical framework for infinite-dimensional linear algebra

TL;DR

This paper introduces a practical, adaptive framework for solving infinite-dimensional linear equations, including differential equations with boundary conditions, using efficient algorithms implemented in Julia.

Contribution

It presents a novel data structure and adaptive QR algorithm for solving broad classes of infinite-dimensional linear equations efficiently.

Findings

Achieves $O(n^{ m opt})$ complexity for solving linear equations.

Handles tensor product PDEs with $O(n_y^2 n_x^{ m opt})$ operations.

Implemented in Julia's ApproxFun package with competitive performance.

Abstract

We describe a framework for solving a broad class of infinite-dimensional linear equations, consisting of almost banded operators, which can be used to resepresent linear ordinary differential equations with general boundary conditions. The framework contains a data structure on which row operations can be performed, allowing for the solution of linear equations by the adaptive QR approach. The algorithm achieves $O(n^{\rm opt})$ complexity, where $n^{\rm opt}$ is the number of degrees of freedom required to achieve a desired accuracy, which is determined adaptively. In addition, special tensor product equations, such as partial differential equations on rectangles, can be solved by truncating the operator in the $y$-direction with $n_y$ degrees of freedom and using a generalized Schur decomposition to upper triangularize, before applying the adaptive QR approach to the $x$-direction, requiring $O(n_y^2 n_x^{\rm opt})$ operations. The framework is implemented in the ApproxFun package written in the Julia programming language, which achieves highly competitive computational costs by exploiting unique features of Julia.