Computing with functions in spherical and polar geometries II. The disk

TL;DR

This paper introduces algorithms for efficiently approximating and computing with smooth functions on the unit disk, enabling fast operations like differentiation, integration, and solving Poisson equations using low rank structures and FFT techniques.

Contribution

It develops a structure-preserving, adaptive low rank approximation method for functions on the disk, combining the double Fourier sphere approach with Gaussian elimination, and introduces a fast disk Poisson solver.

Findings

Converges geometrically for certain analytic functions.

Produces stable, near-optimal low rank approximations.

Enables fast differentiation, integration, and Poisson solving on the disk.

Abstract

A collection of algorithms is described for numerically computing with smooth functions defined on the unit disk. Low rank approximations to functions in polar geometries are formed by synthesizing the disk analogue of the double Fourier sphere method with a structure-preserving variant of iterative Gaussian elimination that is shown to converge geometrically for certain analytic functions. This adaptive procedure is near-optimal in its sampling strategy, producing approximants that are stable for differentiation and facilitate the use of FFT-based algorithms in both variables. The low rank form of the approximants is especially useful for operations such as integration and differentiation, reducing them to essentially 1D procedures, and it is also exploited to formulate a new fast disk Poisson solver that computes low rank approximations to solutions. This work complements a companion paper (Part I) on computing with functions on the surface of the unit sphere.