Approximating smooth, multivariate functions on irregular domains

TL;DR

This paper presents a polynomial frame approximation method for smooth multivariate functions on irregular domains, providing error estimates, sample complexity analysis, and demonstrating its effectiveness in high dimensions.

Contribution

The paper introduces a simple polynomial frame approximation technique that works on irregular domains without prior domain knowledge, with comprehensive error and sample complexity analysis.

Findings

Error estimates for functions with Sobolev regularity

Sample complexity is quadratic in polynomial space dimension, independent of dimension d

Effective regularization yields well-conditioned approximations

Abstract

In this paper, we introduce a method known as polynomial frame approximation for approximating smooth, multivariate functions defined on irregular domains in $d$ dimensions, where $d$ can be arbitrary. This method is simple, and relies only on orthogonal polynomials on a bounding tensor-product domain. In particular, the domain of the function need not be known in advance. When restricted to a subdomain, an orthonormal basis is no longer a basis, but a frame. Numerical computations with frames present potential difficulties, due to the near-linear dependence of the truncated approximation system. Nevertheless, well-conditioned approximations can be obtained via regularization, for instance, truncated singular value decompositions. We comprehensively analyze such approximations in this paper, providing error estimates for functions with both classical and mixed Sobolev regularity, with the latter being particularly suitable for higher-dimensional problems. We also analyze the sample complexity of the approximation for sample points chosen randomly according to a probability measure, providing estimates in terms of the corresponding \textit{Nikolskii inequality} for the domain. In particular, we show that the sample complexity for points drawn from the uniform measure is quadratic (up to a log factor) in the dimension of the polynomial space, independently of $d$, for a large class of nontrivial domains. This extends a well-known result for polynomial approximation in hypercubes.