Approximating smooth, multivariate functions on irregular domains

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 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.

Frame approximation with bounded coefficients

“…The work of [3] on numerical frame approximation originated with the study of Fourier extensions [5], which are motivated by embedding methods for PDEs [4,7,9,13,14]. More recently, fast algorithms have been developed [10,11,12], as well as approaches suitable for higherdimensional problems [1]. Frames are well-known tools in modern signal and image processing, coding theory and sampling theory.…”

“…This frame is particularly useful for approximating smooth functions over Ω, when constructing an orthonormal basis over Ω is difficult (e.g. Ω is an irregular domain) [1].…”

“…Orthogonal or wellconditioned bases can be difficult to construct in certain problems, hence moving towards near-redundant systems can offer substantial flexibility. In a series of recent works [1,2,3], a framework for approximation in certain redundant systems of functions arising as so-called frames of Hilbert spaces has been developed. Frames can provide viable alternatives in approximation problems where it can be challenging or possibly infeasible to devise good orthonormal bases.…”

“…Frames in particular always lead to ill-conditioned linear systems (for large enough values of the truncation parameter), and this ill-conditioning can be arbitrarily bad [3]. Fortunately, as proposed in [1,2,3], this ill-conditioning can be addressed through regularization, for instance, Truncated Singular Value Decompositions (TSVD). The result is a well-conditioned approximation with provable error guarantees.…”

Estimates of the asymptotic Nikolskii constants for spherical polynomials

“…The constant L * (d) can be used to obtain some lower tight-bounds for spherical designs (see [28]). Moreover, the Nikolskii constants play an important role in approximation of smooth, multivariate functions defined on irregular domains by polynomial frame approximation method [7]. More detailed historical comments on the constant L * (d) and related background information will be given in Section 2.…”