Near-Optimal Sampling Strategies for Multivariate Function Approximation on General Domains

Abstract

Many problems arising in computational science and engineering can be described in terms of approximating a smooth function of d variables, defined over an unknown domain of interest Ω ⊂ R d , from sample data. Here both the underlying dimensionality of the problem (in the case d 1) as well as the lack of domain knowledge-with Ω potentially irregular and/or disconnected-are confounding factors for sampling-based methods. Naïve approaches for such problems often lead to wasted samples and inefficient approximation schemes. For example, uniform sampling can result in upwards of 20% wasted samples in some problems considered herein. In applications such as surrogate model construction in computational uncertainty quantification (UQ), the high cost of computing samples necessitates a more efficient sampling procedure. Over the last several years methods for computing such approximations from sample data have been studied in the case of irregular domains, and the advantages of computing sampling measures depending on an approximation space P of dim(P ) = N have been shown. More specifically, such approaches confer advantages such as stability and well-conditioning, with an asymptotically optimal sample complexity scaling O(N log(N )). The recently-proposed adaptive sampling for general domains (ASGD) strategy is one such technique to construct these sampling measures. The main contribution of this paper is a procedure to improve upon the ASGD approach by adaptively updating the sampling measure in the case of unknown domains. We achieve this by first introducing a general domain adaptivity strategy (GDAS), which computes an approximation of the function and domain of interest from the sample points. Second, we propose an adaptive sampling strategy, termed adaptive sampling for unknown domains (ASUD), which generates sampling measures over a domain that may not be known in advance, based on the ideas introduced in the ASGD approach. We then derive (weighted) least squares and augmented least squares techniques for polynomial approximation on unknown domains. We present numerical experiments demonstrating the efficacy of the adaptive sampling techniques with least squares-based polynomial approximation schemes. Our results show that the ASUD approach can reduce the computational cost by as much as 50% when compared with uniform sampling for such problems.

Approximating Smooth, Multivariate Functions on Irregular Domains

“…and µ is the measure given by (3.2). Moreover, the coefficients c of f Υ,Λ, satisfy 5) and the absolute ( 2 , L 2 )-condition number of the reconstruction operator L Υ,Λ, :…”

“…One solution to this problem is to employ discrete measures, supported over a fine grid that suitably fills Ω. This strategy, which uses ideas of [25], has been recently developed in [5,36]. Yet this procedure requires the domain Ω to be known in advance, and requires a fine grid to first be generated.…”

Optimal pointwise sampling for $L^2$ approximation

“…. , L n ), either by performing a first discretization of D with a large number of points, or by using a hierarchical method on a sequence of nested spaces [1,3,5,8,15,16]. These additional steps have complexities O(K n n 2 ) and O(n 4 ) respectively, where K n is the maximal value of the inverse Christoffel function n j=1 |L j | 2 which might grow more than linearly with n for certain choices of spaces V n .…”

“…As a final remark, let us to emphasize that although the results presented in our paper are mainly theorical and not practically satisfactory, due both to the computational complexity of the sparsification, and to the high values of the numerical constants C and K in Theorem 1, they provide some intuitive justification to the boosted least-squares methods presented in [8], which consist in removing points from the initial sample as long as the corresponding Gram matrix G X remains well conditioned. For instance, Lemma 4 allows to keep splitting the sample even after L steps, if one still has a framing 1 2 I G X 3 2 I and a sufficiently large ration |X| n . Nevertheless, it would be of much interest to find a randomized version of [4] giving a bound of the form (50), since this would give algorithmic tractability, smaller values for C and K, and the possibility to balance these constants in Theorem 1.…”

“…The reconstruction map R is obtained through a weighted least-squares method introduced in [6], which has already been discussed in several papers, see [1,5,8,9,7,15,16]. The weights involved are given by the expression…”