An Adaptive Sampling and Domain Learning Strategy for Multivariate Function Approximation on Unknown Domains

Abstract. Many problems arising in computational science and engineering can be described in terms of approximating a smooth function of [Formula: see text] variables, defined over an unknown domain of interest [Formula: see text], from sample data. Here both the underlying dimensionality of the problem (in the case [Formula: see text]) as well as the lack of domain knowledge—with [Formula: see text] 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 upward of 20% wasted samples in some problems considered herein. In applications such as surrogate model construction in computational uncertainty quantification, 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 [Formula: see text] of [Formula: see text] have been shown. More specifically, such approaches confer advantages such as stability and well-conditioning, with an asymptotically optimal sample complexity scaling [Formula: see text]. 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, 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 consistently achieves errors the same as or smaller than uniform sampling, but using fewer, and often significantly fewer, function evaluations.

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