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Many problems in computational science and engineering can be described interms of approximating a smooth function of $d$ variables, defined over anunknown domain of interest $\Omega\subset \mathbb{R}^d$, from sample data. Hereboth the curse of dimensionality ($d\gg 1$) and the lack of domain knowledgewith $\Omega$ potentially irregular and/or disconnected are confounding factorsfor sampling-based methods. Na\"{i}ve approaches often lead to wasted samplesand inefficient approximation schemes. For example, uniform sampling can resultin upwards of 20\% wasted samples in some problems. In surrogate modelconstruction in computational uncertainty quantification (UQ), the high cost ofcomputing samples needs a more efficient sampling procedure. In the last years,methods for computing such approximations from sample data have been studied inthe case of irregular domains. The advantages of computing sampling measuresdepending on an approximation space $P$ of $\dim(P)=N$ have been shown. Inparticular, such methods confer advantages such as stability andwell-conditioning, with $\mathcal{O}(N\log(N))$ as sample complexity. Therecently-proposed adaptive sampling for general domains (ASGD) strategy is onemethod to construct these sampling measures. The main contribution of thispaper is to improve ASGD by adaptively updating the sampling measures overunknown domains. We achieve this by first introducing a general domainadaptivity strategy (GDAS), which approximates the function and domain ofinterest from sample points. Second, we propose adaptive sampling for unknowndomains (ASUD), which generates sampling measures over a domain that may not beknown in advance. Our results show that the ASUD approach consistently achievesthe same or smaller errors as uniform sampling, but using fewer, and oftensignificantly fewer evaluations.

An Adaptive sampling and domain learning strategy for multivariate function approximation on unknown domains