  Many problems 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 $\Omega\subset \mathbb{R}^d$, from sample data. Here
both the curse of dimensionality ($d\gg 1$) and the lack of domain knowledge
with $\Omega$ potentially irregular and/or disconnected are confounding factors
for sampling-based methods. Na\"{i}ve approaches often lead to wasted samples
and inefficient approximation schemes. For example, uniform sampling can result
in upwards of 20\% wasted samples in some problems. In surrogate model
construction in computational uncertainty quantification (UQ), the high cost of
computing samples needs a more efficient sampling procedure. In the last years,
methods for computing such approximations from sample data have been studied in
the case of irregular domains. The advantages of computing sampling measures
depending on an approximation space $P$ of $\dim(P)=N$ have been shown. In
particular, such methods confer advantages such as stability and
well-conditioning, with $\mathcal{O}(N\log(N))$ as sample complexity. The
recently-proposed adaptive sampling for general domains (ASGD) strategy is one
method to construct these sampling measures. The main contribution of this
paper is to improve ASGD by adaptively updating the sampling measures over
unknown domains. We achieve this by first introducing a general domain
adaptivity strategy (GDAS), which approximates the function and domain of
interest from sample points. Second, we propose adaptive sampling for unknown
domains (ASUD), which generates sampling measures over a domain that may not be
known in advance. Our results show that the ASUD approach consistently achieves
the same or smaller errors as uniform sampling, but using fewer, and often
significantly fewer evaluations.

Numerical Analysis;Numerical Analysis