Optimal sampling strategies for multivariate function approximation on general domains
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In this paper, we address the problem of approximating a multivariate function defined on a general domain in d dimensions from sample points. We consider weighted least-squares approximation in an arbitrary finite-dimensional space P from independent random samples taken according to a suitable measure. In general, least-squares approximations can be inaccurate and ill conditioned when the number of sample points M is close to N = (P). To counteract this, we introduce a novel method for sampling in general domains which leads to provably accurate and well-conditioned weighted least-squares approximations. The resulting sampling measure is discrete, and therefore straightforward to sample from. Our main result shows near optimal sample complexity for this procedure; specifically, M = O(N (N)) samples suffice for a well conditioned and accurate approximation. Numerical experiments on polynomial approximation in general domains confirm the benefits of this method over standard sampling.
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