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首页> 外文期刊>Journal of complexity >Optimal pointwise sampling for L~2 approximation
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Optimal pointwise sampling for L~2 approximation

机译：Optimal pointwise sampling for L~2 approximation

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摘要

Given a function u is an element of L-2 = L-2(D, mu), where mu is a measure on a set D, and a linear subspace V-n subset of L-2 of dimension n, we show that near-best approximation of u in V-n can be computed from a near-optimal budget of Cn pointwise evaluations of u, with C 1 a universal constant. The sampling points are drawn according to some random distribution, the approximation is computed by a weighted least-squares method, and the error is assessed in expected L-2 norm. This result improves on the results in [6, 8] which require a sampling budget that is sub-optimal by a logarithmic factor, thanks to a sparsification strategy introduced in [17,18]. As a consequence, we obtain for any compact class K subset of L-2 that the sampling number rho(rand)(Cn) (K)(L2) in the randomized setting is dominated by the Kolmogorov n-width d(n)(K)(L2). While our result shows the existence of a randomized sampling with such near-optimal properties, we discuss remaining issues concerning its generation by a computationally efficient algorithm. (C) 2021 Elsevier Inc. All rights reserved.

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