Optimal weighted least-squares methods
The SMAI journal of computational mathematics, Volume 3 (2017) , pp. 181-203.

We consider the problem of reconstructing an unknown bounded function $u$ defined on a domain $X\subset {ℝ}^{d}$ from noiseless or noisy samples of $u$ at $n$ points ${\left({x}^{i}\right)}_{i=1,\cdots ,n}$. We measure the reconstruction error in a norm ${L}^{2}\left(X,d\rho \right)$ for some given probability measure $d\rho$. Given a linear space ${V}_{m}$ with $\mathrm{dim}\left({V}_{m}\right)=m\le n$, we study in general terms the weighted least-squares approximations from the spaces ${V}_{m}$ based on independent random samples. It is well known that least-squares approximations can be inaccurate and unstable when $m$ is too close to $n$, even in the noiseless case. Recent results from [6, 7] have shown the interest of using weighted least squares for reducing the number $n$ of samples that is needed to achieve an accuracy comparable to that of best approximation in ${V}_{m}$, compared to standard least squares as studied in [4]. The contribution of the present paper is twofold. From the theoretical perspective, we establish results in expectation and in probability for weighted least squares in general approximation spaces ${V}_{m}$. These results show that for an optimal choice of sampling measure $d\mu$ and weight $w$, which depends on the space ${V}_{m}$ and on the measure $d\rho$, stability and optimal accuracy are achieved under the mild condition that $n$ scales linearly with $m$ up to an additional logarithmic factor. In contrast to [4], the present analysis covers cases where the function $u$ and its approximants from ${V}_{m}$ are unbounded, which might occur for instance in the relevant case where $X={ℝ}^{d}$ and $d\rho$ is the Gaussian measure. From the numerical perspective, we propose a sampling method which allows one to generate independent and identically distributed samples from the optimal measure $d\mu$. This method becomes of interest in the multivariate setting where $d\mu$ is generally not of tensor product type. We illustrate this for particular examples of approximation spaces ${V}_{m}$ of polynomial type, where the domain $X$ is allowed to be unbounded and high or even infinite dimensional, motivated by certain applications to parametric and stochastic PDEs.

Published online:
DOI: https://doi.org/10.5802/smai-jcm.24
Classification: 41A10,  41A25,  41A65,  62E17,  93E24
Keywords: multivariate approximation, weighted least squares, error analysis, convergence rates, random matrices, conditional sampling, polynomial approximation.
@article{SMAI-JCM_2017__3__181_0,
author = {Albert Cohen and Giovanni Migliorati},
title = {Optimal weighted least-squares methods},
journal = {The SMAI journal of computational mathematics},
pages = {181--203},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {3},
year = {2017},
doi = {10.5802/smai-jcm.24},
mrnumber = {3716755},
zbl = {1416.62177},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.24/}
}
Albert Cohen; Giovanni Migliorati. Optimal weighted least-squares methods. The SMAI journal of computational mathematics, Volume 3 (2017) , pp. 181-203. doi : 10.5802/smai-jcm.24. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.24/

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