Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Optimal weighted least-squares methods   We consider the problem of reconstructing an unknown bounded function $u$
defined on a domain $X\subset \mathbb{R}^d$ from noiseless or noisy samples of
$u$ at $n$ points $(x^i)_{i=1,\dots,n}$. We measure the reconstruction error in
a norm $L^2(X,d\rho)$ for some given probability measure $d\rho$. Given a
linear space $V_m$ with ${\rm dim}(V_m)=m\leq n$, we study in general terms the
weighted least-squares approximations from the spaces $V_m$ based on
independent random samples. 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. The present
analysis covers also cases where the function $u$ and its approximants from
$V_m$ are unbounded, which might occur for instance in the relevant case where
$X=\mathbb{R}^d$ 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.

30
The categories of this paper are: q-alg
22
The categories of this paper are: chao-dyn
19
The categories of this paper are: eess
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
23
The categories of this paper are: solv-int
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The categories of this paper are: q-fin
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The categories of this paper are: math-ph
13
The categories of this paper are: quant-ph
10
The categories of this paper are: q-bio
8
The categories of this paper are: hep-ex
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The categories of this paper are: ao-sci
25
The categories of this paper are: patt-sol
7
The categories of this paper are: hep-th
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The categories of this paper are: comp-gas