Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Discrete least-squares approximations over optimized downward closed
  polynomial spaces in arbitrary dimension   We analyze the accuracy of the discrete least-squares approximation of a
function $u$ in multivariate polynomial spaces $\mathbb{P}_\Lambda:={\rm span}
\{y\mapsto y^\nu \,: \, \nu\in \Lambda\}$ with $\Lambda\subset \mathbb{N}_0^d$
over the domain $\Gamma:=[-1,1]^d$, based on the sampling of this function at
points $y^1,\dots,y^m \in \Gamma$. The samples are independently drawn
according to a given probability density $\rho$ belonging to the class of
multivariate beta densities, which includes the uniform and Chebyshev densities
as particular cases. We restrict our attention to polynomial spaces associated
with \emph{downward closed} sets $\Lambda$ of \emph{prescribed} cardinality
$n$, and we optimize the choice of the space for the given sample. This
implies, in particular, that the selected polynomial space depends on the
sample. We are interested in comparing the error of this least-squares
approximation measured in $L^2(\Gamma,d\rho)$ with the best achievable
polynomial approximation error when using downward closed sets of cardinality
$n$. We establish conditions between the dimension $n$ and the size $m$ of the
sample, under which these two errors are proven to be comparable. Our main
finding is that the dimension $d$ enters only moderately in the resulting
trade-off between $m$ and $n$, in terms of a logarithmic factor $\ln(d)$, and
is even absent when the optimization is restricted to a relevant subclass of
downward closed sets, named {\it anchored} sets. In principle, this allows one
to use these methods in arbitrarily high or even infinite dimension. Our
analysis builds upon [2] which considered fixed and nonoptimized downward
closed multi-index sets. Potential applications of the proposed results are
found in the development and analysis of numerical methods for computing the
solution to high-dimensional parametric or stochastic PDEs.

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The categories of this paper are: eess
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The categories of this paper are: nucl-ex
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The categories of this paper are: dg-ga
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The categories of this paper are: astro-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: cond-mat
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ph
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The categories of this paper are: q-bio
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The categories of this paper are: chao-dyn
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The categories of this paper are: cmp-lg
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The categories of this paper are: acc-phys
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The categories of this paper are: solv-int
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The categories of this paper are: hep-lat
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The categories of this paper are: plasm-ph
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The categories of this paper are: comp-gas