Publication

Discrete least-squares approximations over optimized downward closed polynomial spaces in arbitrary dimension

Fabio Nobile, Giovanni Migliorati
2017
Journal Articles

Abstract

We analyze the accuracy of the discrete least-squares approximation of a function $u$ in multivariate polynomial spaces $P_\Lambda:=span\{y\mapsto y^\nu \,: \, \nu\in \Lambda\}$ with $\Lambda\subset N_0^d$ 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 density as a particular case. Motivated by recent results in high-dimensional parametric and stochastic PDEs, 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 $L^2(\Gamma,\rho)$ 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. We show 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. In principle, this allows one to use these methods in high dimension. Our analysis builds upon (Chkifa et al. in ESAIM Math Model Numer Anal 49(3):815–837, 2015), which considered fixed and non-optimized downward closed multi-index sets. Potential applications of the proposed results are found in the development and analysis of efficient numerical methods for computing the solution of high-dimensional parametric or stochastic PDEs, but is not limited to this area.

Official source

https://infoscience.epfl.ch/entities/publication/3cf49e40-91e2-4474-bb14-9b3f84d05a4d

Fabio Nobile, Giovanni Migliorati
2017
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/3cf49e40-91e2-4474-bb14-9b3f84d05a4d

Related concepts (32)

Polynomial ring

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers.

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In mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n, Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles.

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In statistics, quality assurance, and survey methodology, sampling is the selection of a subset or a statistical sample (termed sample for short) of individuals from within a statistical population to estimate characteristics of the whole population. Statisticians attempt to collect samples that are representative of the population. Sampling has lower costs and faster data collection compared to recording data from the entire population, and thus, it can provide insights in cases where it is infeasible to measure an entire population.

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