Multivariate Approximation in Downward Closed Polynomial Spaces

Abstract

The task of approximating a function of d variables from its evaluations at a given number of points is ubiquitous in numerical analysis and engineering applications. When d is large, this task is challenged by the so-called curse of dimensionality. As a typical example, standard polynomial spaces, such as those of total degree type, are often uneffective to reach a prescribed accuracy unless a prohibitive number of evaluations is invested. In recent years it has been shown that, for certain relevant applications, there are substantial advantages in using certain sparse polynomial spaces having anisotropic features with respect to the different variables. These applications include in particular the numerical approximation of high-dimensional parametric and stochastic partial differential equations. We start by surveying several results in this direction, with an emphasis on the numerical algorithms that are available for the construction of the approximation, in particular through interpolation or discrete least-squares fitting. All such algorithms rely on the assumption that the set of multi-indices associated with the polynomial space is downward closed. In the present paper we introduce some tools for the study of approximation in multivariate spaces under this assumption, and use them in the derivation of error bounds, sometimes independent of the dimension d, and in the development of adaptive strategies.

Uncertainty Quantification for Low-Frequency, Time-Harmonic Maxwell Equations with Stochastic Conductivity Models

“…The best N -term approximation is obtained by replacing F with an index set Λ N with #(Λ n ) = N , which corresponds to the N largest norms f ν S . The work in [6] (see also [29]) proves that under certain assumptions, best N -term Hermite approximations converge in L 2 (U, S, γ G ) with dimension independent rates. In our case, we are interested in the closely related convergence of the SQ approximation.…”

“…Attempts to approximate these integrals are then faced with the challenge of achieving low computational costs and fast convergence rates that are affected as little as possible by the curse of high dimensionality. An approach that can be competitive in this regard, is high-dimensional polynomial approximation of parametric PDEs, which has been analysed in a number of publications [28,27,26,6,24,25,29]. The theoretical results show that sparsity or summability in the parametric representation of the input translates into sparse polynomial approximations for the output that converge with dimension-independent rates which depend on the level of sparsity in the input.…”

Compressive Hermite Interpolation: Sparse, High-Dimensional Approximation from Gradient-Augmented Measurements

“…Lower sets (also known as monotone or downward closed sets) have been studied extensively in the context of multivariate polynomial approximation [10,11,17]. In particular, for functions arising as solutions of a broad class of parametric PDEs it has been shown that there exist sequences of lower sets of cardinality s which achieve the same approximation error bounds as those of the best s-term approximation [12].…”

Pluripotential Numerics

“…This is motivated by the growing interest that Pluripotential Theory is achieving in applications during the last years. We mention, among the others, the quest for nearly optimal sampling in least squares regression [30,42,22], random polynomials [14,50] and estimation of approximation numbers of a given function [46].…”

“…Lastly, we mention an application of our methods that is ready at hand. Very recently polynomial spaces with non-standard degree ordering (e.g., not total degree nor tensor degree) start to attract a certain attention in the framework of random sampling [22], Approximation Theory [46], and Pluripotential Theory [18]. For instance, one can consider spaces of polynomials of the form P k q := span{z α , α i ∈ N n , q(α) < k}, where q is any norm or even, more generally, P k P := span{z α , α i ∈ N n , α ∈ kP} for any P ⊂ R n + closed and star-shaped with respect to 0 such that ∪ k∈N kP = R n + .…”