Related papers: Randomized least-squares with minimal oversampling…
We study polynomial approximation on a $d$-cube, where $d$ is large, and compare interpolation on sparse grids, aka Smolyak's algorithm (SA), with a simple least squares method based on randomly generated points (LS) using standard…
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…
In this work, we propose a novel sampling method for Design of Experiments. This method allows to sample such input values of the parameters of a computational model for which the constructed surrogate model will have the least possible…
We obtain a result concerning the stability under the interpolation with functional parameter method for the approximation spaces of Lorentz-Marcinkiewicz type and also for the approximation spaces generated by symmetric norming functions…
Functions of interest are often smooth and sparse in some sense, and both priors should be taken into account when interpolating sampled data. Classical linear interpolation methods are effective under strong regularity assumptions, but…
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…
The sparse polynomial approximation of continuous functions has emerged as a prominent area of interest in function approximation theory in recent years. A key challenge within this domain is the accurate estimation of approximation errors.…
We present a new rational approximation algorithm based on the empirical interpolation method for interpolating a family of parametrized functions to rational polynomials with invariant poles, leading to efficient numerical algorithms for…
This paper concerns the approximation of smooth, high-dimensional functions from limited samples using polynomials. This task lies at the heart of many applications in computational science and engineering - notably, some of those arising…
We propose extensions and improvements of the statistical analysis of distributed multipoles (SADM) algorithm put forth by Chipot et al. in [6] for the derivation of distributed atomic multipoles from the quantum-mechanical electrostatic…
We propose and analyze a weighted greedy scheme for computing deterministic sample configurations in multidimensional space for performing least-squares polynomial approximations on $L^2$ spaces weighted by a probability density function.…
We consider the problem of approximating an unknown function from point evaluations. This problem is a crucial subproblem in many modern (nonlinear) approximation schemes. When obtaining these point evaluations is costly, minimising the…
Least squares approximation is a technique to find an approximate solution to a system of linear equations that has no exact solution. In a typical setting, one lets $n$ be the number of constraints and $d$ be the number of variables, with…
Compressive sampling has become a widely used approach to construct polynomial chaos surrogates when the number of available simulation samples is limited. Originally, these expensive simulation samples would be obtained at random locations…
We consider the problem of reconstructing an unknown function $u\in L^2(D,\mu)$ from its evaluations at given sampling points $x^1,\dots,x^m\in D$, where $D\subset \mathbb R^d$ is a general domain and $\mu$ a probability measure. The…
We propose and analyse numerical algorithms based on weighted least squares for the approximation of a real-valued function on a general bounded domain $\Omega \subset \mathbb{R}^d$. Given any $n$-dimensional approximation space $V_n…
The randomized row method is a popular representative of the iterative algorithm because of its efficiency in solving the overdetermined and consistent systems of linear equations. In this paper, we present an extended randomized multiple…
Given any domain $X\subseteq \mathbb{R}^d$ and a probability measure $\rho$ on $X$, we study the problem of approximating in $L^2(X,\rho)$ a given function $u:X\to\mathbb{R}$, using its noiseless pointwise evaluations at random samples. For…
We propose AAA rational approximation as a method for interpolating or approximating smooth functions from equispaced data samples. Although it is always better to approximate from large numbers of samples if they are available, whether…
We present an algorithm for approximating a function defined over a $d$-dimensional manifold utilizing only noisy function values at locations sampled from the manifold with noise. To produce the approximation we do not require any…