In this paper, we address the problem of approximating a multivariate function defined on a general domain in $d$ dimensions from sample points. We consider weighted least-squares approximation in an arbitrary finite-dimensional space $P$ from independent random samples taken according to a suitable measure. In general, least-squares approximations can be inaccurate and ill-conditioned when the number of sample points $M$ is close to $N = \dim(P)$. To counteract this, we introduce a novel method for sampling in general domains which leads to provably accurate and well-conditioned approximations. The resulting sampling measure is discrete, and therefore straightforward to sample from. Our main result shows near-optimal sample complexity for this procedure; specifically, $M = \mathcal{O}(N \log(N))$ samples suffice for a well-conditioned and accurate approximation. Numerical experiments on polynomial approximation in general domains confirm the benefits of this method over standard sampling.
We develop a general framework for estimating the $L_\infty(\mathbb{T}^d)$ error for the approximation of multivariate periodic functions belonging to specific reproducing kernel Hilbert spaces (RHKS) using approximants that are trigonometric polynomials computed from sampling values. The used sampling schemes are suitable sets of rank-1 lattices that can be constructed in an extremely efficient way. Furthermore, the structure of the sampling schemes allows for fast Fourier transform (FFT) algorithms. We present and discuss one FFT algorithm and analyze the worst case $L_\infty$ error for this…
Many problems in computational science and engineering can be described in terms of approximating a smooth function of $d$ variables, defined over an unknown domain of interest $Ω\subset \mathbb{R}^d$, from sample data. Here both the curse of dimensionality ($d\gg 1$) and the lack of domain knowledge with $Ω$ potentially irregular and/or disconnected are confounding factors for sampling-based methods. Naïve approaches often lead to wasted samples and inefficient approximation schemes. For example, uniform sampling can result in upwards of 20\% wasted samples in some problems. In surrogate mode…
We prove a.s. (almost sure) unisolvency of interpolation by continuous random sampling with respect to any given density, in spaces of multivariate a.e. (almost everywhere) analytic functions. Examples are given concerning polynomial and RBF approximation.
General multivariate distributions are notoriously expensive to sample from, particularly the high-dimensional posterior distributions in PDE-constrained inverse problems. This paper develops a sampler for arbitrary continuous multivariate distributions that is based on low-rank surrogates in the tensor-train format. We construct a tensor-train approximation to the target probability density function using the cross interpolation, which requires a small number of function evaluations. For sufficiently smooth distributions the storage required for the TT approximation is moderate, scaling linea…
Phase retrieval consists in the recovery of an unknown signal from phaseless measurements of its usually complex-valued Fourier transform. Without further assumptions, this problem is notorious to be severe ill posed such that the recovery of the true signal is nearly impossible. In certain applications like crystallography, speckle imaging in astronomy, or blind channel estimation in communications, the unknown signal has a specific, sparse structure. In this paper, we exploit these sparse structure to recover the unknown signal uniquely up to inevitable ambiguities as global phase shifts, tr…
The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. We use recent general results on sampling discretization to derive a new Marcinkiewicz type discretization theorem for the multivariate trigonometric polynomials with frequencies from the hyperbolic crosses. It is shown that recently developed techniques allow us to improve the known results in this direction.
Approximation properties of periodic quasi-projection operators with matrix dilations are studied. Such operators are generated by a sequence of functions $φ_j$ and a sequence of distributions/functions $\widetildeφ_j$. Error estimates for sampling-type quasi-projection operators are obtained under the periodic Strang-Fix conditions for $φ_j$ and the compatibility conditions for $φ_j$ and $\widetildeφ_j$. These estimates are given in terms of the Fourier coefficients of approximated functions and provide analogs of some known non-periodic results. Under some additional assumptions error estima…
We tensorize the Faber spline system from [14] to prove sequence space isomorphisms for multivariate function spaces with higher mixed regularity. The respective basis coefficients are local linear combinations of discrete function values similar as for the classical Faber Schauder system. This allows for a sparse representation of the function using a truncated series expansion by only storing discrete (finite) set of function values. The set of nodes where the function values are taken depends on the respective function in a non-linear way. Indeed, if we choose the basis functions adaptively…
For the multivariate trigonometric polynomials we study convolution with the corresponding the de la Vallee Poussin kernel from the point of view of discretization. In other words, we replace the normalized Lebesgue measure by a discrete measure in such a way, which preserves the convolution properties and provides sampling discretization of integral norms. We prove that in the two-variate case the Fibonacci point sets provide an ideal (in the sense of order) solution. We also show that the Korobov point sets provide a suboptimal (up to logarithmic factors) solution for an arbitrary number of …
