On the reconstruction of functions from values at subsampled quadrature points

TL;DR

This paper introduces a subsampling method for structured quadrature points that maintains optimal function reconstruction accuracy, reduces computational costs, and is applicable in high dimensions, demonstrated on rank-1 lattices.

Contribution

It connects structured and unstructured sampling approaches, proposing a subsampling technique that preserves stability and optimal convergence rates in function reconstruction.

Findings

Subsampling of quadrature points retains optimal error decay.

The method is dimension-independent and computationally efficient.

Numerical experiments confirm theoretical results.

Abstract

This paper is concerned with function reconstruction from samples. The sampling points used in several approaches are (1) structured points connected with fast algorithms or (2) unstructured points coming from, e.g., an initial random draw to achieve an improved information complexity. We connect both approaches and propose a subsampling of structured points in an offline step. In particular, we start with structured quadrature points (QMC), which provide stable $L_2$ reconstruction properties. The subsampling procedure consists of a computationally inexpensive random step followed by a deterministic procedure to further reduce the number of points while keeping its information. In these points functions (belonging to a RKHS of bounded functions) will be sampled and reconstructed from whilst achieving state of the art error decay. Our method is dimension-independent and is applicable as soon as we know some initial quadrature points. We apply our general findings on the $d$-dimensional torus to subsample rank-1 lattices, where it is known that full rank-1 lattices lose half the optimal order of convergence (expressed in terms of the size of the lattice). In contrast to that, our subsampled version regains the optimal rate since many of the lattice points are not needed. Moreover, we utilize fast and memory efficient Fourier algorithms in order to compute the approximation. Numerical experiments in several dimensions support our findings.