Quasi-interpolation with random sampling centers

TL;DR

This paper introduces a stochastic quasi-interpolation framework using Monte Carlo methods, providing dimension-independent error bounds and validating results through numerical simulations.

Contribution

It develops a novel stochastic quasi-interpolation method with new concentration inequalities, improving error estimates and addressing the curse of dimensionality.

Findings

Dimension-independent $L^1$ concentration inequalities

Enhanced $L^ Infty$ error estimates

Numerical validation of theoretical results

Abstract

We propose and study a general quasi-interpolation framework for stochastic function approximation, which stems and draws motivation from convolution-type solutions for certain practical weighted variational problems. We obtain our quasi-interpolants using Monte Carlo discretization of the pertinent integrals and establish a family of $L^p$-McDiarmid-type concentration inequalities for $1\leq p\leq \infty$, which resulted in verifiable expected error estimates for the stochastic quasi-interpolants. The $L^1$-version of these concentration inequalities is dynamically-independent of dimensions, which offers a partial stochastic mitigation of the so called ``curse of dimensionality". The $L^\infty$-version of these concentration inequalities strengthens the existing expected $L^\infty$-error estimates in the literature. Numerical simulation results are provided at the end of the paper to validate the underlying theoretical analysis.