Quasi-interpolation with random sampling centers

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 $1\le p\le \infty$, 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.