Multivariate Monte Carlo Approximation Based on Scattered Data

Abstract

We propose and study a new multivariate stochastic scattered data quasi-interpolation scheme that is reminiscent of the classical Monte Carlo method for estimating integrals. We first employ a convolution operator to approximate (deterministically) Sobolev space functions and use a result of Cheney, Light, and Xu [On kernels and approximation orders, in Approximation Theory, Lecture Notes in Pure Appl. Math. 138, Dekker, 1992, pp. 227--242] and Cheney and Lei [Quasi-interpolation on irregular points, in Approximation and Computation, Internat. Ser. Numer. Math. 119, Birkhäuser Boston, 1994, pp. 121--135] to obtain an approximation error estimate in terms of moment conditions. We then approximate (stochastically) the convolution integral using a Monte Carlo method and derive the maximal mean squared error (M-MSE) estimate and mean $L^p$-error estimate on bounded domains which are in line with those obtained by the classical Monte Carlo method for estimating multivariate integrals. The introduction of convolution operators is solely for the purpose of facilitating error analysis. The implementation of this scheme does not require any numerical handling of the convolution integral involved. Our final approximant is in the form of scattered data quasi-interpolation. It enjoys a simple construction and optimal convergence rate, yet it provides an efficient tool in various computing environments. Asymptotic normality and confidence interval test results show that the scheme is computationally stable. Numerical simulation results show that the scheme is robust in the presence of noise.