OPTIMALITY AND REGULARIZATION PROPERTIES OF QUASI-INTERPOLATION: DETERMINISTIC AND STOCHASTIC APPROACHES

摘要

Probabilistic numerics aims to study numerical algorithms from a stochastic perspective. This field has recently evolved into a surging interdisciplinary research area (between numerical approximation and probability theory) attracting much attention from the data science community at large. Motivated by this development, we incorporate a stochastic viewpoint into our study of multivariate quasi-interpolation for irregularly spaced data, a subject traditionally investigated in the realm of deterministic function approximation. We first construct quasi-interpolants directly from irregularly spaced data and show their optimality in terms of a certain quadratic functional on a weighted Hilbert space. We then derive the approximation order of our quasi-interpolants via a two-step procedure. In the first step, we approximate a target function by scaled integral operators (with an error term referred to as bias). In the second step, we discretize the underlying convolution integral at the irregularly spaced data sites (with an error term called variance). The final approximation order is obtained as an optimal trade-off between bias and variance. We also show that the scale parameter of the integral operators governs the regularization effect of the quasi-interpolation scheme, and find an optimal parameter value range to fine-tune the subtle balance between bias and variance under some additional assumptions on the distribution of the data sites. It is worth noting that evaluation of integrals is not needed in the implementation of our quasi-interpolation scheme, and that our quasi-interpolants are easy to construct. Numerical simulation results, including approximating the classical bore hole test function in eight dimensional space, provide evidence that our quasi-interpolation scheme is robust and capable of providing accurate generalizations.