Kernel-based Approximation Methods for Generalized Interpolations: A Deterministic or Stochastic Problem?

TL;DR

This paper introduces a kernel-based stochastic approach to solve generalized interpolation problems and elliptic PDEs, providing new estimators and error analysis tools within a probabilistic framework.

Contribution

It develops a novel kernel-based probability measure on Banach spaces for deterministic interpolation and PDE approximation, integrating stochastic methods with meshfree techniques.

Findings

Provides a new kernel-based estimator for interpolation

Offers error analysis for noisy and noise-free data

Enables kernel-based PDE solutions similar to meshfree methods

Abstract

In this article, we solve a deterministically generalized interpolation problem by a stochastic approach. We introduce a kernel-based probability measure on a Banach space by a covariance kernel which is defined on the dual space of the Banach space. The kernel-based probability measure provides a numerical tool to construct and analyze the kernel-based estimators conditioned on non-noise data or noisy data including algorithms and error analysis. Same as meshfree methods, we can also obtain the kernel-based approximate solutions of elliptic partial differential equations by the kernel-based probability measure.