Adaptive LOOCV-based kernel methods for solving time-dependent BVPs

TL;DR

This paper introduces an adaptive kernel-based method using LOOCV for efficiently solving time-dependent boundary value problems, improving accuracy in regions with steep solution variations.

Contribution

It presents a novel adaptive scheme combining kernel methods with LOOCV error estimation for better spatial resolution in time-dependent BVPs.

Findings

Effective detection of high-error regions

Enhanced solution accuracy through adaptive refinement

Numerical experiments confirm method efficacy

Abstract

In this paper we propose an adaptive scheme for the solution of time-dependent boundary value problems (BVPs). To solve numerically these problems, we consider the kernel-based method of lines that allows us to split the spatial and time derivatives, dealing with each separately. This adaptive algorithm is based on a leave-one-out cross validation (LOOCV) technique, which is employed as an error indicator. By this scheme, we can first detect the domain areas where the error is estimated to be too large -- generally due to steep variations or quick changes in the solution -- and then accordingly enhance the numerical solution by applying a two-point refinement strategy. Numerical experiments show the efficacy and performance of our adaptive refinement method.