Randomized sampling for basis functions construction in generalized finite element methods

TL;DR

This paper investigates randomized sampling strategies for constructing basis functions in generalized finite element methods, demonstrating that Gaussian and smooth boundary sampling yield the best numerical results.

Contribution

It introduces and compares new random sampling strategies for basis function construction in generalized finite element methods, providing a quantitative criterion for evaluation.

Findings

Random Gaussian sampling performs well in basis function approximation.

Smooth boundary sampling yields the best numerical accuracy.

The proposed criterion effectively compares sampling strategies.

Abstract

In the framework of generalized finite element methods for elliptic equations with rough coefficients, efficiency and accuracy of the numerical method depend critically on the use of appropriate basis functions. This work explores several random sampling strategies that construct approximations to the optimal set of basis functions of a given dimension, and proposes a quantitative criterion to analyze and compare these sampling strategies. Numerical evidence shows that the best results are achieved by two strategies, Random Gaussian and Smooth boundary sampling.