Randomized sampling for basis functions construction in generalized finite element methods
For researchers in numerical methods for PDEs, this work provides a quantitative criterion to compare sampling strategies, though the improvement is incremental.
The paper explores random sampling strategies for constructing basis functions in generalized finite element methods for elliptic equations with rough coefficients, finding that Random Gaussian and Smooth boundary sampling achieve the best results.
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.