Randomized Sampling for Basis Functions Construction in Generalized Finite Element Methods

In the context of generalized finite element methods for elliptic equations with rough coefficients a(x), efficiency and accuracy of the numerical method depend critically on the use of appropriate basis functions. This work explores several random sampling strategies for construction of basis functions, and proposes a quantitative criterion to analyze and compare these sampling strategies. Numerical evidence shows that the optimal basis functions can be well approximated by a random projection of generalized eigenvalue problem onto subspace of a-harmonic functions.