Randomized Sampling for Basis Function 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.

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