Sparse operator compression of higher-order elliptic operators with rough coefficients
This work provides a theoretical framework for compressing elliptic operators, benefiting numerical methods for PDEs and sparse PCA for Matérn covariance functions.
The paper introduces a sparse operator compression method for higher-order elliptic operators with rough coefficients, achieving optimal compression rates using localized basis functions with support diameter O(h log(1/h)).
We introduce the sparse operator compression to compress a self-adjoint higher-order elliptic operator with rough coefficients and various boundary conditions. The operator compression is achieved by using localized basis functions, which are energy-minimizing functions on local patches. On a regular mesh with mesh size $h$, the localized basis functions have supports of diameter $O(h\log(1/h))$ and give optimal compression rate of the solution operator. We show that by using localized basis functions with supports of diameter $O(h\log(1/h))$, our method achieves the optimal compression rate of the solution operator. From the perspective of the generalized finite element method to solve elliptic equations, the localized basis functions have the optimal convergence rate $O(h^k)$ for a $(2k)$th-order elliptic problem in the energy norm. From the perspective of the sparse PCA, our results show that a large set of Matérn covariance functions can be approximated by a rank-$n$ operator with a localized basis and with the optimal accuracy.