Sparse operator compression of higher-order elliptic operators with rough coefficients

TL;DR

This paper presents a method for compressing higher-order elliptic operators with rough coefficients using localized basis functions, achieving optimal compression rates and convergence in finite element solutions.

Contribution

It introduces a novel sparse operator compression technique with localized basis functions for higher-order elliptic operators with rough coefficients.

Findings

Achieves optimal compression rate of the solution operator.

Localized basis functions have supports of diameter O(h log(1/h)).

Convergence rate of O(h^k) for energy norm in elliptic problems.

Abstract

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 Mat\'{e}rn covariance functions can be approximated by a rank-$n$ operator with a localized basis and with the optimal accuracy.