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

Thomas Y Hou,Pengchuan Zhang

doi:10.1186/s40687-017-0113-1

Thomas Y Hou, Pengchuan Zhang

Open Access

https://doi.org/10.1186/s40687-017-0113-1

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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(hlog (1/h)) and give optimal compression rate of the solution operator. We show that by using localized basis functions with supports of diameter O(hlog (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.

Highlights

1.1 Main objectives and the problem setting The main purpose of this paper is to develop a general strategy to compress a class of self-adjoint higher-order elliptic operators by localized basis functions that give optimal approximation property of the solution operator
9 Concluding remarks In this paper, we have developed a general strategy to compress a class of self-adjoint higher-order elliptic operators by minimizing the energy norm of the localized basis functions
These energy-minimizing localized basis functions are obtained by solving decoupled local quadratic optimization problems with linear constraints, and they give optimal approximation property of the solution operator

Summary

Background

1.1 Main objectives and the problem setting The main purpose of this paper is to develop a general strategy to compress a class of self-adjoint higher-order elliptic operators by localized basis functions that give optimal approximation property of the solution operator. 2. If we use a generalized finite element method [1,10,22,44] to solve the elliptic equations, we achieve the optimal convergence rate in the energy norm, i.e., L−1f − Ψ locL−n 1(Ψ loc)T f H ≤ Ce λn(L−1) f 2 ∀f ∈ L2(D),. 3. For the sparse operator compression problem, we achieve the optimal approximation error up to a constant, i.e., Eoc(Ψ loc; L−1) ≤ Ce2λn(L−1),. We will focus on the theoretical analysis of the approximation accuracy (1.5) and the localization of the basis functions (1.4)

Our construction

We have

An inverse energy estimation by scaling

Localization of the basis functions