Sparse operator compression of higher-order elliptic operators with rough coefficients
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.
© 2017 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Received: 9 December 2016; Accepted: 26 June 2017; Published: 4 December 2017. The research was in part supported by NSF Grants DMS 1318377 and DMS 1613861. We would like to thank Professor Lei Zhang and Venkat Chandrasekaran for several stimulating discussions, and Professor Houman Owhadi for valuable comments. Dedication: Honor of Bjorn Engquist on the occasion of his 70th birthday. Publisher's Note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Published - 10.1186_2Fs40687-017-0113-1.pdf
Submitted - 1708.02701.pdf