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, that are energy
minimizing functions on local patches. On a regular mesh with mesh size, 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 our method achieves the
optimal compression rate of the solution operator. We then discuss how to generalize
this operator compression to develop a fast solver for graph Laplacians with rough
coefficients using a novel energy decomposition method. This decomposition
framework naturally reflects the intrinsic geometric information of the operator that
inherits the localities of the topological structure.
Utilizing this information, we proposea multiresolution operator compression scheme for the
inverse operator of a symmetric positive definite matrix with controllable compression error
and condition number.
This is a joint work with Pengchuan Zhang, De Huang, and Ka Chun Lam.