Multiresolution Approximate Inverse Preconditioners

We introduce a new preconditioner for elliptic PDE’s on unstructured meshes. Using a wavelet-inspired basis we compress the inverse of the matrix, allowing an effective sparse approximate inverse by solving the sparsity vs. accuracy conflict. The key issue in this compression is to use second-generation wavelets which can be adapted to the unstructured mesh, the true boundary conditions, and even the PDE coefficients. We also show how this gives a new perspective on multiresolution algorithms such as multigrid, interpreting the new preconditioner as a variation on node-nested multigrid. In particular, we hope the new preconditioner will combine the best of both worlds: fast convergence when multilevel methods can succeed, but with robust performance for more difficult problems. The rest of the paper discusses the core issues for the preconditioner: ordering and construction of a factored approximate inverse in the multiresolution basis, robust interpolation on unstructured meshes, automatic mesh coarsening, and purely algebraic alternatives. Some exploratory numerical experiments suggest the superiority of the new basis over the standard basis for several tough problems, including discontinuous anisotropic coefficients, strong convection, and indefinite reaction problems on unstructured meshes, with scalability like hierarchical basis methods achieved.