Fast Eigenpairs Computation with Operator Adapted Wavelets and Hierarchical Subspace Correction

Abstract

We present a method for the fast computation of the eigenpairs of a bijective positive symmetric linear operator $\mathcal{L}$. The method is based on a combination of operator adapted wavelets (gamblets) with hierarchical subspace correction. First, gamblets provide a raw but fast approximation of the eigensubspaces of $\mathcal{L}$ by block-diagonalizing $\mathcal{L}$ into sparse and well-conditioned blocks. Next, the hierarchical subspace correction method computes the eigenpairs associated with the Galerkin restriction of $\mathcal{L}$ to a coarse (low-dimensional) gamblet subspace and then corrects those eigenpairs by solving a hierarchy of linear problems in the finer gamblet subspaces (from coarse to fine, using multigrid iteration). The proposed algorithm is robust to the presence of multiple (a continuum of) scales and is shown to be of near-linear complexity when $\mathcal{L}$ is an (arbitrary local, e.g., differential) operator mapping $\mathcal{H}^s_0(\Omega)$ to $\mathcal{H}^{-s}(\Omega)$ (e.g., an elliptic PDE with rough coefficients).