On the spectral equivalence of hierarchical matrix preconditioners for elliptic problems

Abstract:

We will discuss the spectral equivalence of hierarchical matrix approximations for second order elliptic problems. Our theory will show that a modified variant of the hierarchical matrix Cholesky decomposition which preserves test vectors while truncating blocks to lower rank will lead to a spectrally equivalent approximation when using an adapted truncation threshold. Our theory also covers the usual hierarchical Cholesky decomposition which does not preserve test vectors but expects a significantly more restrictive threshold adaption to obtain a spectrally equivalent approximation. Numerical experiments indicate that the adaption of the truncation parameter seems to be necessary for the traditional hierarchical Cholesky preconditioner to obtain mesh-independent convergence while the variant which preserves test vectors works in practice quite well even with a fixed parameter.