Construction of Data-Sparse H2-Matrices by Hierarchical Compression

Discretizing an integral operator by a standard finite element or boundary element method typically leads to a dense matrix. Since its storage complexity grows quadratically with the number of degrees of freedom, the standard representation of the matrix as a two-dimensional array cannot be applied to large problem sizes. H2-matrix techniques use a multilevel approach to represent the dense matrix in a more efficient data-sparse format. We consider the challenging task of finding a good multilevel representation of the matrix without relying on a priori information of its contents. This paper presents a relatively simple algorithm that can use any of the popular low-rank approximation schemes (e.g., cross approximation) to find an “initial guess” and constructs a matching multilevel structure on the fly. Numerical experiments show that the resulting technique is as fast as competing methods and requires far less storage for large problem dimensions.