Combining Kronecker Product Approximation with Discrete Wavelet Transforms to Solve Dense, Function-Related Linear Systems

A new solution technique is proposed for linear systems with large dense matrices of a certain class including those that come from typical integral equations of potential theory. This technique combines Kronecker product approximation and wavelet sparsification for the Kronecker product factors. The user is only required to supply a procedure for computation of each entry of the given matrix. The main sources of efficiency are the incomplete cross approximation procedure adapted from the mosaic-skeleton method of the second author and data-sparse preconditioners (the incomplete LU decomposition with dynamic choice of the fill-in structure with a prescribed threshold and the inverse Kronecker product preconditioner) constructed for the sum of Kronecker products of sparsified finger-like matrices computed by the Discrete Wavelet Transform. In some model, but quite representative, examples the new technique allowed us to solve dense systems with more than 1 million unknowns in a few minutes on a personal computer with 1 Gbyte operative memory.