LU Factorization of Non-standard Forms and Direct Multiresolution Solvers

In this paper we introduce the multiresolution LU factorization of non-standard forms (NS-forms) and develop fast direct multiresolution methods for solving systems of linear algebraic equations arising in elliptic problems. The NS-form has been shown to provide a sparse representation for a wide class of operators, including those arising in strictly elliptic problems. For example, Green’s functions of such operators (which are ordinarily represented by dense matrices, e.g., of size N by N) may be represented by 0log erN coefficients, where e is the desired accuracy. The NS-form is not an ordinary matrix representation and the usual operations such as multiplication of a vector by the NS-form are different from the standard matrix–vector multiplication. We show that (up to a fixed but arbitrary accuracy) the sparsity of the LU factorization is maintained on any finite number of scales for self-adjoint strictly elliptic operators and their inverses. Moreover, the condition number of matrices for which we compute the usual LU factorization at different scales is O(1) . The direct multiresolution solver presents, therefore, an alternative to a multigrid approach and may be interpreted as a multigrid method with a single V-cycle.