LU Factorization of Non-standard Forms and Direct Multiresolution Solvers

AbstractIn this paper we introduce the multiresolution LU factorization of non-standard forms (NS-forms) and develop fastdirect multiresolutionmethods 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 sizeNbyN) may be represented by −log ϵ·Ncoefficients, where ϵ is the desired accuracy.The NS-form is not an ordinary[fn9] 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 isO(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.For self-adjoint strictly elliptic operators the multiresolution LU factorization requires onlyO((−log ϵ)2·N) operations. Combined withO(N) procedures of multiresolution forward and back substitutions, it yields a fast direct multiresolution solver. We also describe direct methods for solving matrix equations and demonstrate how to construct the inverse inO(N) operations (up to a fixed but arbitrary accuracy). We present several numerical examples which illustrate the algorithms developed in the paper. Finally, we outline several directions for generalization of our algorithms. In particular, we note that the multidimensional versions of the multiresolution LU factorization maintain sparsity, unlike the usual LU factorization

Random matrix over a DVR and LU factorization

Let R be a discrete valuation ring (DVR) and K be its fraction field. If M is a matrix over R admitting a LU decomposition, it could happen that the entries of the factors L and U do not lie in R, but just in K. Having a good control on the valuations of these entries is very important for algorithmic applications. In the paper, we prove that in average these valuations are not too large and explain how one can apply this result to provide an efficient algorithm computing a basis of a coherent sheaf over A^1 from the knowledge of its stalks.Comment: 23 page

Use of LU Factorization in Solving a Real-World Problem

We applied the LU factorization to find the temperature distribution in a two dimensional flat rectangular metallic plate. The temperatures on the boundary are known and the interior temperature distribution will be determined. The problem will require solving a linear system: Ax = bThis system can also be solved by finding the inverse of A. However, we will show that for this problem, the LU factorization technique is more suitable.https://ecommons.udayton.edu/stander_posters/2816/thumbnail.jp