Preconditioning in a wavelet basis and its application to some boundary value problems

Abstract

Standard finite difference methods applied to the boundary value problem a(x)u" (x) + b(x)u'(x) + c(x)u(x) = f (x), u(0) = 0, u(1) = 0, lead to linear systems with large condition numbers. Solving a system, i.e. finding the inverse of a matrix with a large condition number can be achieved by some iterative procedure in a large number of iteration steps. By projecting the matrix of the system into the wavelet basis, and applying a diagonal pre-conditioner, we obtain a matrix with a small condition number. Computing the inverse of such a matrix requires fewer iteration steps, and that number does not grow significantly with the size of the system. Numerical examples, with various operators, are presented to illustrate the effect preconditioners have on the condition number, and the number of iteration steps.