The Use of Wavelets for the Acceleration of Iteration Schemes

Abstract

In general, large systems of linear equations cannot be solved directly. Both the storage requirements and the number of O(n ³) necessary operations let direct solvers appear inappropriate.

In many cases iterative solvers can be designed in such a way that only a small fraction of the necessary information is held in the computers memory and the rest can be computed on-the-fly during the iteration process.

Unfortunately, iterative solvers as SOR or conjugate gradients have a notoriously slow convergence rate, which in the worst case can prevent convergence at all, due to the unavoidable rounding errors.

The paper aims at a demonstration that wavelet can be used as a systematic tool for the construction of iteration accelerators such as multi-grid solvers or preconditioners for the conjugate gradient iteration scheme.

The paper starts with a theoretical explanation of the links between wavelets multi-grid solvers and preconditioning of conjugate gradient iteration schemes. Two numerical examples will demonstrate the efficiency of wavelet based multi-grid solvers for the planar Stokes problem.