A Wavelet Approach for the Construction of Multi-Grid Solvers for Large Linear Systems

Abstract

In general, large systems of linear equations cannot be solved directly. An iterative solver has to be applied instead. Unfortunately, iterative solvers have a notouriously slow convergence rate, which in the worst case can prevent convergence at all, due to the inavoidable rounding errors.

Multi-grid iteration schemes are meant to guarantee a sufficiently high convergence rate, independent from the dimension of the linear system. The idea behind the multi-grid solvers is that the traditional iterative solvers eliminate only the short-wavelength error constituents in the initial guess for the solution. For the elimination of the remaining long-wavelength error constituents a much coarser grid is sufficient. On the coarse grid the dimension of the problem is much smaller so that the elimination can be done by a direct solver.

The paper shows that wavelet techniques successfully can be applied for the following steps of a multi-grid procedure:

• Generation of an approximation of the proplem on a coarse grid from a given approximation on the fine grid.

• Restriction of a signal on a fine grid to its approximation on a coarse grid.

• Uplift of a signal from the coarse to the fine grid.

The paper starts with a theoretical explanation of the links between wavelets and multi-grid solvers. Based on this investigation the class of operators, which are suitable for a multi-grid solution strategy can be characterized. Two numerical examples will demonstrate the efficiency of wavelet based multi-grid solvers for the planar Stokes problem and for satellite gravity gradiometry.