Incomplete discrete wavelet transform and its application to a Poisson equation solver

This paper introduces an incomplete discrete wavelet transform (iDWT), which is applied to a preconditioning method for linear equation systems discretized from differential equations. The linear systems can be solved with a matrix solver, but the convergence speed becomes worse with increase of condition number, which exponentially increases with the scale magnification. The use of wavelets in linear systems has an advantage in that a diagonal rescaling makes the number become bounded by a limited. value, and the advantage is utilized in a matrix solver presented by G. Beylkin. The method, however, has several problems and is difficult to apply to the real numerical analysis. To solve the problems, we introduce the iDWT method that approximates the discrete wavelet transform and is easy to implement in the computational analysis. The effects and advantages of the iDWT preconditioning are confirmed with one- and two-dimensional boundary value problems of elliptic equations. On Cray C94D vector computer, the iDWT preconditioned CG method can solve 2-D Poisson equation, discretized with 1,024x1,024 grid points, about 14 times faster than the ICCG method.