Parallel iterative solvers involving fast wavelet transforms for the solution of BEM systems Patricia Gonza ´lez a, * , Jose ´ C. Cabaleiro b , Toma ´s F. Pena b a Departamento Electronics and Systems, Universidad A Corun ˜a, 15071 A Corun ˜a, Spain b Departamento Electronics and Computer Science, Universidad Santiago de Compostela, 15782 Santiago de Compostela, Spain Received 15 November 2000; accepted 1 July 2002 Abstract This paper describes the parallelization of a strategy to speed up the convergence of iterative methods applied to boundary element method (BEM) systems arising from problems with non-smooth boundaries and mixed boundary conditions. The aim of the work is the application of fast wavelet transforms as a black box transformation in existing boundary element codes. A new strategy was proposed, applying wavelet transforms on the interval, so it could be used in case of non-smooth coefficient matrices. Here, we describe the parallel iterative scheme and we present some of the results we have obtained. q 2002 Civil-Comp Ltd and Elsevier Science Ltd. All rights reserved. Keywords: Dense and sparse linear systems; Boundary element method (BEM); Wavelet transform; Lifting scheme; Iterative solvers; Distributed-memory multiprocessors 1. Introduction One of the focuses of our work is to develop parallel effective tools and algorithms to help us in solving very large systems of equations. Currently, we are working on solving systems of equations arising from the boundary element method (BEM) [1,2]. The idea of using wavelets is presented as an alternative way to speed up the process of solving large systems. If the fast wavelet transform is applied to the coefficient matrix, then, depending upon the nature of the discontinuities, the resulting dense transformed matrix may have a sparsity pattern (obtained by means of a suitable thresholding process) that can be exploited to speed up the solution process using iterative methods. Problems arising from the application of BEM formulation to non- smooth boundaries and mixed boundary conditions present systems of equations with non-smooth zones, and, as a consequence, difficult to compress using standard wavelet transforms. Our proposal consisted in a combination of fast 1D wavelet transforms within the iterative method in order to solve the system of equations avoiding the use of the 2D transform in the non-smooth zone. In this paper, we present results of the parallel iterative scheme. The structure of the paper is as follows. In Section 2, we briefly review wavelet transforms and their main charac- teristics; in Section 3, we describe the strategy proposed for the solution of dense linear systems using fast wavelet transform and describe the parallel iterative scheme; in Section 4, we describe the parallel implementation of the wavelet transform and in Section 5, we present some numerical results to test the proposed strategy. 2. Wavelet transform 2.1. Discrete wavelet transform Wavelets form a versatile tool for representing general functions or data sets. They are capable of quickly capturing the essence of a data set with only a small number of coefficients. This is based on the fact that most groups of data have correlation both in time and frequency. Wavelet functions c j;m are traditionally defined as the dyadic translates and dilates of one particular L 2 function, the mother wavelet c : c j;m ðxÞ¼ cð2 j x 2 mÞ: Such wavelets are referred to as first generation wavelets. Fast wavelet transforms can be implemented with a specifically designed pair of finite impulse response (FIR) 0965-9978/02/$ - see front matter q 2002 Civil-Comp Ltd and Elsevier Science Ltd. All rights reserved. PII: S0965-9978(02)00047-9 Advances in Engineering Software 33 (2002) 417–426 www.elsevier.com/locate/advengsoft * Corresponding author.