This thesis has had as double objective the study of the parallel computation of Discrete Wavelet Transform (DWT), and the study of the applications of the DWT in the field of numerical linear algebra.

Firstly a study is made of different variants of parallelization of the DWT, and a new parallel variant is proposed, in distributed memory, with data distributions oriented to blocks of matrices, like the two-dimensional block cyclic distribution (2DBC) used in the ScaLAPACK library. The idea is that, in many cases, the DWT is an intermediate operation and must be adjusted to the distributions of data that are being used. A new strategy has been proposed for computation of the multilevel parallel DWT, based on determining exactly how many elements must send and receive each processor so that all the wavelet coefficients can be calculated of independent form (no need of later communications). Finally another strategy has been proposed for the efficient calculation of the DWT-2D when it is applied as a previous step to the resolution of a system of equations distributed with the 2DBC distribution. This proposal consists of performing a permutation of rows and columns of the system that reduces the communications.

Another contribution of this thesis is to consider as a typical case, the calculation of the nonstandard DWT-2D of sparse matrices; we propose algorithms to conduct this operation without building the DWT matrix. In addition we consider the "fill-in" phenomenon that happens when the DWT is applied to a sparse matrix. The classic methods of reordering matrices, minimum degree and reduction to band algorithms, have been tested for this problem. It has been suggested that these reorderings can influence the convergence of the multigrid methods (proposed in the last chapter) since they cause a redistribution of the norm of the matrix towards the lower levels of the multi-scale representation, which would guarantee a better compression.

The main field of application of wavelet transform proposed is the solution of large systems of linear equations. In this thesis we will propose two specific applications: parallelization of preconditioners of linear systems based on the DWT, and the efficient calculation of the DWT-2D in sparse matrices, in conjunction with the multi-resolution analysis inherent to wavelets and the multigrid methods. The Algebraic Wavelet Multigrid Methods are studied (Wavelet Algebraic Multigrid Methods, WAMG), these are algorithms that combine the multigrid methods with DWT, and do not need any detailed knowledge of the problem to solve; only  the matrix of coefficients and the right part of the system are needed. Two new variants of WAMG algorithms have been proposed. The first is based on the decomposition of the linear system when applying the DWT, and the second reduces to the computational cost of the cycles of the multigrid algorithm by skipping operations in some levels or meshes. Finally the application of the WAMG to the efficient resolution of shifted linear systems is studied.