Víctor M. García

Variants of algebraic wavelet-based multigrid methods: Application to shifted linear systems

Abstract In this paper, we describe some new variants and applications of the wavelet algebraic multigrid method. This method combines the algebraic multigrid method (a well known family of multilevel techniques for solving linear systems, without use of knowledge of the underlying problem) and the discrete wavelet transform. These two techniques can be combined in several ways, obtaining different methods for solution of linear systems; these can be used alone or as preconditioners for Krylov iterative methods. These methods can be applied for solution of linear systems with shifted matrices of the form A - hI , whose efficient solution is very important for implicit ODE methods, unsteady PDEs, computation of eigenvalues of large sparse matrices and other important problems.

Parallel resolution of the two-group time dependent neutron diffusion equation with public domain ODE codes

In this paper it is shown how specialised codes for the resolution of ordinary differential equations (FCVODE[3], DASPK[2]) can be used to solve efficiently the time dependent neutron diffusion equation. Using these codes as basis, several new codes have been developed, combining the sequential and parallel versions of DASPK and FCVODE with different preconditioners. Their performance has been assessed using a two-dimensional benchmark (The TWIGL reactor).

Numerical experiments on the solution of the inverse additive singular value problem

The work presented here is an experimental study of four iterative algorithms for solving the Inverse Additive Singular Value Problem (IASVP). The algorithms are analyzed and evaluated with respect to different points of view: memory requirements, convergence, accuracy and execution time, in order to observe their behaviour with different problem sizes and to identify those capable to solve the problem efficiently.

A gradient-based ordering for MIMO decoding

The ordering of the columns of the channel matrix has a deep impact in the performance of many decoding methods for MIMO problems. The most popular algorithms for computation of good orderings use as only input the channel matrix (such as the V-BLAST ordering). However, there are other interesting algorithms that compute the ordering as a function of the matrix and of the signal to be decoded. Here we propose a new gradient-based scheme to obtain a good ordering that uses as input the channel matrix and the signal to be decoded. In this paper, this new ordering method is compared experimentally with other algorithms, showing that it is an interesting method, specially in connection with Maximum Likelihood decoding algorithms such as Sphere Decoding algorithms.

Parallel approach to NNMF on multicore architecture

We tackle the parallelization of Non-Negative Matrix Factorization (NNMF), using the Alternating Least Squares and Lee and Seung algorithms, motivated by its use in audio source separation. For the first algorithm, a very suitable technique is the use of active set algorithms for solving several non-negative inequality constraints least squares problems. We have addressed the NNMF for dense matrix on multicore architectures, by organizing these optimization problems for independent columns. Although in the sequential case, the method is not as efficient as the block pivoting variant used by other authors, they are very effective in the parallel case, producing satisfactory results for the type of applications where is to be used. For the Lee and Seung method, we propose a reorganization of the algorithm steps that increases the convergence speed and a parallelization of the solution. The article also includes a theoretical and experimental study of the performance obtained with similar matrices to that which arise in applications that have motivated this work.

Parallel Implementation in PC Clusters of a Lanczos-based Algorithm for an Electromagnetic Eigenvalue Problem

This paper describes a parallel implementation of a Lanczos-based method to solve generalised eigenvalue problems related to the modal computation of arbitrarily shaped waveguides. This efficient implementation is intended for execution in moderate-low cost workstations (2 to 4 processors). The problem under study has several features: the involved matrices are sparse with a certain structure, and all the eigenvalues needed are contained in a given interval. The novel parallel algorithms proposed show excellent speed-up for small number of processors

Parallel optimization methods based on direct search

This paper is focused in the parallelization of Direct Search Optimization methods, which are part of the family of derivative-free methods. These methods are known to be quite slow, but are easily parallelizable, and have the advantage of achieving global convergence in some problems where standard Newton-like methods (based on derivatives) fail. These methods have been tested with the Inverse Additive Singular Value Problem, which is a difficult highly nonlinear problem. The results obtained have been compared with those obtained with derivative methods; the efficiency of the parallel versions has been studied.

Implementation and tuning of a parallel symmetric Toeplitz eigensolver

In a previous paper (Vidal et al., 2008, [21]), we presented a parallel solver for the symmetric Toeplitz eigenvalue problem, which is based on a modified version of the Lanczos iteration. However, its efficient implementation on modern parallel architectures is not trivial. In this paper, we present an efficient implementation on multicore processors which takes advantage of the features of this architecture. Several optimization techniques have been incorporated to the algorithm: improvement of Discrete Sine Transform routines, utilization of the Gohberg-Semencul formulas to solve the Toeplitz linear systems, optimization of the workload distribution among processors, and others. Although the algorithm follows a distributed memory parallel programming paradigm that is led by the nature of the mathematical derivation, special attention has been paid to obtaining the best performance in multicore environments. Hybrid techniques, which merge OpenMP and MPI, have been used to increase the performance in these environments. Experimental results show that our implementation takes advantage of multicore architectures and clearly outperforms the results obtained with LAPACK or ScaLAPACK.

A deterministic lower bound for the radius in sphere decoding search

In this paper we examine the problem of MIMO detection with sphere decoding (SD) methods. These methods obtain the Maximum Likelihood solution in a reasonable time, through restriction of the space search to an hypersphere of a given radius, with center in the received signal. The performance of the sphere decoding algorithms will be acceptable only if the initial radius estimate is close to the final, optimal radius. Here we give a nontrivial lower bound for the radius, based only on the received signal and on the channel matrix, being then a purely deterministic estimate. This lower bound can be successfully integrated in some SD algorithms, providing a substantial decrease of the computational cost of the search.

Partial data replication as a strategy for parallel computing of the multilevel discrete wavelet transform

In this paper we propose a strategy of partial data replication for efficient parallel computing of the Discrete Wavelet Transform in a distributed memory environment. The key is to avoid the communications needed between computation of different wavelet levels, by replicating part of the data and part of the computations, avoiding completely communications (except maybe at the setup phase). A similar idea was proposed in a paper by Chaver et al.; however, they proposed to replicate completely the data, which can require too much memory in each processor. In this work we have determined exactly how many data items shall be needed for each processor, in order to compute the DWT without extra communications and using only the memory strictly necessary.