Filomena D. d'Almeida

An L 1 refined projection approximate solution of the radiation transfer equation in stellar atmospheres

This paper deals with the numerical approximation of the solution of a weakly singular integral equation of the second kind which appears in Astrophysics. The reference space is the complex Banach space of Lebesgue integrable functions on a bounded interval whose amplitude represents the optical thickness of the atmosphere. The kernel of the integral operator is defined through the first exponential-integral function and depends on the albedo of the media. The numerical approximation is based on a sequence of piecewise constant projections along the common annihilator of the corresponding local means. In order to produce high precision solutions without solving large scale linear systems, we develop an iterative refinement technique of a low order approximation. For this scheme, parallelization of matrix computations is suitable.

Performance evaluation of a parallel algorithm for a radiative transfer problem

The numerical approximation and parallelization of an algorithm for the solution of a radiative transfer equation modeling the emission of photons in stellar atmospheres will be described. This is formulated in the integral form yielding a weakly singular Fredholm integral equation defined on a Banach space. The main objective of this work is to report on the performance of the parallel code. AMS Subject Classification: 32A55, 45B05, 65D20, 65R20, 68W10.

Preconditioners for Nonsymmetric Linear Systems in Domain Decomposition Applied to a Coupled Discretization of Navier-Stokes Equations

We will consider the linear system generated by a coupled discretization and linearization method for the Navier-Stokes equations. This method consists of a discretization of the momentum equations to obtain the velocities and pressure at the faces of a finite volume, in terms of the values of these variables at the grid points followed by the integration of the momentum and continuity equations in the finite volumes.

A Jacobi---Davidson type method with a correction equation tailored for integral operators

We propose two iterative numerical methods for eigenvalue computations of large dimensional problems arising from finite approximations of integral operators, and describe their parallel implementation. A matrix representation of the problem on a space of moderate dimension, defined from an infinite dimensional one, is computed along with its eigenpairs. These are taken as initial approximations and iteratively refined, by means of a correction equation based on the reduced resolvent operator and performed on the moderate size space, to enhance their quality. Each refinement step requires the prolongation of the correction equation solution back to a higher dimensional space, defined from the infinite dimensional one. This approach is particularly adapted for the computation of eigenpair approximations of integral operators, where prolongation and restriction matrices can be easily built making a bridge between coarser and finer discretizations. We propose two methods that apply a Jacobi–Davidson like correction: Multipower Defect-Correction (MPDC), which uses a single-vector scheme, if the eigenvalues to refine are simple, and Rayleigh–Ritz Defect-Correction (RRDC), which is based on a projection onto an expanding subspace. Their main advantage lies in the fact that the correction equation is performed on a smaller space while for general solvers it is done on the higher dimensional one. We discuss implementation and parallelization details, using the PETSc and SLEPc packages. Also, numerical results on an astrophysics application, whose mathematical model involves a weakly singular integral operator, are presented.

Nonoverlapping Domain Decomposition Applied to a Computational Fluid Mechanics Code

The purpose of this paper is the description of the development and implementation of the linear part of a numerical algorithm for the simulation of a Newtonian fluid flow and the parallelization of that code on several computer architectures. The test problem treated is the steady state, laminar, incompressible, isothermic, 2D fluid flow (extendible to 3D case), the Navier-Stokes equations being discretized by a fully coupled finite volume method. For this problem, sparse data structures, nonstationary iterative methods and several preconditioners are applied. The numerical results allow the conclusion that the fully coupled version can compete with the decoupled classic SIMPLE method (Semi-Implicit Method for Pressure-Linked Equations), by using the Krylov subspace methods. Parallel versions of the coupled method based on nonoverlapping domain decomposition are discussed.

Columnwise block LU factorization using blas kernels on VAX 6520/2VP

Abstract The LU factorization of a matrix A is a widely used algorithm, for instance in the solution of linear systems Ax = b . The increasing capacities of high performance computers allow us to use direct methods for systems of large and dense matrices. To build portable and efficient LU codes for vector and parallel computers, this method is rewritten in block versions and BLAS (Basic Linear Algebra Subprograms) kernels are used to mask the architectural details and allow good performance of codes such as the LAPACK (Linear Algebra PACKage) library. In the references it was proved that this strategy leads to portability and efficiency of codes using tuned BLAS kernels. After a short description of the block versions we will present some results obtained on the VAX 6520/2VP, comparing the block algorithm versus point algorithm, and vectorized versions versus scalar versions. The three columnwise versions of the block algorithm showed similar performance for this computer and large matrix dimensions. The block size used is a crucial parameter for these algorithms and the results show that the best performance is obtained with block size 64 (for large matrices) which is the vector registered size of the machine used.

Performance of a QR algorithm implementation on a multicluster of transputers

Abstract Some results of an implementation of the QR factorization by Householder reflectors, on a multicluster transputer system with distributed memory are presented, that show how important is the communication time between processor in the performance of the algorithm. The QR factorization was chosen as test method because it is required for many real life applications, for instance in least squares problems. We use a version of Householder transformation that is the basis for numerically stable QR factorization. The machine used was the MultiCluster 2 model of Parsytec which is distributed memory system with 16 Inmos T800 processors. The Helios operating system was chosen because it provides transparency in CPU management. However it limits the sets of connecting topologies to be used. The results are presented in terms of speedup and efficiency, showing the importance of the communication time on the total elapsed time.

Is the iterative refinement of eigenelements an expensive technique

Abstract Iterative refinement techniques to improve the accuracy of approximate eigenelements of compact integral operators appeared as an alternative to discretization methods to overcome the problem of having to solve large full eigenvalue problems, but other approaches are sometimes possible such as Pade techniques. Here we compare the applicability conditions of these methods and their number of operations for some cases where both approaches are possible.