Fabienne Jézéquel

CADNA : a library for estimating round-off error propagation

The CADNA library enables one to estimate round-off error propagation using a probabilistic approach. With CADNA the numerical quality of any simulation program can be controlled. Furthermore by detecting all the instabilities which may occur at run time, a numerical debugging of the user code can be performed. CADNA provides new numerical types on which round-off errors can be estimated. Slight modifications are required to control a code with CADNA, mainly changes in variable declarations, input and output. This paper describes the features of the CADNA library and shows how to interpret the information it provides concerning round-off error propagation in a code.

Numerical 'health check' for scientific codes: the CADNA approach

Scientific computation has unavoidable approximations built into its very fabric. One important source of error that is difficult to detect and control is round-off error propagation which originates from the use of finite precision arithmetic. We propose that there is a need to perform regular numerical ‘health checks’ on scientific codes in order to detect the cancerous effect of round-off error propagation. This is particularly important in scientific codes that are built on legacy software. We advocate the use of the CADNA library as a suitable numerical screening tool. We present a case study to illustrate the practical use of CADNA in scientific codes that are of interest to the Computer Physics Communications readership. In doing so we hope to stimulate a greater awareness of round-off error propagation and present a practical means by which it can be analyzed and managed.

GREMLINS: a large sparse linear solver for grid environment

Traditional large sparse linear solvers are not suited in a grid computing environment as they require a large amount of synchronization and communication penalizing the performance on this architecture. This paper presents some features of the solver designed during the current GREMLINS (GRid Efficient Method for LINear Systems) project. The GREMLINS solver limits the amount of communication as it is based on a coarse grained iterative method called multisplitting method. Moreover, the solver can be executed either in a synchronous or an asynchronous mode. In the latter case, iterations are desynchronized and there is no more synchronization at all. It may result in a faster execution time compared to the synchronous case. Some experiments presented in this paper with the GRID5000 architecture, a nation wide experimental grid in France, allowed us to highlight interesting features of this solver.

A validated parallel across time and space solution of the heat transfer equation

In this paper, we present a parallel across time and space algorithm, based on an implicit collocation method, for the heat transfer equation. The solution is approximated by polynomials, the coefficients of which are computed from a block-tridiagonal linear system. The optimal degree of the polynomials is the highest degree for which all the coefficients are significant. It is obtained with the use of the CADNA library. If the time step and the space step increase, this optimal degree may increase and the time interval where the solution can be computed in parallel becomes larger. Once the polynomials have been determined for a given time interval, the solution can be accurately computed in parallel at any point of this interval. If the number of points where the solution is computed in the considered time interval is sufficient, it appears that the run-time performances of the implicit collocation method on a parallel machine are better than those of an explicit finite difference method. Numerical experiments on an MIMD architecture are presented.

E-collisions using e-science

We describe a computational science research program primarily aimed at engineering numerically robust software that can exploit high performance on distributed computers in the study of electron collisions with atoms and ions. In particular, we describe the development of 2DRMP-G, a Grid aware 2-dimensional R-matrix propagator and its numerical validation using CADNA, a software tool based on discrete stochastic arithmetic.

Ixaru's extended frequency dependent quadrature rules for slater integrals: Parallel computation issues

We describe recent progress of an ongoing research programme aimed at producing computational science software that can exploit high performance architectures in the atomic physics application domain. We examine the computational bottleneck of matrix construction in a suite of two-dimensional R-matrix propagation programs, 2DRMP, that are aimed at creating virtual electron collision experiments on HPC architectures. We build on Ixarus extended frequency dependent quadrature rules (EFDQR) for Slater integrals and examine the challenge of constructing Hamiltonian matrices in parallel across an m-processor compute node in a block cyclic distribution for subsequent diagonalization by ScaLAPACK.

Stochastic arithmetic and verification of mathematical models

Stochastic arithmetic enables one to estimate round-off error propagation using a probabilistic approach. With Stochastic arithmetic, the numerical quality of any simulation program can be controlled. Furthermore by detecting all the instabilities which may occur at run time, a numerical debugging of the user code can be performed. Stochastic arithmetic can be used to dynamically control approximation methods. Such methods provide a result which is affected by a truncation error inherent to the algorithm used and a round-off error due to the finite precision of the computer arithmetic. If the discretization step decreases, the truncation error also decreases, but the round-off error increases. Therefore it can be difficult to control these two errors simultaneously. In order to obtain with an approximation method a result for which the global error (consisting of both the truncation error and the round-off error) is minimal, a strategy, based on a converging sequence computation, has been proposed. Computation is carried out until the difference between two successive iterates has no exact significant digit. Then it is possible to determine which digits of the result obtained are in common with the exact solution. This strategy can apply to the computation of integrals using the trapezoidal rule, Simpsons rule, Rombergs method or the Gauss—Legendre method.

A Sparse Linear System Solver Used in a Distributed and Heterogenous Grid Computing Environment

Many scientific applications need to solve very large sparse linear systems in order to simulate phenomena close to the reality. Grid computing is an answer to the growing demand of computational power. In a grid computing environment, communication times are significant and the bandwidth is variable, therefore frequent synchronizations slow down performances. Thus it is desirable to reduce the number of synchronizations in a parallel direct algorithm. Inspired from multisplitting techniques, the GREMLINS (GRid Efficient Methods for LINear Systems) solver we developed consists of solving several linear problems obtained by splitting. The principle of the balancing algorithm is presented, and experimental results are given.

For Reliable and Powerful Scientific Computations

Some arithmetics for numerical validation have been developed on sequential computer architectures: interval arithmetic, discrete stochastic arithmetic or multi-precision arithmetic. Today, a lot of powerful computations are performed on vector computer architectures. We present here the first vector version of the CADNA software based on discrete stochastic arithmetic which combines powerful and reliable computations.

A Time and Space Parallel Algorithm for the Heat Equation: The Implicit Collocation Method

A collocation method used as a parallel method across time is proposed for solving parabolic partial differential equations. With this method, the solution of the heat equation is approximated by polynomials, the coefficients of which are computed from a block-tridiagonal linear system. Once the polynomials have been determined for a given time interval, the solution can be computed in parallel at any point of this interval. If the number of points where the solution is computed in the considered time interval is sufficient, it appears that the performances of the implicit collocation method on a parallel machine are better than those of finite difference methods. Numerical experiments on a SIMD architecture are reported.