Jean-Marie Chesneaux

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.

Reflection from a corrugated surface revisited

The problem of scattering of a plane sonic wave from a soft surface with periodic (sinusoidal) unevenness along one direction is examined by means of the Rayleigh plane‐wave expansion and the Waterman extinction methods, numerically implemented by Fourier projection and expansion, respectively. The computations are done with real, double‐precision, stochastic arithmetic instead of the usual complex, double‐precision floating‐point arithmetic in order to precisely evaluate the numerical accuracy of the results conditioned by round‐off errors. It is shown that the low‐order plane‐wave coefficients obtained by the Rayleigh and Waterman methods are identical when obtained from matrix systems that are large enough to give ‘‘convergence’’ of these coefficients. For the same matrix size, the higher‐order coefficients differ the higher the diffraction order. It is also shown that the Waterman (Fourier‐series) computation of the near field is generally meaningful, whereas that of Rayleigh, involving summation of t...

Comparison of four software packages applied to a scattering problem

We investigate characteristic features of four different software packages by applying them to the numerical solution of a non-trivial physical problem in computer simulation, viz., scattering of waves from a sinusoidal boundary. The numerical method used is based on boundary collocation. This leads to highly ill-conditioned linear systems of equations, such that ensuing results may lose significant digits. The packages under consideration, each of which is based on a specific computer arithmetic, are the following: CADNA, PROFIL, MAPLE and MATLAB.

Response to ‘‘Comments on ‘Reflection from a corrugated surface revisited [J. Acoust. Soc. Am. 96, 1116–1129 (1994)]’ ’’

The comments of Kazandjian [J. Acoust. Soc. Am. 98, 1813–1814 (1995)] concerning the paper [J. Acoust. Soc. Am. 96, 1116–1129 (1994)] are discussed, notably those concerning the equivalence of the Rayleigh and extinction methods, the identity and convergence of perturbation series deriving from these methods, and the pertinence of numerical computations as indicators of the validity of theoretical predictions.

Estimation of Round-Off Errors on Several Computers Architectures

Numerical validation of computed results in scientific computation is always an essential problem as well on sequential architecture as on parallel architecture. The probabilistic approach is the only one that allows to estimate the round-off error propagation of the floating point arithmetic on computers. We begin by recalling the basics of the CESTAC method (Controle et Estimation STochastique des Arrondis de Calculs). Then, the use of the CADNA software (Control of Accuracy and Debugging For Numerical Applications) is presented for numerical validation on sequential architecture. On parallel architecture, we present two solutions for the control of round-off errors. The first, one is the combination of CADNA and the PVM library. This solution allows to control round-off errors of parallel codes with the same architecture. It does not need more processors than the classical parallel code. The second solution is represented by the RAPP prototype. In this approach, the CESTAC method is directly parallelized. It works both on sequential and parallel programs. The essential difference is that this solution requires more processors than the classical codes. These different approaches are tested on sequential and parallel programs of multiplication of matrices.

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.

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.

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.