René Alt

A-stable one-step methods with step-size control for stiff systems of ordinary differential equations

Abstract Two efficient third-and fourth-order processes for solving the initial value problem for ordinary differential equations are studied. Both are A-stable and so recommended for stiff systems. An economic and efficient way of step-size control is given for each of them. Numerical examples are considered.

On the algebraic Properties of Stochastic Arithmetic. Comparison to Interval Arithmetic

Interval arithmetic and stochastic arithmetic have been both developed for the same purpose, i. e. to control errors coming from floating point arithmetic of computers. Interval arithmetic delivers guaranteed bounds for numerical results whereas stochastic arithmetic provides confidence intervals with known probability. The algebraic properties of stochastic arithmetic are studied with an emphasis on the structure of the set of stochastic numbers. Some new properties of stochastic numbers are obtained based on the comparison with interval arithmetic in midpoint-radius form.

Theoretical and computational studies of some bioreactor models

We study certain classical basic models for bioreactor simulation in case of batch mode with decay. It is shown that in many cases the two-dimensional differential system describing the dynamics of the substrate and biomass concentrations can be reduced to an algebraic equation for the biomass together with a single differential equation for the substrate. Then from an analogy with the Henri-Michaelis-Menten enzyme kinetic mechanism a simple model is proposed for a bioreactor in batch mode with decay. Two more models are also proposed taking into account the phases of microbial growth. Some properties of these two models are studied and compared to classical Monod type models using computer simulations.

Stochastic arithmetic: s-spaces and some applications

It has been recently shown that computation with stochastic numbers as regard to addition and multiplication by scalars can be reduced to computation in familiar vector spaces. In this work we show how this can be used for the algebraic solution of linear systems of equations with stochastic right-hand sides. On several examples we compare the algebraic solution with the simulated solution using the CADNA package.

Error propagation in fourier transforms

The Fourier transform has long been of great use in simulating mathematical or physical phenomena, especially in signal theory. However the finite length representation of numbers introduces round-off errors in computing. Here, developing a new point of view on the topic, we give an evaluation of the total relative mean square error in the computation of direct and fast Fourier transforms using floating point artihmetic. Thus we show that in direct Fourier transforms the output noise-to-signal ratio is equivalent to N or N2 according to whether the arithmetic is a rounding or a chopping one, whereas for fast Fourier transforms it is equivalent to log2(N) or [log2(N)]2, with N being the number of points of the signal. Good agreement with numerical results is observed.

On the numerical solution to linear problems using stochastic arithmetic

It has been recently shown that computation with stochastic numbers as regard to addition and multiplication by scalars can be reduced to computation in familiar vector spaces. This result allows us to solve certain practical problems with stochastic numbers and to compare algebraically obtained results with practical applications of stochastic numbers, such as the ones provided by the CESTAC method. Such comparisons give additional information related to the stochastic behavior of random roundings in the course of numerical computations. A number of original numerical experiments are presented that agree with the expected theoretical results.

Numerical study of algebraic solutions to linear problems involving stochastic parameters

We formulate certain numerical problems with stochastic numbers and compare algebraically obtained results with experimental results provided by the CESTAC method. Such comparisons give additional information related to the stochastic behavior of random roundings in the course of numerical computations. The good coincidence between theoretical and experimental results confirms the adequacy of our algebraic model and its possible application in the numerical practice. AMS Subject Classification: 65C99, 65G99, 93L03.

On the solution to numerical problems using stochastic arithmetic

We investigate some algebraic properties of the system of stochastic numbers with the arithmetic operations addition and multiplication by scalars and the relation inclusion and point out certain practically important consequences from these properties. Our idea is to start from a minimal set of empirically known properties and to study these properties by an axiomatic approach. Based on this approach we develop an algebraic theory of stochastic numbers. A numerical example based on the Lagrange polynomial demonstrates the consistency between the CESTAC method and the presented theory of stochastic numbers.

Parallel Integration Across Time of Initial Value Problems Using PVM

This paper shows the efficiency of PVM in the solution of initial value problems on distributed architectures. The method used here is a collocation method showing a large possibility of parallelism across time. The solution of the linear or non linear system, which is a part of the method is obtained with Picards iterations using divided differences. This solution is also obtained in parallel as well as the values of the approximated solution at different times. Numerical examples are considered. The tests have been performed on a network of workstations and on the Connection Machine CM5.