Béatrice Tombuyses

Solving Markovian systems of O.D.E. for availability and reliability calculations

Abstract The computation of availability or reliability in a Markovian approach involves the solution of O.D.E.: d p d t = L(t)p , where the transition matrix is time dependent, at least when some aggregation of the states is introduced to reduce the size of the problem. Methods of solution like uniformization are in this case unapplicable and we investigate here four explicit and six implicit R.K. methods from the point of view of stability, amount of numerical work and accuracy. The test problem chosen allows an analytical solution, a uniformization method (when L(t) is constant) and a fortiori all R.K. methods. The implicit trapezoidal rule used with a variable step scheme appears to be the best compromise between accuracy and computational work.

The use of the component influence graph to reduce the size of Markovian availability problems

Abstract In this paper we will develop a new method of solution of large Markovian problems by means of auxiliary problems of smaller size. The influence graph of the physical components is built, each edge of the digraph symbolizing the influence (if any) of the composant (node of the graph) on the rate of failure or repair of another component. The analysis of the digraph introduces a natural ordering and the buildup of approximate differential systems which are solved recursively for the conditional probabilities. Two examples are given to illustrate the method.

Contribution to the study of elastic electron backscattering from polycrystalline metallic targets by Monte Carlo simulation

Low energy elastic electron backscattering has been used recently as a new tool for surface analysis. The theoretical description of this phenomenon has been mostly made by Monte Carlo simulation. Recently, several analytical approaches have been developed based on the Boltzmann equation for electron transport. However, these approaches do not give a clear image of the elastic electron backscattering phenomenon. It is precisely the aim of this paper to contribute to the study of this phenomenon by Monte Carlo simulation and to propose a very simple expression for the backscattering yield as well as a simple model for the angular distribution of elastically reflected electrons.

Computational efficiency of the continuous cell-to-cell mapping technique as a function of integration schemes

Abstract The continuous cell-to-cell mapping technique (CCCMT) is a recently proposed Markovian approach which can be used for the dynamic reliability and safety analysis of process control systems, as well as for the global analysis of nonlinear dynamic systems in general. A parametric study is performed on the computational efficiency of CCCMT as a function of various integration schemes. The results show that a variable stepsize scheme is desirable with a sufficient required precision. Among the integration schemes considered, a fourth order Runge-Kutta scheme seems to be preferable for short term simulations and the mid-point implicit scheme seems to be preferable for long term simulations or to obtain the steady-state system behavior.

Aggregation methods for reliability and availability calculations

Aggregation is a way to reduce the size of Markovian reliability and availability problems. Since exact aggregation is only possible in exceptional cases, we introduce a canonical partition for the Markovian states for systems with 2-state components that allows a systematic study of approximate aggregation. Restriction and prolongation operators are introduced to define aggregation. Four methods are proposed to define approximate systems of birth and death type. Examples are given for k-out-of-4 systems and for a 7 component system.

Backward Monte Carlo for probabilistic dynamics

Probabilistic dynamics studies the behavior of a system constituted of components and physical variables interacting together. Monte Carlo simulation is one way to solve the associated mathematical problem for a realistic system; however, Monte Carlo simulations are inefficient and biasing techniques are needed. Most research efforts have been aimed at the improvement of forward Monte Carlo schemes. A backward Monte Carlo simulation associated to the adjoint problem is studied in this paper. It can use any approximate forward solution to perform the importance biasing and can be exploited as a diagnostic approach. The method is illustrated on an example.

Reduction of the Markovian system by the influence graph method: error bound and reliability computation

Abstract In a previous article, a new method allowing the treatment of large Markovian problems was presented. Based on a graph describing the influences between the components of the system, it performs successive approximate aggregations on the exact Markovian system to reduce its size. The main drawback of this method, as of any approximate method, is to assess its validity. That is why we develop a new presentation of the method here and we define, from this presentation, error bounds for the approximate results. They are then tested for two applications, one being very large with more than 10 30 states for the exact Markovian system. We also extend the method, initially defined for availability problems, to reliability problems.

S/P images of upper triangular M‐matrices

Preconditioning by approximate factorizations is widely used in iterative methods for solving linear systems such as those arising from the finite element formulation of many engineering problems. The influence of the ordering of the unknowns on their convergence behaviour has been the subject of recent investigations because of its particular relevance for the parallel implementation of these methods. Consistent orderings are attractive for parallel implementations and subclasses of these orderings have been shown to also enhance the convergence properties of the associated preconditioned iteration scheme. The present contribution is concerned with one such class of orderings, called S/P consistent orderings. More precisely, we review here their known properties and we propose a new definition which enlarges their scope of application. A device, called S/P image of an upper triangular M‐matrix, provides a criterion for checking S/P consistency and a means to compute a relevant parameter, called maximal reduction ratio. All known properties of S/P consistent orderings are generalized to the new definition. Copyright