Volume 10, Issue 2

The Smoluchowski equation is a nonlinear integro-differential equation describing the evolution of the concentration μ t ( dx ) of particles of mass in ( x , x + dx ) in an infinite particle system where coalescence occurs. We introduce a class of algorithms, which allow, under some conditions, to simulate exactly a stochastic process ( X t ) t ≥0 , whose time marginals are given by ( xμ t ( dx )) t ≥0 .

Discrepancy of van der Corput sequences constructed by piecewise linear transformations with same slope β > 1 is discussed. If there is no eigenvalues of the Perron-Frobenius operator associated with the dynamical system in | z | > 1/ β except 1, the discrepancy equals (log N ) k +1 / N , where k is the number of the non-Markov endpoints.

Splitting is a widely known Monte Carlo variance reduction method (VRM). It has been successfully applied for a long time in Monte Carlo applications to neutral particles transport in Nuclear Engineering. In this field splitting is usually referred to as a Geometric Splitting method since it relies on geometric properties of the medium, namely distance and direction. Splitting is remarkable in the sense that it is a "safe" method. Unlike VRMs based on biasing it can not cause the sow tooth phenomenon in which a wrong result appears with very small statistical error. The only risk of using this method is increasing computation time. This makes the application of this method to such areas as safety-critical systems reliability and risk assessment a very attractive option. Suggested work concerns application of splitting to systems reliability problems. The system's structure based metric is used instead of the geometric approach. This metric takes into account importance of each system's component for the target event. Multiple disjunctive surfaces splitting, and splitting combined with the Cluster-Event biasing are considered. The optimal splitting parameters are predicted using the mathematical apparatus developed for applications of splitting in neutral particles transport. Provided numerical examples show considerable benefit of the method.

We propose a two-step simulation scheme for the solution of a singular stochastic differential equation with exploding drift. First we estimate the strong order of the Yosida approximation. Then we use a semi-implicit Euler scheme to discretize the approximate solution. Numerical experiments are displayed for the paths of Brownian particles with strong repulsive interaction. We also present two simple simulation schemes for Bessel processes with arbitrary dimension.

The methods which track particles to simulate concentrations are successfully used in problems for large scale transport in groundwater, mainly when the aquifer properties are spatially heterogeneous and the process is advection dominated. These methods, sometimes called "analogical Monte Carlo methods", are not concerned with the numerical diffusion occurring in finite difference/element schemes. The limitations of classical random walk methods are due to large computation time and memory necessary to achieve statistically reliable results and accurate concentration fields. To overcome these computational limitations a "global random walk" (GRW) algorithm was developed. Unlike in the usual approach where the trajectory of each particle is simulated and stored, in GRW all the particles from a given grid node are scattered, at a given time, using a single numerical procedure, based on Bernoulli distribution, partial-deterministic, or deterministic rules. Because no restrictions are imposed for the maximum number of particles to be used in a simulation, the Monte Carlo repetitions are no longer necessary to achieve the convergence. It was proved that for simple diffusion problems GRW converges to the finite difference scheme and that for large scale transport problems in groundwater, GRW produces stable and statistically reliable simulations. A 2-dimensional transport problem was modeled by simulating local diffusion processes in realizations of a random velocity field. The behavior over 5000 days of the effective diffusion coefficients, concentration field and concentration fluctuations were investigated using 2560 realizations of the velocity field and 10 10 particles in every realization. The results confirm the order of magnitude of the effective diffusion coefficients predicted by stochastic theory but the time needed to reach the asymptotic regime was found to be thousands times larger. It is also underlined that the concentration fluctuations and the dilution of contaminant solute depend essentially on local diffusion and boundary conditions.