Karl K. Sabelfeld

Integral Formulation of the Boundary Value Problems and the Method of Random Walk on Spheres

We present probabilistic representations for some systems of elliptic equations. They are constructed as expectations of functionals of some specific Markov chains, in particular the walk on spheres processes. These representations rely on converse mean value theorems that we show for the equations under analysis, especially the Lame equation of elasticity theory. We deduce Monte Carlo procedures, whose convergence is proven and cost is analyzed.

A Lagrangian Stochastic Model for the Transport in Statistically Homogeneous Porous Media

A new type of stochastic simulation models is developed for solving transport problems in saturated porous media which is based on a generalized Langevin stochastic differential equation. A detailed derivation of the model is presented in the case when the hydraulic conductivity is assumed to be a random field with a lognormal distribution, being statistically isotropic in space. To construct a model consistent with this statistical information, we use the well-mixed condition which relates the structure of the Langevin equation and the probability density function of the Eulerian velocity field. Numerical simulations of various statistical characteristics like the mean displacement, the displacement covariance tensor and the Lagrangian correlation function are presented. These results are compared against the conventional Direct Simulation Method.

Iterative procedure for multidimensional Euler equations

s — A numerical iterative scheme is suggested to solve the Euler equations in two and three dimensions. The step of the iteration procedure consists in integration over the velocity which is here carried out by three different approximate integration methods, and in particular, by a special Monte Carlo technique. In the Monte Carlo integration, we suggest a dependent sampling technique which ensures that the statistical errors are quite small and uniform in space and time. Comparisons of the Monte Carlo calculations with the trapezoidal rule and a gaussian integration method show good agreement.