Department of Mathematics Colloquium
Center for Fluid Mechanics Seminar
Abstract: The talk is about the rheological behavior of sheared gas-solid suspensions with finite particle inertia. The fluid inertia, Brownian diffusion and gravity are neglected. The respective role of the Stokes drag and collisions on the macroscopic behavior is investigated through the simulation of discrete particle trajectories (agitation, kinetic and collisional stresses, and self-diffusion). When the Stokes number based on the shear rate increases from 1 to 10, the suspension evolves from weakly to strongly agitated regime. The transition between the two states is investigated for low to moderately concentrated suspensions. Results of numerical simulations are compared to models based on the kinetic theory adapted to moderate Stokes numbers. The effect of inelastic collisions will be briefly discussed. Finally, a modification of the Force Coupling Method is proposed to model simultaneously the particle inertia and hydrodynamic interactions.
***Joint Stochastic / Scientific Computing Seminar***
Abstract: Dijkstra's and Dial's methods are the well-known "label-setting" algorithms for finding deterministic shortest paths on graphs with positive edge-costs. The stochastic shortest path (SSP) problem is a natural generalization, for which such efficient (non-iterative) methods are usually unavailable. In SSPs, the current state of the system is still described by a specific node of a directed graph. At each stage of the process we select one from a (possibly node-dependent) compact set of available control values. The chosen control determines the cost incurred at that stage and the probability distribution over the successor-nodes for the next transition. The process continues until a transition to one of the "exit nodes". The goal is to find a policy minimizing the expected value of the total cost up to the termination. Bellman's optimality for the value function results in a system of coupled nonlinear equations, which has to be solved iteratively and generally requires an infinite number of iterations. Additional structure of an optimal policy can sometimes be used to prove that only a finite number of iterations is needed and that label-setting algorithms are, in fact, applicable. We will introduce a class of "multimode" SSPs, for which such a structure is guaranteed to be present in all optimal policies. SSPs provide a natural discretization for static Hamilton-Jacobi-Bellman PDEs of continuous optimal control. Hence, label-setting for the former yields efficient (non-iterative) numerical methods for the latter. We will illustrate this point using time-optimal control problems on a continuous domain.
PDE Seminar
Department of Mathematics Colloquium
<--- 2007 Index