Graduate Student Seminar

Speaker 1: Serap Tay

Title

A Stochastic Binary Opinion Model and Asymptotic Analysis of Stationary Probabilities

Abstract

In this study we analyze a stochastic binary opinion dynamics model where we consider “yes” and “no” to be possible opinions obtained by an agent at each moment. In this model, the number of agents with opinion “yes” at time t is regarded as a birth-death process and the configuration of the opinions at each moment is determined by a probability distribution.  We consider that the rate an agent changes his/her opinion is a function of environmental influence and agent’s own thinking. The former is represented by a rate function and a coefficient of “conformity”, alpha. On the other hand, the coefficient of "self-motivation,” beta, represents the latter. We give an asymptotic analysis of the stationary probabilities when the rate function is C^1 and present computational results. Moreover, we consider a discontinuous rate function and obtain asymptotic approximations of stationary probabilities when the number of agents gets very large. We give approximations when “conformity” dominates “self-motivation” that is alpha > beta and also when alpha < beta. We present computational results supporting the obtained approximations.

Speaker 2: Mona Hajghassem

Title

Multigrid preconditioners for boundary control of elliptic-constrained optimal control problems

Abstract

The goal of this project is to devise efficient multigrid algorithms for the boundary control of elliptic equations. Using a reduced formulation, our focus is on designing optimal order multigrid preconditioners for the Hessian of the reduced cost functional. Ideally, the preconditioners should approximate the reduced Hessian with optimal order with respect to the discretization of the elliptic equation. We show that for Dirichlet boundary control of elliptic equations the preconditioner is of suboptimal quality, though still efficient.

This project is part of a larger research program on developing efficient solution methods for optimal control problems with PDE constraints.