This project is to study numerical solutions for time-dependent stochastic partial differential equations (SPDEs). In particular the investigator will construct fast, practical numerical algorithms. At the same time, A solid theoretical basis for these algorithms based on proper error analysis will be provided. The project intends to concentrate on stochastic parabolic partial differential equations as our model problem. However, it is expected that the algorithms and analysis will be extended to many other types of time dependent SPDEs. A concerted and comprehensive effort will be made to develop efficient, accurate, and robust computational methodologies, which from the very beginning incorporate uncertainty effects. The research has three major components: 1. Study of finite element approximations for high dimensional parabolic SPDEs with a forcing term involving either multiplicative colored noise or a random field; 2. Investigation of fast collocation methods to numerically evaluate statistical moments of parabolic SPDEs with random boundary input data; 3. Construction of enhanced Monte Carlo methods, using sensitivity derivatives, for SPDEs with random parameters such as diffusion coefficients.
Scientists have discovered that there is a significant amount of uncertainty in all physical systems; not just when something is measured, but even when an attempt is made to describe how the system changes. No computer calculation can possibly consider every slight variation in the measurements and dynamics of a system under study. And yet it is known that, at least in some cases, small amounts of uncertainty can lead to significant, and even disastrous, errors in the computed results. The proposed research intends to make an effort to understand, quantify and control the effect of uncertainties through numerical computations. The underling mathematical equations describe basic physical phenomena such as heat transfer, diffusion processes and fluid flow dynamics. An important bonus of this research is the involvement of a group of undergraduate and graduate students, some of them from under represented groups. It is expected that their participation in this project will expose them to scientific research, induce them to pursue further training, and to consider a scientific career as a future.
