## Differential Equations Seminar: graduate students

#### Steve Bjerkaas, Sumaya Alzuhairy, Randy Price

Monday, May 7, 2018 · 11 AM - Noon

**Speaker**

**:**Randy Price

**Title**

**:**Stochastic finite element methods for the time-dependent heat equation

**Abstract**

**:**In this talk we will briefly go over three methods for implementing Stochastic Finite Elements: Monte Carlo, Stochastic Collocation, and Stochastic Galerkin. We will talk about how to apply these methods to my model problem,

the time-dependent heat equation. Finally, we will cover some numerical results that show that for our model problem Stochastic Collocation is a more efficient method than Monte Carlo.

**Speaker:**Steve Bjerkaas

**Multigrid methods applied to stochastic diffusion problems**

**Title:****Abstract:**Numerical solutions of stochastic partial differential equations have been explored for many years. We study the benefits of applying multigrid methods to the numerical solution of the stochastic diffusion problem. We compare the residual and computation cost and time of various forms of multigrid to include V-cycles, and full multigrid method. We conclude with describing future research involving incorporating the stochastic random variable space into the coarsening strategy of the multigrid algorithm in addition to coarsening of the physical grid that is implemented in existing algorithms.

**Speaker:**Sumaya Alzuhairy

**Title:**Efficient multilevel methods for optimal control of elliptic equations with stochastic coefficients.

**Abstract:**A common strategy for solving optimal control of stochastic PDEs relies on stochastic collocation, which reduces the problem to multiple solves of optimal control problems constrained by deterministic PDEs. In this work we investigate an alternative approach where we use a stochastic Galerkin formulation and discretization of the PDE prior to solving the optimal control problem. Ultimately this requires solving a potentially very large linear system, which we then solve using specially designed multilevel algorithms.

*Acknowledgement*: The research was supported in part by the National Science Foundation (DMS-1521563).