DEPARTMENT OF MATHEMATICAL SCIENCES

APPLIED AND COMPUTATIONAL MATHEMATICS SEMINAR

**Speaker:**Drew Kouri, Sandia National Laboratories

**Title: ***
A Data-Driven Approach to PDE-Constrained Optimization under Uncertainty
*

**Abstract:**
Many science and engineering applications require the control or
design of a physical system governed by partial differential equations
(PDEs). More often then not, PDE inputs such as coefficients,
boundary
conditions, or initial conditions are unknown and estimated
from experimental data. In this talk, I will discuss some
theoretical challenges associated with such PDE-constrained
optimization problems, including their mathematical
formulation and their efficient numerical solution.
First, I will assume that we know the probability
distributions that characterize the uncertain PDE inputs.
For this case, I will introduce the notion of a risk measure
as a means to quantify the `hazard' associated with large
objective function values. Often risk-averse quantities
are not differentiable in the classic sense and thus not
appropriate for derivative-based optimization. To
circumvent this issue, I will present a theory for smooth
risk measures in the context of PDE-constrained optimization.
Next, to handle the situation of an unknown probability
distribution, I will introduce and analyze a
distributionally-robust formulation for the optimization
problem. To enable numerical solutions, I will present
a novel discretization for the unknown probability measure
and provide rigorous error bounds for this approximation.
I will conclude with numerical results confirming the
aforementioned error bounds.

**Time:** Friday, October 23, 2015, 1:30-2:30 p.m.

**Place:** Exploratory Hall, Room 4106

Department of Mathematical Sciences

George Mason University

4400 University Drive, MS 3F2

Fairfax, VA 22030-4444

http://math.gmu.edu/

Tel. 703-993-1460, Fax. 703-993-1491