# Applied Math Seminar: A Data-Driven Approach to PDE-Constrained Optimization under Uncertainty

### Event Description:

Abstract:

Many science and engineering applications require the optimal control or design of a physical system governed by partial differential equations (PDEs). More often then not, PDE inputs such as coefficients, boundary conditions and initial conditions are unknown and estimated from noisy, incomplete data. In this talk, I will discuss the theoretical challenges associated with such PDE-constrained optimization problems, including their mathematical formulation and their efficient numerical solution. First, I will assume that the probability distributions characterizing the uncertain PDE inputs are known. For this case, I will review risk measures as a means for quantifying the "hazard" associated with large objective function values. Next, to handle the situation of unknown probability distributions, 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 bounds.

Coffee and tea will be served in the lounge at 15.00

### Event Contact

**Contact Name: **Deborah Sulsky