Title: A Brief Tour of Risk-Averse PDE-Constrained Optimization: From models to data-driven stability
Abstract: Over the past few decades, risk-averse optimization has emerged as a crucial framework for decision-making under uncertainty in systems governed by partial differential equations (PDEs). Because uncertainty in PDE parameters introduces profound theoretical and computational challenges, addressing these problems requires a synthesis of functional analysis, numerical optimization, and statistics. In this plenary, I will survey recent advances in risk-averse PDE-constrained optimization, emphasizing theoretical foundations, algorithmic innovations, and asymptotic analysis.
The talk will focus on three central themes. First, I will present rigorous formulations that integrate risk measures into PDE-constrained settings to accurately capture the behavior of risk-averse decision-makers. Second, I will discuss algorithmic strategies designed to handle nonsmooth risk measures, ensuring that large-scale computations remain both tractable and robust. Finally, I will explore stability and asymptotic behaviors, characterizing how solutions respond to perturbations in data, discretization, and sampling, which ultimately provides statistical guarantees for these risk-averse models.
Together, these directions demonstrate how risk-averse optimization enriches the mathematical landscape of PDE-constrained problems, offering a principled and reliable framework for decision-making in the face of uncertainty.