Mathematics

Parametric optimal control problems aim to narrow the gap between collected data and mathematical models, enabling more reliable and accurate simulations. These problems are common in various scientific contexts; however, their computational complexity often limits their applicability, especially in uncertain parametric settings involving multiple evaluations for many parameters. Realistic problems typically involve parameters affected by uncertainty due to scattered or missing data and noisy measurements.This presentation will focus on optimization problems governed by parametric partial differential equations under the influence of random variables. To estimate statistical quantities, such as moments of the solutions, Monte Carlo estimators are employed. These estimators compute the average over numerous optimal solutions for various outcomes of the random parametric instance. This approach requires many simulations for different parameters, making standard discretization techniques unfeasible due to their time-consuming and computationally expensive nature. To address this challenge, we employ reduced-order models. These models exploit the parametric structure of the problem, identifying a low-dimensional representation known as the reduced space. Through Galerkin projection in this reduced space, the problem can be solved faster without compromising accuracy. These strategies accelerate standard statistical analysis techniques. Specifically, we focus on weighted ROMs (wROMs), tailored reduced strategies based on prior knowledge about the distribution of the random variable. Enhancing the reduced model with previous distribution information further accelerates simulations for new parametric instances, surpassing the capabilities of standard ROMs. This presentation will start with an introductory overview of optimal control for parametric PDEs and standard ROMs. Then, it will delve into wROMs and explore their applications in environmental sciences and convection-dominated flows in both steady and time-dependent settings.