Efficient Algorithms for Optimization Problems with PDE Constraints
Abstract
Simulation-constrained optimization arises in most areas of science and engineering, including design of electromagnetic materials and devices for use in surveillance and warfighter applications and control of ventilation systems to mitigate the spread of contagions such as COVID-19. These problems are often fraught with uncertainty due to unknown physical parameters and environments. A common thread between these applications is the need to efficiently optimize physical systems given by partial differential equations (PDEs). Such problems falls in the category of PDE Constrained Optimization. The proposal aims to consider generic optimization problems constrained by PDEs with or without uncertainty.Many challenges exist in studying such problems. Firstly, the existing optimization algorithms are mostly focused on finite dimensional problems. However, the PDE constrained optimization problems are infinite dimensional, nonsmooth and nonconvex. Indeed using off-the-shelf optimization algorithms lead to mesh dependent behavior, i.e., the optimization iterations grow with mesh refinement. Secondly, the existing discretization error estimates focus on continuous and discrete solutions and fail to account for optimization solvers. From a practical point of view, this is not a useful setting. Thirdly, many of the PDE constrains are dynamic, one evaluation of gradient using adjoint method requires storing the entire solution trajectory. Since the optimization algorithms are iterative, therefore these approaches without compression are intractable. Finally, the traditional approaches to optimization problems with uncertainty typically computes expectation of the random variable objective function. This is an #average-based# approach and does not account for rare but catastrophic events. In addition, all these problems are high dimensional optimization problems (due to random variable) and are intractable for the existing sampling based methods.The proposal aims to tackle all these challenges. In case of uncertainty, a risk-averse optimization based framework will be created that can account for rare but high consequence events. Novel augmented Lagrangian based algorithms will be developed with subproblem solves given by adaptive inexact trust region optimization methods. Inexactness arise from tensor train (TT) approximations and iterative linear system solves, and reduced order modeling techniques such as matrix sketching leading to inexact objective and gradient. Novel criticality based discretization error estimates will be developed which will also guide the inexactness and adaptivity while movingtowards optimality.The proposal will develop these fundamental mathematical concepts and algorithms which have many potential applications for Navy. In particular, discussions will be carried out with the US Naval Research Lab in Washington DC to understand the applicability of these mathematical concepts to the electromagnetic (EM) signature problems. Several students and postdocs will be trained under the project to build next generation of workforce.Approved for Public Release
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Mar 08, 2024
- Source ID
- N000142412147
Entities
People
- Harbir Antil
Organizations
- George Mason University
- Office of Naval Research
- United States Navy