Adaptive Model Hyperreduction to Accelerate Optimization Problems Governed by Partial Differential Equations

Optimization problems constrained by partial differential equations (PDEs) are prevalent across modern science and engineering. They are crucial in the optimal design and control of multiphysics systems, as well as in nondestructive evaluation, detection, and inverse problems. Solving these optimization problems often requires numerous numerical solutions of the governing equations. When these problems involve complex physical interactions in intricate domains, obtaining solutions can be computationally demanding, both in terms of time and resources, making the optimization process challenging or even infeasible. This dissertation first introduces a numerical method to efficiently solve optimization problems governed by large-scale nonlinear systems of equations, including discretized partial differential equations, using projection-based reduced-order models accelerated with hyperreduction (empirical quadrature) and embedded in a trust-region framework that guarantees global convergence. The proposed framework constructs a hyperreduced model on-the-fly during the solution of the optimization problem, which completely avoids an offline training phase. This ensures all snapshot information is collected along the optimization trajectory, which avoids wasting samples in remote regions of the parameters space that are never visited, and inherently avoids the curse of dimensionality of sampling in a high-dimensional parameter space. At each iteration of the proposed algorithm, a reduced basis and empirical quadrature weights are constructed precisely to ensure the global convergence criteria of the trust-region method are satisfied, ensuring global convergence to a local minimum of the original (unreduced) problem. The proposed trust-region method is then integrated into an augmented Lagrangian framework to solve generally constrained problems. At each major augmented Lagrangian iteration, the expensive optimization subproblem involving the full nonlinear system is replaced by an empirical quadrature-based hyperreduced model constructed on the fly. To ensure convergence of these inexact augmented Lagrangian subproblems, we develop a bound-constrained trust-region method that accommodates inexact gradient evaluations and is specialized for our setting, leveraging hyperreduced models.