A globally convergent method to accelerate large-scale optimization using on-the-fly model hyperreduction: Application to shape optimization

• Published in: Journal of computational physics 2023-07, Vol.484, p.112082, Article 112082
• Main Authors: Wen, Tianshu, Zahr, Matthew J.
• Format: Article
• Language: English
• Subjects: Hyperreduction; On-the-fly sampling; PDE-constrained optimization; Reduced-order model; Shape optimization; Trust-region method
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2023.112082

We present 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. Numerical experiments are performed on two fluid shape optimization problems to verify the global convergence of the method and demonstrate its computational efficiency; speedups over 18× (accounting for all computational cost, even cost that is traditionally considered “offline” such as snapshot collection and data compression) relative to standard optimization approaches that do not leverage model reduction are shown. •Accelerated large-scale optimization using hyperreduced models and trust regions.•Global convergence theory for the proposed accelerated trust-region method.•Empirical quadrature with additional constraints tailored to the optimization setting.•Acceleration of physics-based mesh motion using linear model reduction.•Speedups over 18× obtained for fluid shape optimization problems.