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

TL;DR

This paper introduces a novel on-the-fly hyperreduction method integrated with a trust-region framework to efficiently solve large-scale nonlinear optimization problems, ensuring global convergence without offline training.

Contribution

The method constructs hyperreduced models dynamically during optimization, avoiding offline training and high-dimensional sampling, while guaranteeing convergence to a local minimum.

Findings

Achieves over 18x speedup compared to standard methods.

Ensures global convergence to a local minimum.

Demonstrates effectiveness on fluid shape optimization problems.

Abstract

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 18x (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.