Progressive construction of a parametric reduced-order model for PDE-constrained optimization

TL;DR

This paper introduces an adaptive, progressive reduced-order modeling approach for PDE-constrained optimization that iteratively refines the model, significantly reducing high-dimensional model queries while maintaining high accuracy in solutions.

Contribution

It presents a novel progressive framework that combines adaptive sampling, trust-region methods, and sensitivity integration for efficient PDE-constrained optimization.

Findings

Reduces HDM queries by a factor of 4-5.

Achieves optimal solutions with errors less than 0.1%.

Demonstrates effectiveness in aerodynamic shape optimization.

Abstract

An adaptive approach to using reduced-order models as surrogates in PDE-constrained optimization is introduced that breaks the traditional offline-online framework of model order reduction. A sequence of optimization problems constrained by a given Reduced-Order Model (ROM) is defined with the goal of converging to the solution of a given PDE-constrained optimization problem. For each reduced optimization problem, the constraining ROM is trained from sampling the High-Dimensional Model (HDM) at the solution of some of the previous problems in the sequence. The reduced optimization problems are equipped with a nonlinear trust-region based on a residual error indicator to keep the optimization trajectory in a region of the parameter space where the ROM is accurate. A technique for incorporating sensitivities into a Reduced-Order Basis (ROB) is also presented, along with a methodology for computing sensitivities of the reduced-order model that minimizes the distance to the corresponding HDM sensitivity, in a suitable norm. The proposed reduced optimization framework is applied to subsonic aerodynamic shape optimization and shown to reduce the number of queries to the HDM by a factor of 4-5, compared to the optimization problem solved using only the HDM, with errors in the optimal solution far less than 0.1%.