A trust-region algorithm for PDE-constrained optimization with bound constraints using reduced-order modeling

Abstract

We present a trust-region algorithm for large-scale PDE-constrained optimization with bound constraints that uses proper orthogonal decomposition (POD) to accelerate the iterative solution. The algorithm samples high-fidelity state and Lagrange-multiplier data along the optimization trajectory and automatically computes and updates the POD basis used to construct reduced-order models, which replace computationally intensive high-fidelity finite element evaluations. A trust-region acceptance criterion detects loss of predictive accuracy in the reduced model and triggers adaptive enrichment of the POD basis, providing a sound mathematical metric to control inexactness during optimization. Bound constraints on the control variables are handled natively through a projected-gradient method.

We also present a linear Hessian formulation for compliance minimization in topology optimization. The quadratic model used inside the trust-region sub-problem requires second-order derivative information; a nonlinear Hessian formulation incurs additional finite element evaluations at each inner iteration. The proposed linear formulation eliminates these extra solves. On three benchmark topology optimization problems, namely the symmetric MBB beam, the Michell beam, and the cantilever beam, it is shown to deliver substantial speedups over the high-fidelity baseline without degrading the optimal topology.