🤖 AI Summary
This work addresses the high computational cost of gradient-based optimization for transient convection–diffusion problems using reduced-order models (ROMs). We propose a “optimize-then-reduce” coupled framework wherein the primal and adjoint systems are simultaneously solved within the reduced space at each time step. To alleviate the dependency on high-dimensional adjoint snapshots, we devise an efficient adjoint snapshot collection strategy. Furthermore, we overcome conventional limitations in adjoint basis construction by introducing an adaptive adjoint basis selection method guided by energy decay, iteration count, and computational time. Numerical experiments demonstrate that the proposed approach significantly reduces the computational overhead of adjoint system solves and overall simulation time, while maintaining controllable error levels. The method enables real-time, high-fidelity optimization in ROM–ROM coupled settings.
📝 Abstract
Optimization-based coupling (OBC) is an attractive alternative to traditional Lagrange multiplier approaches in multiple modeling and simulation contexts. However, application of OBC to time-dependent problems has been hindered by the computational cost of finding the stationary points of the associated Lagrangian, which requires primal and adjoint solves. This issue can be mitigated by using OBC in conjunction with computationally efficient reduced order models (ROM). To demonstrate the potential of this combination, in this paper we develop an optimization-based ROM-ROM coupling for a transient advection-diffusion transmission problem. We pursue the ``optimize-then-reduce'' path towards solving the minimization problem at each timestep and solve reduced-space adjoint system of equations, where the main challenge in this formulation is the generation of adjoint snapshots and reduced bases for the adjoint systems required by the optimizer. One of the main contributions of the paper is a new technique for efficient adjoint snapshot collection for gradient-based optimizers in the context of optimization-based ROM-ROM couplings. We present numerical studies demonstrating the accuracy of the approach along with comparison between various approaches for selecting a reduced order basis for the adjoint systems, including decay of snapshot energy, average iteration counts, and timings.