Optimal control of transport dominated problems with reduced order modeling
Optimal control for transport-dominated systems becomes tractable and practical when paired with carefully constructed reduced-order models. Transport phenomena (shocks, moving vortices, concentration fronts) concentrate energy in low-dimensional, moving coherent structures that standard global bases struggle to represent. However, nonlinear model order reduction techniques like the sPOD-Galerkin method capture those features with far fewer degrees of freedom. That dramatic reduction in state dimension cuts the cost of forward and adjoint solves by orders of magnitude, enabling faster gradient evaluation, real-time or many-query optimization, and easier incorporation of constraints and parameter dependencies. When designed and validated carefully, ROM-based optimal control preserves essential physics (stability, conservation) while making control synthesis, sensor/actuator placement, and uncertainty-aware design feasible for problems that would otherwise be computationally intractable.
Our work presents an in-depth theoretical and numerical analysis of an example transport problem applied in an optimal control context and is shown in our paper
related publications
2024
OCsPOD
Optimal control for a class of linear transport-dominated systems via the shifted proper orthogonal decomposition
Tobias Breiten, Shubhaditya Burela, and Philipp Schulze
Solving optimal control problems for transport-dominated partial differential equations (PDEs) can become computationally expensive, especially when dealing with high-dimensional systems. To overcome this challenge, we focus on developing and deriving reduced-order models that can replace the full PDE system in solving the optimal control problem. Specifically, we explore the use of the shifted proper orthogonal decomposition (POD) as a reduced-order model, which is particularly effective for capturing high-fidelity, low-dimensional representations of transport-dominated phenomena. Furthermore, we propose two distinct frameworks for addressing these problems: one where the reduced-order model is constructed first, followed by optimization of the reduced system, and another where the original PDE system is optimized first, with the reduced-order model subsequently applied to the optimality system. We consider a 1D linear advection equation problem and compare the computational performance of the shifted POD method against the conventional methods like the standard POD when the reduced-order models are used as surrogates within a backtracking line search.
@article{BrBuSc,title={Optimal control for a class of linear transport-dominated systems via the shifted proper orthogonal decomposition},author={Breiten, Tobias and Burela, Shubhaditya and Schulze, Philipp},journal={arXiv},year={2024},doi={https://doi.org/10.48550/arXiv.2412.18950},}