Solving optimal control problems governed by random Navier-Stokes equations using low-rank methods
For researchers in computational fluid dynamics and uncertainty quantification, this provides a method to tackle stochastic optimal flow control problems that were previously intractable due to computational cost.
The paper addresses optimal control of time-dependent Navier-Stokes equations with uncertain inputs, which is computationally prohibitive due to high dimensionality. By using low-rank Tensor Train decomposition, they solve the optimality equations directly in low-rank form, achieving modest rank growth even for high Reynolds numbers, enabling reduced-order modeling.
Many problems in computational science and engineering are simultaneously characterized by the following challenging issues: uncertainty, nonlinearity, nonstationarity and high dimensionality. Existing numerical techniques for such models would typically require considerable computational and storage resources. This is the case, for instance, for an optimization problem governed by time-dependent Navier-Stokes equations with uncertain inputs. In particular, the stochastic Galerkin finite element method often leads to a prohibitively high dimensional saddle-point system with tensor product structure. In this paper, we approximate the solution by the low-rank Tensor Train decomposition, and present a numerically efficient algorithm to solve the optimality equations directly in the low-rank representation. We show that the solution of the vorticity minimization problem with a distributed control admits a representation with ranks that depend modestly on model and discretization parameters even for high Reynolds numbers. For lower Reynolds numbers this is also the case for a boundary control. This opens the way for a reduced-order modeling of the stochastic optimal flow control with a moderate cost at all stages.