Low-rank solvers for unsteady Stokes–Brinkman optimal control problem with random data

We consider the numerical simulation of an optimal control problem constrained by the unsteady Stokes–Brinkman equation involving random data. More precisely, we treat the state, the control, the target (or the desired state), as well as the viscosity, as analytic functions depending on uncertain parameters. This allows for a simultaneous generalized polynomial chaos approximation of these random functions in the stochastic Galerkin finite element method discretization of the model. The discrete problem yields a prohibitively high dimensional saddle point system with Kronecker product structure. We develop a new alternating iterative tensor method for an efficient reduction of this system by the low-rank Tensor Train representation. Besides, we propose and analyze a robust Schur complement-based preconditioner for the solution of the saddle-point system. The performance of our approach is illustrated with extensive numerical experiments based on two- and three-dimensional examples, where the full problem size exceeds one billion degrees of freedom. The developed Tensor Train scheme reduces the solution storage by two–three orders of magnitude, depending on discretization parameters.