Low‐rank solution of an optimal control problem constrained by random Navier‐Stokes equations

Abstract

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.

Stochastic Discontinuous Galerkin Methods for Robust Deterministic Control of Convection Diffusion Equations with Uncertain Coefficients

“…We address this issue by applying a lowrank variant of generalized minimal residual (GMRES) [30] with suitable preconditioners, which reduce both the storage requirements and the computational complexity by exploiting a Kronecker-product structure of system matrices; see, e.g., [31,32,33]. Low-rank approximation for the optimal control problems with uncertain inputs have been also studied in [14,34,35,36] for unconstrained control problems and in [36] for control constraint problems. In the aforementioned studies, randomness is generally defined in the diffusion parameter however we here consider the randomness both in diffusion or convection parameters by using also discontinuous Galerkin method in spatial domain.…”

A Low-Rank Solver for the Navier--Stokes Equations with Uncertain Viscosity

“…As a result, the lowrank method achieves greater computational savings for the problems with larger correlation length. Next, we examine the performances of the low-rank approximation method for varying CoV , which is defined in (3). In this experiment, we fix the value of Re 0 = 100 and the variance of the random σ ν is controlled.…”

“…In particular, we consider a random viscosity affinely dependent on a set of random variables as suggested in [19] (and in [23], which considers a gPC approximation of the lognormally distributed viscosity). The stochastic Galerkin formulation of the stochastic Navier-Stokes equations is also considered in [3], which studies an optimal control problem constrained by the stochastic Navier-Stokes problem and computes an approximate solution using a low-rank tensor-train decomposition [17]. Related work [26] extends a Proper Generalized Decomposition method [16] for the stochastic Navier-Stokes equations, where a low-rank approximate solution is built from successively computing rank-one approximations.…”

Parallel algorithms for computing the tensor-train decomposition

“…A general tensor X ∈ C n1ו••×n d requires d j=1 n j degrees of freedom to store, which scales exponentially with the order d. Therefore, it is often essential to approximate or represent large tensors with data-sparse formats so that storing and computing with them is feasible. The tensor-train (TT) decomposition [31] is a tensor format with a storage cost that can scale linearly in n j and d. The TT format is used in molecular simulations [33], high-order correlation functions [26], and partial differential equation (PDE) constrained optimization [4,14]. In practice, one tries to replace X by a tensor X with a data-sparse TT format such that…”