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Abstract
We develop a lowrank tensor decomposition algorithm for the numerical solution of a distributed optimal control problem constrained by twodimensional timedependent NavierStokes equations with a stochastic inflow. The goal of optimization is to minimize the flow vorticity. The inflow boundary condition is assumed to be an infinitedimensional random field, which is parametrized using a finite (but high) dimensional Fourier expansion and discretized using the stochastic Galerkin finite element method. This leads to a prohibitively large number of degrees of freedom in the discrete solution. Moreover, the optimality conditions in a timedependent problem require solving a coupled saddlepoint system of nonlinear equations on all time steps at once. For the resulting discrete problem, we approximate the solution by the tensortrain (TT) decomposition and propose a numerically efficient algorithm to solve the optimality equations directly in the TT representation. This algorithm is based on the alternating linear scheme (ALS), but in contrast to the basic ALS method, the new algorithm exploits and preserves the block structure of the optimality equations. We prove that this structure preservation renders the proposed block ALS method well posed, in the sense that each step requires the solution of a nonsingular reduced linear system, which might not be the case for the basic ALS. Finally, we present numerical experiments based on two benchmark problems of simulation of a flow around a von Kármán vortex and a backward step, each of which has uncertain inflow. The experiments demonstrate a significant complexity reduction achieved using the TT representation and the block ALS algorithm. Specifically, we observe that the highdimensional stochastic timedependent problem can be solved with the asymptotic complexity of the corresponding deterministic problem.
Original language  English 

Journal  International Journal for Numerical Methods in Fluids 
Early online date  13 May 2020 
DOIs  
Publication status  Epub ahead of print  13 May 2020 
Keywords
 iterative methods
 lowrank solution
 PDEconstrained optimization
 preconditioning
 saddlepoint system
 Schur complement
 stochastic Galerkin system
 tensor train format
ASJC Scopus subject areas
 Computational Mechanics
 Mechanics of Materials
 Mechanical Engineering
 Computer Science Applications
 Applied Mathematics
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Projects
 1 Finished

Tensor product numerical methods for highdimensional problems in probability and quantum calculations
1/01/16 → 31/12/18
Project: Research council