Low‐rank solution of an optimal control problem constrained by random Navier‐Stokes equations,International Journal for Numerical Methods in Fluids

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Low‐rank solution of an optimal control problem constrained by random Navier‐Stokes equations
International Journal for Numerical Methods in Fluids ( IF 1.7 ) Pub Date : 2020-04-20 , DOI: 10.1002/fld.4843
Peter Benner ₁ , Sergey Dolgov ₂ , Akwum Onwunta _{3,

4} , Martin Stoll ₅

Affiliation

We develop a low‐rank tensor decomposition algorithm for the numerical solution of a distributed optimal control problem constrained by two‐dimensional time‐dependent Navier‐Stokes equations with a stochastic inflow. The goal of optimization is to minimize the flow vorticity. The inflow boundary condition is assumed to be an infinite‐dimensional 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 time‐dependent problem require solving a coupled saddle‐point system of nonlinear equations on all time steps at once. For the resulting discrete problem, we approximate the solution by the tensor‐train (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 high‐dimensional stochastic time‐dependent problem can be solved with the asymptotic complexity of the corresponding deterministic problem.

中文翻译：

随机Navier-Stokes方程约束的最优控制问题的低秩解

针对具有随机流入的二维时变Navier-Stokes方程约束的分布式最优控制问题的数值解，我们开发了一种低秩张量分解算法。优化的目的是使流动涡度最小化。假定流入边界条件是一个无限维随机场，它使用有限（但高维）维傅立叶展开进行参数化，并使用随机Galerkin有限元方法离散化。这导致离散解决方案中的自由度过高。此外，与时间有关的问题中的最优条件要求在所有时间步上同时求解非线性方程的鞍点耦合系统。对于由此产生的离散问题，我们通过张量-火车（TT）分解来近似解，并提出一种数值有效的算法来直接在TT表示中求解最优性方程。该算法基于交替线性方案（ALS），但是与基本ALS方法相反，该新算法利用并保留了最优性方程的块结构。从每个步骤都需要求解非奇异线性系统的解的意义上说，我们证明了这种结构的保留使所提出的ALS块方法具有很好的适用性，对于基本ALS可能不是这种情况。最后，我们基于两个基准问题提出了数值实验，该问题模拟了围绕vonKármán涡流和后退台阶的流动，每一个都有不确定的流入。实验表明，使用TT表示和块ALS算法可以显着降低复杂度。具体来说，我们观察到高维随机时变问题可以通过相应确定性问题的渐近复杂性来解决。

更新日期：2020-04-20

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