Mean-Variance Risk-Averse Optimal Control of Systems Governed by PDEs with Random Parameter Fields Using Quadratic Approximations
Mean-Variance Risk-Averse Optimal Control of Systems Governed by PDEs with Random Parameter Fields Using Quadratic Approximations
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DOI：
10.1137/16m106306x
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发表时间：
2017-01-01
影响因子：
2
通讯作者：
Ghattas, Omar
中科院分区：
文献类型：
--
作者：
Alexanderian, Alen;Petra, Noemi;Ghattas, Omar
提出了一种求解参数不确定的偏微分方程（PDE）系统最优控制的方法。我们考虑一个目标函数，涉及的均值和方差的控制目标，导致风险厌恶的最优控制问题。传统的数值优化方法在不确定性是禁止应用于这个问题。为了使最优控制问题易于处理，我们调用一个二次泰勒级数近似的控制目标的不确定参数字段。这使得推导出显式表达式的平均值和方差的控制目标的梯度和海森相对于不确定的参数。风险厌恶的最优控制问题，然后制定为一个偏微分方程约束的优化问题的约束给出的前向和伴随偏微分方程定义这些梯度和海森。二次近似下的控制目标的均值和方差的表达式涉及（预处理）Hessian的迹，因此无法进行评估。为了克服这个困难，我们采用跟踪估计，这只需要一个适度的海森向量产品。我们说明了我们的方法与两个具体的问题：控制的半线性椭圆型偏微分方程的不确定边界源项，和控制的线性椭圆型偏微分方程的不确定系数字段。对于后一个问题，我们推导出基于伴随的表达式，用于有效计算风险厌恶目标相对于控制的梯度。沿着的二次近似和跟踪估计，这确保了计算风险厌恶目标的成本和其相对于PDE解的数量中测量的控制的梯度是独立的（离散化的）参数和控制维度，并且仅取决于跟踪估计中采用的随机向量的数量，从而导致用于解决最优控制问题的有效的拟牛顿方法。最后，我们提出了一个全面的数值研究的最优控制问题的流体在多孔介质中的不确定渗透率场。
We present a method for optimal control of systems governed by partial differential equations (PDEs) with uncertain parameter fields. We consider an objective function that involves the mean and variance of the control objective, leading to a risk-averse optimal control problem. Conventional numerical methods for optimization under uncertainty are prohibitive when applied to this problem. To make the optimal control problem tractable, we invoke a quadratic Taylor series approximation of the control objective with respect to the uncertain parameter field. This enables deriving explicit expressions for the mean and variance of the control objective in terms of its gradients and Hessians with respect to the uncertain parameter. The risk-averse optimal control problem is then formulated as a PDE-constrained optimization problem with constraints given by the forward and adjoint PDEs defining these gradients and Hessians. The expressions for the mean and variance of the control objective under the quadratic approximation involve the trace of the (preconditioned) Hessian and are thus prohibitive to evaluate. To overcome this difficulty, we employ trace estimators, which only require a modest number of Hessian-vector products. We illustrate our approach with two specific problems: the control of a semilinear elliptic PDE with an uncertain boundary source term, and the control of a linear elliptic PDE with an uncertain coefficient field. For the latter problem, we derive adjoint-based expressions for efficient computation of the gradient of the risk-averse objective with respect to the controls. Along with the quadratic approximation and trace estimation, this ensures that the cost of computing the risk-averse objective and its gradient with respect to the control measured in the number of PDE solves-is independent of the (discretized) parameter and control dimensions, and depends only on the number of random vectors employed in the trace estimation, leading to an efficient quasi-Newton method for solving the optimal control problem. Finally, we present a comprehensive numerical study of an optimal control problem for fluid flow in a porous medium with an uncertain permeability field.