We discretize a risk-neutral optimal control problem governed by a linear elliptic partial differential equation with random inputs using a Monte Carlo sample-based approximation and a finite element discretization, yielding finite dimensional control problems. We establish an exponential tail bound for the distance between the finite dimensional problems' solutions and the risk-neutral problem's solution. The tail bound implies that solutions to the risk-neutral optimal control problem can be reliably estimated with the solutions to the finite dimensional control problems. Numerical simulations illustrate our theoretical findings.

翻译：我们采用基于蒙特卡罗样本的近似和有限元离散方法，对由带随机输入的线性椭圆型偏微分方程支配的风险中性最优控制问题进行离散化，从而得到有限维控制问题。我们建立了有限维问题解与风险中性问题解之间距离的指数尾界。该尾界表明，风险中性最优控制问题的解可以通过有限维控制问题的解进行可靠估计。数值模拟验证了我们的理论结果。