论文标题
抛物线PDE在不确定性下使用Quasi-Monte Carlo Integration在不确定性下受到熵风险度量的最佳控制
Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration
论文作者
论文摘要
我们研究了在不确定性下，量身定制的准蒙特卡洛（QMC）方法在一类受抛物线偏微分方程（PDE）约束的最佳控制问题上的应用：我们设置的状态是用随机的热扩散系数的抛物线PDE解决方案，由对照函数进行操作。为了说明最佳控制问题中不确定性的存在，目标函数由风险度量组成。我们专注于两项风险度量，均涉及随机变量上的高维积分：期望值和（非线性）熵风险度量。高维积分是使用特殊设计的QMC方法进行数值计算的，并且在输入随机字段的中等假设下，错误率显示出本质上是线性的，与问题的随机维度无关，从而优于普通的蒙特卡洛方法。数值结果证明了我们方法的有效性。
We study the application of a tailored quasi-Monte Carlo (QMC) method to a class of optimal control problems subject to parabolic partial differential equation (PDE) constraints under uncertainty: the state in our setting is the solution of a parabolic PDE with a random thermal diffusion coefficient, steered by a control function. To account for the presence of uncertainty in the optimal control problem, the objective function is composed with a risk measure. We focus on two risk measures, both involving high-dimensional integrals over the stochastic variables: the expected value and the (nonlinear) entropic risk measure. The high-dimensional integrals are computed numerically using specially designed QMC methods and, under moderate assumptions on the input random field, the error rate is shown to be essentially linear, independently of the stochastic dimension of the problem -- and thereby superior to ordinary Monte Carlo methods. Numerical results demonstrate the effectiveness of our method.
