论文标题
Mg/opt和MLMC用于强大的PDE
MG/OPT and MLMC for Robust Optimization of PDEs
论文作者
论文摘要
提出了一种算法来解决基于所谓的MG/OPT框架，该算法由具有不确定系数的部分微分方程限制的强大控制问题。像往常一样，此mg/opt层次结构中的级别对应于PDE的离散水平。对于随机问题，相关数量（例如梯度）都包含这些级别的预期价值运算符。它们是使用多级蒙特卡洛法估计的，其细节取决于mg/opt水平。然后，每个优化水平都包含多个基础多级蒙特卡洛水平。 MG/OPT层次结构允许该算法利用PDE中固有的结构，从而加快收敛性到达最佳。相比之下，存在多级蒙特卡洛层次结构，以利用问题的随机维度中存在的结构。证明了有关该算法收敛速率的声明，并讨论了一些其他属性。在三个测试用例中，对算法的性能进行了数值研究。可以观察到昂贵水平上所需的样品数量的减少，因此可以观察到计算时间。
An algorithm is proposed to solve robust control problems constrained by partial differential equations with uncertain coefficients, based on the so-called MG/OPT framework. The levels in this MG/OPT hierarchy correspond to discretization levels of the PDE, as usual. For stochastic problems, the relevant quantities (such as the gradient) contain expected value operators on each of these levels. They are estimated using a multilevel Monte Carlo method, the specifics of which depend on the MG/OPT level. Each of the optimization levels then contains multiple underlying multilevel Monte Carlo levels. The MG/OPT hierarchy allows the algorithm to exploit the structure inherent in the PDE, speeding up the convergence to the optimum. In contrast, the multilevel Monte Carlo hierarchy exists to exploit structure present in the stochastic dimensions of the problem. A statement about the rate of convergence of the algorithm is proven, and some additional properties are discussed. The performance of the algorithm is numerically investigated for three test cases. A reduction in the number of samples required on expensive levels and therefore in computational time can be observed.
