论文标题
由随机PDES约束的最佳控制问题的组合技术
A combination technique for optimal control problems constrained by random PDEs
论文作者
论文摘要
我们为随机变量的空间近似值和正交公式的混合差异提供了一种组合技术，以有效地求解一类受随机部分微分方程约束的最佳控制问题（OCP）。该方法需要为目标功能求解OCP的几个低保真空间网格和正交公式。然后将所有计算的解决方案线性合并，以获得最终近似值，在适当的规律性假设下，该近似值可保持精细张量产品近似的准确性，同时大大降低了计算成本。该组合技术仅涉及张量产品正交公式，因此离散的OCPS保留了连续OCP的（可能）凸度。因此，该组合技术避免了多级蒙特卡洛和/或稀疏网格接近的不便，但仍然适合于高维问题。手稿提出了一种A-Priori程序，可以选择最重要的混合差异和渐近复杂性分析，该分析指出，渐近复杂性仅由空间求解器确定。数值实验验证结果。
We present a combination technique based on mixed differences of both spatial approximations and quadrature formulae for the stochastic variables to solve efficiently a class of Optimal Control Problems (OCPs) constrained by random partial differential equations. The method requires to solve the OCP for several low-fidelity spatial grids and quadrature formulae for the objective functional. All the computed solutions are then linearly combined to get a final approximation which, under suitable regularity assumptions, preserves the same accuracy of fine tensor product approximations, while drastically reducing the computational cost. The combination technique involves only tensor product quadrature formulae, thus the discretized OCPs preserve the (possible) convexity of the continuous OCP. Hence, the combination technique avoids the inconveniences of Multilevel Monte Carlo and/or sparse grids approaches, but remains suitable for high dimensional problems. The manuscript presents an a-priori procedure to choose the most important mixed differences and an asymptotic complexity analysis, which states that the asymptotic complexity is exclusively determined by the spatial solver. Numerical experiments validate the results.
