Advanced Computational Stochastic Dynamic Programming for Continuous Time Problems

研究者和同事开发了大规模的最优随机控制问题的计算程序。大规模并行处理器和并行算法的进步对于管理大量计算和内存需求至关重要，特别是在地下水污染修复和制造系统中的大规模应用。先进的计算技术的实现，如并行化、图形可视化和高效的数据结构，使解决更大维度的问题成为可能。这里的重点是正式证明和有用的计算方法，而不是有限的高度严格的结果。目标是发展与大规模并行处理相结合的快速算法，用于一般连续时间非线性随机动力系统的最优反馈控制。这些随机系统包括高斯白噪声和分布泊松白噪声，因此是混合随机模型。对这些系统进行先进的计算处理，特别是用泊松噪声来模拟灾难性事件，是本研究的独特之处。计算方法在地下水污染模型和多级制造系统上进行了测试，但适用于各种应用，为计算方法提供了更高水平的鲁棒性。数值方法直接处理随机动态规划的偏微分方程。包括有限元和随机模拟在内的新算法被开发出来，以减轻“维度诅咒”带来的内存和计算密集型需求。研究人员和同事开发了地下水污染和制造问题的不确定性的最佳计算解决方案。地下水和制造业问题分别是重大和全国性挑战问题的例子。在地下水的情况下，不确定性的原因是意外引入的污染和变化的环境参数，而在制造系统的情况下，它是具有波动的制造参数的机器的故障和维修。寻求最优解的原因是使成本最小化。由于控制物理方程和基本数值近似的复杂性，大规模计算是必要的。由于在可行的情况下，清理受污染的地下水场址需要花费数十亿美元，因此最优解决方案可以比非最优方法节省数百万美元。类似的节省也可以通过管理最小化成本的准时制制造系统获得，从而增强我们的全球竞争力。先进的计算技术的发展是利用大规模并行计算机的高性能计算来获得最优解。图形化可视化对于使环境资源或制造工厂经理（他们将是主要用户）可以使用巨大的输出非常重要。

The investigator and associates develop large scalecomputational procedures for optimal stochastic control problems.Massively parallel processor and parallel algorithm advances areessential for managing massive computational and memory demands,with particular emphasis on large scale applications in groundwater pollution remediation and manufacturing systems.Implementation of advanced computational techniques, such asparallelization, graphical visualization, and efficient datastructures make possible the solution of larger dimensionalproblems. The emphasis here is on formally justified and usefulcomputational methods rather than limited highly rigorousresults. The objective is to develop fast algorithms coupledwith massively parallel processing for the optimal feedbackcontrol of general continuous-time nonlinear stochastic dynamicalsystems. These stochastic systems include both Gaussian anddistributed Poisson white noise, thus hybrid stochastic models.Advanced computational treatment of these systems, especiallywith Poisson noise for modeling disastrous events, is aparticularly unique feature of this research. The computationalmethods are tested on ground water pollution models andmultistage manufacturing systems, but are applicable to a widevariety of applications, yielding a higher level of robustnessfor the computational methods. The numerical approach directlytreats the partial differential equation of stochastic dynamicprogramming. New algorithms, including finite element and randomsimulations, are developed to alleviate both memory andcomputationally intensive demands from the "curse ofdimensionality." The investigator and associates develop optimalcomputational solutions to ground water pollution andmanufacturing problems perturbed by uncertainty. Ground waterand manufacturing problems are examples of grand and nationalchallenge problems, respectively. In the case of ground waterthe cause of uncertainty is the unexpected introduction ofpollution and varying environmental parameters, while in the caseof the manufacturing systems it is the failure and repair ofmachines with fluctuating manufacturing parameters. The reasonfor seeking optimal solutions is to minimize costs. Large scalecomputations are necessary due to the complexity of the governingphysical equations and essential numerical approximations. Sincethe cleanup of polluted ground water sites can cost billionswhere feasible, optimal solutions could save millions overnonoptimal methods. Similar savings can be obtained forjust-in-time manufacturing systems that are managed to minimizecosts, enhancing our globally competitive capabilities. Advancedcomputing techniques are developed to get optimal solutions usinghigh performance computation with massively parallel computers.Graphical visualization is important to make the immense outputusable to the environmental resource or manufacturing plantmanager who would be the prime user.