Advanced Computational Stochastic Dynamic Programming for Continuous Time Problems

本研究的目的是发展一般连续时间随机动力系统最优反馈控制的快速算法。这些随机动力系统包括高斯白噪声和泊松白噪声的扰动。连续时间马尔可夫噪声（包括泊松噪声）下的一般非线性随机微分方程的最优反馈控制的计算处理是本项目的一个特别独特的特点。这些算法正在多状态资源模型上进行测试，但适用于各种各样的应用。数值方法直接处理随机动态规划的偏微分方程。新的数据结构和算法，如有限元和多重网格方法，将被开发出来，以减轻“维数诅咒”对内存和计算的巨大需求。纯并行方法正在为可扩展、大规模并行处理器和海量内存的超级计算机开发。结果给出了最优反馈控制变量和以状态变量和时间为变量的期望最优性能指标。大规模科学计算对于管理大型应用的最优控制中的计算和内存需求至关重要，例如航空航天动力学，柔性结构，资源，经济学和机器人技术。先进的计算技术的实现，如并行化、向量化和最优数据结构，使解决更大维度的问题成为可能。

The objective of this research is to develop fast algorithms for the optimal feedback control of general continuous time stochastic dynamical systems. These stochastic dynamical systems include perturbations by both Gaussian and Poisson white noise. The computational treatment of the optimal feedback control of general, nonlinear, stochastic differential equations with Markov noise in continuous time, including Poisson noise, is a particularly unique feature of this project. The algorithms are being tested on multi-state resource models, but are applicable to a wide variety of applications. The numerical approach directly treats the partial differential equation of stochastic dynamic programming. New data structures and algorithms, such as finite element and multigrid methods, will be developed to alleviate both memory and computation intensive demands from the "curse of dimensionality". Purely parallel methods are being developed for scalable, massively parallel processors and massive memory supercomputers. Results give the optimal feedback control variables and the expected optimal performance index in terms of state variables and time. Large scale scientific computing is essential for managing the computational and memory demands in the optimal control of large applications, such as aerospace dynamics, flexible structures, resources, economics and robotics. The implementation of advanced computational techniques, such as parallelization, vectorization and optimal data structures, make it possible to solve problems of larger dimension.