Mathematical Sciences: Advanced Computational Stochastic Dynamic Programming for Continuous Time Problems

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Hanson 9626692 The investigator and associates develop advanced computational methods to solve large scale dynamic programming problems in uncertain environments. These methods include parallel computation, graphical visualization, problem mapping, memory management and data restructuring on massively parallel and massive memory processors to solve large dimensional problems that could not be solved by ordinary methods. The objective is develop fast algorithms for the optimal feedback control of large, general, continuous time, nonlinear, stochastic dynamical systems, perturbed by both Gaussian and distributed Poisson white noise. The treatment of the random jumps of Poisson processes for modeling rare, disastrous events is a special and challenging feature of this research. The main application of this research is to optimal remediation for groundwater quality in an uncertain environment. Other applications are to parameter uncertainty in partially observed control systems, i.e., the dual control problem, and to just-in-time manufacturing systems. The numerical approach of the investigators directly treats the partial differential equation of stochastic dynamic programming. New algorithms, such as spatial and state finite elements, are used to alleviate both memory and computational demands. Purely parallel algorithms based on intelligent state space search are also being developed. The development of graphical visualization and data management tools are essential for analyzing massive amounts of output in many dimensions. The investigator and associates are developing optimal solutions to environmental and manufacturing problems when there is some uncertainty involved. Examples of the cause of uncertainty in the problem are the unexpected introduction of pollution and unknown parameters in the case of the environment, while in the case of manufacturing an example would be the random failure of machines. The reason for seeking optimal solutio ns is to minimize costs or wasting precious materials or resources. A large and important application that the investigator studies is the cleanup of groundwater resources where the source of the pollution or the underground state is uncertain. Since polluted groundwater sites can costs from millions to billions and more (there are some sites for which cleanup procedures are unknown), optimal solutions could save millions over existing sub-optimal procedures. Similar savings can be obtained for just-in-time manufacturing systems that are optimally managed, enhancing our globally competitive capabilities. The investigator and associates develop advanced computing techniques to compute the optimal solutions using high performance computing, which is essential when uncertain environments or parameters make the amount of computation for the solution very large. The high performance computing involves using massive computers for fast optimal solutions and graphical visualization to make the immense amount of output understandable, for instance to the environmental resource or manufacturing plant manager who would be the prime user.