Stochastic Variational Problems: Approximation and Modelization Issues

这项研究建议是围绕近似问题随机规划，特别是因为他们出现在两个quitechallenging问题：地下水修复和不确定环境中的平衡问题。 其中第一个要求处理随机程序的“追索权”是由一个系统ofpartial微分方程。“随机”均衡问题增加了一个新的难度，而不仅仅是优化一个必须找到一个机制，以确定一个价格体系下的优化发生。这样的平衡是由不动点定理导出的。 最后，由于随机最优化问题只能离散化求解，因此深入研究近似问题是非常重要的。本文不仅讨论描述不确定性的随机过程的近似问题，而且讨论如何根据现有数据改进随机过程的构造，并分析其对随机规划解的影响。随机规划领域提供了求解和分析不确定性决策模型的数学工具。本项目将涉及两个重要而困难的应用，即随机规划和随机规划。随机规划是随机规划的一个重要分支，它是随机规划的一个近似问题：--选择地下水修复中的一个问题，因为它需要理论和计算的发展。这是一个随机优化问题，系统的状态是通过求解一个偏微分方程获得的，该方程的系数是快速振荡的（非均匀介质）和随机的（介质成分的不确定性）。导出这个问题的非均匀化版本的可能性也将被调查。不确定环境下的Walras均衡问题这个问题被选中，因为它增加了一个维度，随机优化，其中一个还必须找到价格系统（建立一个“均衡”），在这个随机优化必须发生。 --随机规划中的近似问题。提出了动态随机规划问题的参数可靠估计问题。人们期望，一个更全面的方法，利用所有可用的信息，而不仅仅是收集的数据将导致更可靠的解决方案forstochastic规划问题。

This research proposal is centered around approximation issues instochastic programming, in particular as they arise in two quitechallenging problems: groundwater remediation and equilibria problems inan uncertain environment. The first of these requires dealing withstochastic programs where the `recourse' is determined by a system ofpartial differential equations. The `stochastic' equilibrium problemadds a new level of difficulty, rather than just optimizing one mustfind a mechanism to determine a price system under which the optimizationtakes place. Such equilibria have been derived by relying on fixed pointtheorems. Finally, because it's only possible to solve discretizedversions of stochastic optimization problems, it's of paramountimportance to investigate thoroughly approximation issues. Not only thequestion of approximating the stochastic process that describes theuncertainty but also how to improve the construction of this processfrom the available data and to analyze the effect this will have on the solution of the stochastic program.The field of stochastic programming provides mathematical tools forsolving and analyzing models for decision making under uncertainty.This project will be concerned with two significant and difficultapplications and with approximation issues: -- A problem in groundwater remediation which was selected becauseit requires both theoretical and computational developments. It's astochastic optimization problem where the state of the system isobtained by solving a partial differential equation whose coefficientsare rapidly oscillating (heterogeneous media) and stochastic (uncertaintyabout the media composition). The possibility of deriving anhomogenized version of this problem will also be investigated. -- Walras equilibrium problem in an uncertain environment. Thisproblem is selected because it adds a dimension to stochasticoptimization in that one must also find price systems (setting up an`equilibrium') under which this stochastic optimization must take place. -- Approximation issues in stochastic programming. The question ofhaving a reliable estimate for the parameters of a dynamic stochasticprogramming problem is raised. It is expected that a more comprehensive,approach which makes use of all the information available, rather thanjust the collected data will result in more reliable solutions forstochastic programming problems.