Stochastic Variational Problems: Approximation and Modelization Issues

本研究计划以随机规划中的近似问题为中心，特别是当它们出现在两个相当具有挑战性的问题中：地下水修复和不确定环境中的平衡问题。其中第一个需要处理随机程序，其中的“追索权”是由一个偏微分方程组决定的。“随机”均衡问题提出了一个新的难度水平，而不仅仅是优化，一个人必须找到一个机制来确定一个价格体系，在这个价格体系下，优化发生了。这种平衡是依靠不动点定理推导出来的。最后，由于只能解决随机优化问题的离散化版本，因此彻底研究近似问题至关重要。不仅是逼近描述不确定性的随机过程的问题，而且是如何从现有数据中改进该过程的构造以及分析这将对随机程序的解产生的影响。随机规划领域为不确定条件下的决策提供了求解和分析模型的数学工具。该项目将涉及两个重要而困难的应用和近似问题：-地下水修复问题，选择该问题是因为它需要理论和计算的发展。这是一个随机优化问题，通过求解一个系数为快速振荡（异质介质）和随机（介质组成的不确定性）的偏微分方程来获得系统的状态。我们还将探讨导出这个问题的非均匀化版本的可能性。——不确定环境下的瓦尔拉斯平衡问题。选择这个问题是因为它为随机优化增加了一个维度，因为人们还必须找到价格系统（建立一个“均衡”），在这个系统下，这种随机优化必须发生。——随机规划中的近似问题。提出了动态随机规划问题参数的可靠估计问题。人们期望采用一种更全面的方法，利用所有可用的信息，而不仅仅是收集到的数据，将为随机规划问题带来更可靠的解决方案。

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.