Stochastic Variational Problems: Approximation and Modelization Issues

这项研究的建议是围绕近似问题，特别是当它们出现在两个典型的问题：地下水修复和不确定环境中的平衡问题时。其中第一个要求处理随机程序，其中“追索权”是由偏微分方程组确定的。“随机”均衡问题增加了新的难度，而不仅仅是优化，人们必须找到一种机制来确定进行优化的价格系统。这种平衡是通过依赖不动点定理得到的。最后，由于只能解决离散化形式的随机优化问题，所以深入研究逼近问题是至关重要的。不仅是描述不确定性的随机过程的近似问题，而且还包括如何根据现有数据改进这个过程的结构，并分析这将对随机规划解产生的影响。随机规划领域为不确定情况下的决策提供了数学工具。本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，因为它需要理论和计算两方面的发展。这是一个弹性优化问题，其中系统的状态是通过求解一个偏微分方程来获得的，该偏微分方程的系数是快速振荡(非均匀介质)和随机的(关于介质组成的不确定性)。还将研究推导出该问题的非均质版本的可能性。不确定环境中的Walras均衡问题。之所以选择这个问题，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，随机优化必须发生。--随机规划中的逼近问题。提出了动态随机规划问题参数的可靠估计问题。人们期望，一种更全面的方法，利用所有可用的信息，而不仅仅是收集的数据，将导致随机规划问题的更可靠的解决方案。

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.