Mathematical Sciences: Approximation Theory for Variational Problems with Applications to Random Composites and Stochastic Optimization Problems

本文研究了变分（优化）问题的近似理论及其应用。这项研究的一项主要工作将是发展一种定量理论，这种理论将更直接地适用于数值环境，并将提供一种计算误差估计的方法。特别感兴趣的是该理论在随机环境中的应用，包括求解过程的计算机实现。本项目选择了三个应用领域进行详细研究：随机优化问题、统计学中的估计理论和模拟电导率或流过随机复合材料的随机算子方程。本研究是在应用数学的一般领域与应用到统计理论。

This research is concerned with an approximation theory for variational (optimization) problems and its application. A major effort of the research will go to the development of a quantitative theory that would be more directly applicable in a numerical environment and would provide a means for calculating error estimates. Of particular interest is the application of the theory to a stochastic environment, including the computer implementation of solution procedures. Three application areas have been singled out for detailed study in this project: stochastic optimization problems, estimation theory in statistics, and random operator equations that model conductivity or flow through random composite materials. The research is in the general area of applied mathematics with applications to the theory of statistics.