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

本研究涉及变分（优化）问题的近似理论及其应用。 该研究的主要工作将致力于开发一种定量理论，该理论将更直接地适用于数值环境，并提供计算误差估计的方法。 特别令人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。 该项目挑选了三个应用领域进行详细研究：随机优化问题、统计估计理论以及模拟电导率或流经随机复合材料的随机算子方程。 该研究属于应用数学的一般领域，应用于统计理论。

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.