Mathematical Sciences: "Statistical Computation and Computational Geometry"

开发的几何工具将提供 计算统计理论的重要发展， 方法论研究方向是开发和研究 凸概率容度的概率界 通过“抽象管”生成具有特殊结构的多面体 法“考虑到这种几何结构， 在某些情况下，明确和容易计算的误差范围 概率，并在其他情况下，以更高的效率， 误差概率的蒙特-卡罗评估。本研究 它是计算统计学和 计算几何，将产生一个扩展的集合， 构建实用而精确的概率的工具 广阔的边界 一系列应用程序。尤其重点 为了获得新的 的概率内容的界 在更高维度的身体使用的方法， 具有相似项的低维问题的研究 结构 有大量的真实的应用程序， 随机模型用于基于 概率计算。比如说， 概率 计算用于确定模型化的 系统会失败。 这适用于桥梁、 生态系统、人体或制造方法。 的 研究探索了特殊模型结构的方法， 这样的情况可以被利用来生产更多的 高效精确的概率计算 这个新 方法应直接适用于制造业 质量控制，结构和系统可靠性， 环境保护，以及其他一些领域。

Geometric tools are developed that will provide key developments in computational statistical theory and methodology. The research direction is to develop and study probability bounds for the probability content of convex polyhedra with special structure via the "abstract tube method." Accounting for this geometric structure can lead, in some cases, to explicit and easy to calculate bounds for error probabilities, and in other cases to greater efficiency in the Monte-Carlo evaluation of error probabilities. This research which is at the interface between computational statistics and computational geometry, will produce an expanded collection of tools for constructing practical and sharp probability bounds for a vast array of applications. Particular focus is given to obtaining new bounds for the probability content of bodies in higher dimensions using methods that arise from the study of lower dimensional problems with similar structure. There is an abundance of real world applications in which stochastic models are utilized to make critical decisions based on probability calculations. For example, probability calculations are used to determine how likely a modeled system is to fail. This applies to systems such as a bridge, an ecosystem, a human body, or a manufacturing method. The research explores ways in which special model structure for situations such as these can be exploited to produce more efficient and precise probability calculations. This new methodology should be directly applicable to manufacturing and quality control, structural and system reliability, environmental regulation, and a host of other areas.