Mathematical Sciences: "Statistical Computation and Computational Geometry"

几何工具的开发将提供计算统计理论和方法的关键发展。研究方向是利用“抽象管法”发展和研究具有特殊结构的凸多面体的概率内容的概率界。考虑到这种几何结构在某些情况下可以导致错误概率的显式且易于计算的界限，并且在其他情况下可以在错误概率的蒙特卡罗评估中获得更高的效率。这项处于计算统计和计算几何之间的研究，将产生一个扩展的工具集合，用于为大量的应用构建实用和精确的概率界限。特别侧重于利用研究具有相似结构的低维问题所产生的方法来获得高维物体的概率含量的新界限。在现实世界中，有大量的应用，其中随机模型被用来基于概率计算来做出关键决策。例如，概率计算用于确定建模系统发生故障的可能性。这适用于桥梁、生态系统、人体或制造方法等系统。这项研究探索了如何利用针对此类情况的特殊模型结构来产生更有效和更精确的概率计算。这种新的方法应该直接适用于制造和质量控制、结构和系统可靠性、环境监管和许多其他领域。

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.