Computational Methods for Multivariate Orthogonal Polynomials

计算数学在很大程度上依赖于构建基于计算机的模型，这些模型预测物理特性，如电化学电池的预期电容或风力涡轮机的功率输出。构建这种计算机模型的一种计算稳定的方法是构建新的数学函数。该项目旨在为生成多项式提供严格的数学基础和强大的计算工具，并随后设计出使用这些多项式来构建函数的有效算法。这个项目的目标是深入了解分析和应用逼近理论的相关数学领域，并帮助计算科学家建立稳健的模拟模型。从正交多项式的展开建立逼近是应用数学中的经典工具。这些近似常常是计算求解微分方程式和积分方程式的算法的基础。在过去的几十年里，在一个变量中这种展开式的产生和处理一直是大量理论和计算研究的主题，对于大多数感兴趣的问题都有建设性的算法。在多变量非张量情况下所知的要少得多，对于这种情况，只有几个限制性的计算工具来生成多变量正交多项式。本项目的研究重点是在非张量区域上生成具有非张量权重的多元正交多项式的方法的理论发展和计算实现。这个项目的数学研究包括对多元正交多项式理论的基本贡献，以及产生多元正交多项式的稳健算法的设计。从实用的角度来看，本项目中的算法和方法以正交级数形式展开，将在各种工程设计、优化和可靠性方面发挥作用。

Much of computational mathematics relies on the task of constructing computer-based models that predict a physical property, such as expected capacitance of an eletrochemical battery or power output of a wind turbine. A computationally stable way to construct such a computer model is to build new mathematical functions. This project aims to provide rigorous mathematical foundations and robust computational tools for generating polynomials and subsequently devise efficient algorithms for using these polynomials to build functions. The objectives of this project provide insight into the related mathematical fields of analysis and applied approximation theory, and aid computational scientists in building robust simulation models.Approximations built from an expansion in orthogonal polynomials are classical tools in applied mathematics. These approximations are frequently the bedrock of algorithms for computationally solving differential and integral equations. The generation and manipulation of such expansions in one variable has been the subject of a great deal of theoretical and computational research in past decades, and there are constructive algorithms for most problems of interest. Far less is known in the multivariate non-tensorial case, for which there are only a few restrictive computational tools for generation of multivariate orthogonal polynomials. The research of this project focuses on theoretical development and computational implementation of methods for generating multivariate orthogonal polynomials on non-tensorial domains with non-tensorial weights. Mathematical investigations of this project involve fundamental contributions to the theory of multivariate orthogonal polynomials, and design of robust algorithms for generation of multivariate orthogonal polynomials. From a practical standpoint the algorithms and methodologies in this project produce expansions in an orthogonal series and will be useful in various engineering design, optimization, and reliability contexts.