Computational Methods for Multivariate Orthogonal Polynomials

许多计算数学依赖于构建基于计算机的模型来预测物理特性，例如电化学电池的预期电容或风力涡轮机的功率输出。构建这样一个计算机模型的一种计算稳定的方法是建立新的数学函数。本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，并随后设计有效的算法来使用这些多项式来构建函数。该项目的目标是为分析和应用近似理论的相关数学领域提供见解，并帮助计算科学家建立稳健的模拟模型。由正交多项式展开建立的近似是应用数学中的经典工具。这些近似通常是计算求解微分和积分方程的算法的基础。在过去的几十年里，在一个变量中生成和操作这样的展开一直是大量理论和计算研究的主题，并且对于大多数感兴趣的问题都有建设性的算法。在多元非张量的情况下，人们所知甚少，因为只有少数限制性的计算工具用于生成多元正交多项式。本项目的研究重点是在非张量域上用非张量权重生成多元正交多项式的理论发展和计算实现方法。本项目的数学研究涉及对多元正交多项式理论的基础贡献，以及设计生成多元正交多项式的鲁棒算法。从实际的角度来看，本项目中的算法和方法在正交系列中产生扩展，并将在各种工程设计，优化和可靠性环境中发挥作用。

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.