Computational Methods for Multivariate Orthogonal Polynomials

大部分计算数学都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。构建这样的计算机模型的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构建函数的有效算法。该项目的目标是深入了解分析和应用近似理论的相关数学领域，并帮助计算科学家建立强大的仿真模型。从正交多项式展开建立的近似是应用数学中的经典工具。这些近似经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，这种扩展的生成和操作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，已知的要少得多，只有几个限制性的计算工具来生成多元正交多项式。本项目的研究重点是非张量域上具有非张量权的多元正交多项式的生成方法的理论发展和计算实现。本计画的数学研究涉及多元正交多项式理论的基础性贡献，以及多元正交多项式产生的强健演算法的设计。从实用的角度来看，在这个项目中的算法和方法产生的正交系列的扩展，将是有用的各种工程设计，优化和可靠性的情况下。

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.