Computational Methods for Multivariate Orthogonal Polynomials

多元正交多项式的计算方法

• 批准号: 1720416
• 负责人: Akil Narayan
• 金额: $ 10万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-08-01 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1720416&HistoricalAwards=false
• 关键词: Computational; Methods; Multivariate; Orthogonal; Polynomials

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.

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。构建此类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逼近理论的相关数学领域，并帮助计算科学家构建稳健的仿真模型。通过正交多项式展开构建的逼近是应用数学中的经典工具。这些近似通常是计算求解微分方程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纵这种展开式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。对于多元非张量情况知之甚少，对于这种情况，只有少数限制性计算工具用于生成多元正交多项式。该项目的研究重点是在具有非张量权重的非张量域上生成多元正交多项式的方法的理论发展和计算实现。该项目的数学研究涉及对多元正交多项式理论的基础贡献，以及用于生成多元正交多项式的鲁棒算法的设计。从实践的角度来看，该项目中的算法和方法产生了正交级数的扩展，并将在各种工程设计、优化和可靠性环境中发挥作用。

项目成果

Polynomial chaos expansions for dependent random variables

因随机变量的多项式混沌展开

• DOI: 10.1016/j.cma.2019.03.049
• 发表时间: 2019-03-22
• 期刊: Computer Methods in Applied Mechanics and Engineering
• 影响因子: 7.2
• 作者: J. Jakeman;F. Franzelin;A. Narayan;M. Eldred;Dirk Plfueger
• 通讯作者: Dirk Plfueger

On the Computation of Recurrence Coefficients for Univariate Orthogonal Polynomials

关于单变量正交多项式递推系数的计算

• DOI: 10.1007/s10915-021-01586-w
• 发表时间: 2021-09
• 期刊: Journal of Scientific Computing
• 影响因子: 2.5
• 作者: Liu, Zexin;Narayan, Akil
• 通讯作者: Narayan, Akil

Generation of nested quadrature rules for generic weight functions via numerical optimization: Application to sparse grids

通过数值优化生成通用权重函数的嵌套求积规则：在稀疏网格中的应用

• DOI: 10.1016/j.jcp.2019.108979
• 发表时间: 2020-01
• 期刊: Journal of Computational Physics
• 影响因子: 4.1
• 作者: Keshavarzzadeh, Vahid;Kirby, Robert M.;Narayan, Akil
• 通讯作者: Narayan, Akil

Numerical Integration in Multiple Dimensions with Designed Quadrature

设计求积的多维数值积分

• DOI: 10.1137/17m1137875
• 发表时间: 2018-04-17
• 期刊: SIAM J. Sci. Comput.
• 作者: Vahid Keshavarzzadeh;R. Kirby;A. Narayan
• 通讯作者: A. Narayan

Computation of induced orthogonal polynomial distributions

诱导正交多项式分布的计算

• DOI: 10.1553/etna_vol50s71
• 发表时间: 2018-01
• 期刊: ETNA - Electronic Transactions on Numerical Analysis
• 作者: Narayan; Akil
• 通讯作者: Akil