Highly effective representations for surface and solid spherical studies

表面和固体球形研究的高效表示

• 批准号: 0709046
• 负责人: Pencho Petrushev
• 金额: $ 14.39万
• 依托单位: University South Carolina Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-15 至 2010-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0709046&HistoricalAwards=false
• 关键词: Highly; effective; representations; surface; solid

Various practical applications require effective representation of functions on the sphere or on the ball. Examples include the gravitational field modeling, geomagnetism, helioseismology, astronomy, cosmology, and seismology. In most applications the functions on the sphere have been traditionally represented in terms of spherical harmonics. The nature of spherical harmonics as global functions, however, creates problems. The spherical harmonic representations rely on delicate cancellation/interference of spherical harmonics and are slowly convergent. The existing representations on the ball using orthogonal polynomials or other methods have the same drawbacks or are even worse. The primary objective of this project is to develop innovative multiscale data representations on the sphere and on the ball based on newly created wavelet type systems, called "needlets". The needlet system on the sphere consists of almost exponentially localized radial band-limited functions, which are automatically extendable to harmonic functions in the exterior of the sphere and thereby enabling needlet representations to provide a highly effective framework for representation and analysis of harmonic functions such as the gravitational potential. The needlet compatibility with spherical harmonics permits for fast conversions between needlet and spherical harmonic representations. This makes them easy to integrate into the existing models based on spherical harmonics. Furthermore, the superb localization of needlets at fine scales makes needlet-based models highly amenable to efficient local updates, which is a significant advantage over traditional models based on spherical harmonics. The needlet system on the ball has a similar structure and consists of almost exponentially localized algebraic polynomials. Theoretical results show that the new representations are superior to the existing mono- and multiscale methods used in these areas. An important element of the proposed research is the employment of nonlinear approximation methods for effective representation and approximation of functions from needlets. These are multilevel techniques which allow control of the uniform (or other) norm of the error of approximation. Another step forward will be the development of anisotropic elements on the sphere and ball (e.g. curvlets) for better extraction of curvlinear features of the data.The targeted applications of this research are mainly in the domain of geodesy. Most geodetic applications rely on the ability to compute accurately the gravitational (disturbing) potential. This project will pursue the implementation of multiscale needlet representations for modeling of the gravitational potential. Other potential applications of the new representations are in geomagnetism, helioseismology, astronomy, cosmology, seismology, where spherical harmonic representations are widely used. An important goal of this project is to promote the broad utilization of the new representations in other diverse disciplines (from geophysics to high-speed videoendoscopy) and to stimulate interest in younger mathematicians to this area. This research project offers an excellent opportunity for graduate and undergraduate students at the University of South Carolina to participate in testing ideas for further development.

各种实际应用需要球上或球上的函数的有效表示。例如重力场模拟、地磁学、日震学、天文学、宇宙学和地震学。在大多数应用中，球面上的函数传统上用球面调和来表示。然而，球谐函数作为全局函数的性质会产生问题。球谐表示依赖于微妙的球谐抵消/干涉，并且是缓慢收敛的。现有的用正交多项式或其他方法表示球的方法也有同样的缺点，甚至更差。该项目的主要目标是开发基于新创建的小波型系统的球面和球面上的创新的多尺度数据表示法，称为“针”。球上的针尖系统由几乎指数局域的径向带限函数组成，这些函数可以自动扩展到球体外部的调和函数，从而使针尖表示为表示和分析调和函数(如引力势)提供了一个高效的框架。针尖与球谐的兼容性允许在针尖和球谐表示之间进行快速转换。这使得它们很容易集成到现有的基于球谐函数的模型中。此外，针尖在细微尺度上的高度局部化使得基于针尖的模型高度易于进行有效的局部更新，这是基于球谐函数的传统模型的显著优势。球上的小针系统具有类似的结构，并由几乎指数局部化的代数多项式组成。理论结果表明，新的表示法优于现有的单尺度和多尺度方法。所提出的研究的一个重要元素是使用非线性近似方法来有效地表示和逼近来自针尖的函数。这些是允许控制近似误差的统一(或其他)范数的多级技术。另一步是发展球面和球面上的各向异性元素(例如曲线片)，以更好地提取数据的曲率特征。这项研究的目标应用主要是在大地测量领域。大多数大地测量应用依赖于精确计算引力(扰动)势的能力。这个项目将致力于实现多尺度针状表示法来模拟引力势。新表示法的其他潜在应用还包括地磁学、日震学、天文学、宇宙学、地震学，其中球谐表示法得到了广泛的应用。这个项目的一个重要目标是促进在其他不同学科(从地球物理到高速视频内窥镜)中广泛使用新的表示法，并激发对这一领域的年轻数学家的兴趣。这一研究项目为南卡罗来纳大学的研究生和本科生提供了一个极好的机会，让他们参与测试进一步发展的想法。