Highly effective representations for surface and solid spherical studies

各种实际应用需要在球体或球上有效地表示函数。例子包括重力场模拟、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谐函数表示。然而，球谐函数作为全局函数的性质产生了问题。球谐表示依赖于球谐的微妙抵消/干涉，并且缓慢收敛。使用正交多项式或其他方法的球上的现有表示具有相同的缺点或甚至更糟。这个项目的主要目标是开发创新的多尺度数据表示的领域和球的基础上新创建的小波类型的系统，称为“针”。球上的针状系统由几乎指数局部化的径向带限函数组成，这些函数可以自动扩展到球外部的调和函数，从而使针状表示提供了一个高效的框架，用于表示和分析调和函数，如引力势。针状体与球谐函数的兼容性允许针状体和球谐函数表示之间的快速转换。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，在精细尺度下的针状物的极好的定位使得基于针状物的模型非常适合于有效的局部更新，这是基于球谐函数的传统模型的显著优势。球上的Needlet系统具有类似的结构，并且由几乎指数局部化的代数多项式组成。理论结果表明，新的表示是上级优于现有的单尺度和多尺度方法在这些领域中使用。拟议的研究的一个重要组成部分是就业的非线性逼近方法的有效表示和近似的功能从针状。这些是多级技术，允许控制近似误差的均匀（或其他）范数。另一个进步将是在球体和球体上开发各向异性元素（例如曲率），以更好地提取数据的曲线特征。这项研究的目标应用主要是大地测量领域。大多数大地测量应用依赖于精确计算重力（扰动）位的能力。本计画将追求多尺度针状表示法的实作，以模拟重力位。新表象的其他潜在应用是地磁学、日震学、天文学、宇宙学、地震学，其中球谐表象被广泛使用。该项目的一个重要目标是促进新的表示法在其他不同学科（从电子物理学到高速视频内窥镜）中的广泛应用，并激发年轻数学家对这一领域的兴趣。这个研究项目为南卡罗来纳州大学的研究生和本科生提供了一个很好的机会，让他们参与测试进一步发展的想法。

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.