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。例子包括重力场建模、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是用球面谐波表示的。然而，球谐波作为全局函数的性质产生了问题。球面谐波表示依赖于球面谐波的微妙抵消/干涉，并且缓慢收敛。现有的用正交多项式或其他方法在球上的表示有同样的缺点，甚至更差。该项目的主要目标是基于新创建的小波类型系统（称为“针尖”），在球体和球上开发创新的多尺度数据表示。球体上的针尖系统由几乎指数局域化的径向带限函数组成，这些函数可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引力势等调和函数提供了一个非常有效的框架。针尖与球谐波的兼容性允许在针尖和球谐波表示之间快速转换。这使得它们很容易集成到基于球面谐波的现有模型中。此外，针在精细尺度上的出色定位使得基于针的模型非常适合于有效的局部更新，这是基于球面谐波的传统模型的显着优势。球上的针尖系统具有类似的结构，由几乎指数局域代数多项式组成。理论结果表明，该方法优于现有的单尺度和多尺度方法。提出的研究的一个重要组成部分是采用非线性逼近方法来有效地表示和逼近来自针的函数。这些是多层技术，允许控制近似误差的均匀（或其他）范数。另一个进步将是开发球体和球的各向异性元素（例如曲率），以便更好地提取数据的曲线特征。本研究的目标应用主要集中在大地测量学领域。大多数大地测量应用依赖于精确计算引力（扰动）势的能力。本项目将采用多尺度针形表示方法对重力势进行建模。新表示的其他潜在应用是在地磁、日震、天文学、宇宙学、地震学，其中球谐表示被广泛使用。该项目的一个重要目标是促进新表示在其他不同学科（从地球物理学到高速视频内窥镜）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大学的研究生和本科生提供了一个绝佳的机会，让他们参与到进一步发展的想法测试中来。

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.