RUI: Modelling of Scattered Data on Manifolds

DMS奖摘要奖#：0204704PI：Mhaskar，Hrushikes机构：加州州立大学洛杉矶分校计划：应用数学计划经理：Catherine Mavriplis标题：RUI：流形上散乱数据的建模提出者将继续他对球面和其他流形上散乱数据的建模和分析的研究。为了分析数据，他将开发具有小支撑或紧凑支撑的框架，并具有具有其他所需特性的双框架。提出者将研究在流形上使用有限多比特而不是实数系数的函数的最佳表示，并开发基于散乱数据的这些表示的算法。这项研究中的一个重要的新数学工具将是基于流形附近(而不是基于)数据的具有正权重的求积公式，以及基于流形部分附近数据的局部求积公式。这项研究有望在信号处理、卫星数据分析和肿瘤发展研究等领域得到应用。球面上函数的逼近问题是一个非常古老的问题；这一领域的早期工作者之一是高斯。它自然地出现在测地线研究中，例如，研究环境以及地球周围重力场和电磁场的变化。从卫星上获取的数据最自然地被建模为球面上的函数。更多的应用出现在数学生物学中，其中肿瘤的生长是由球面上的函数满足的微分方程组来模拟的，这些微分方程组需要高效而准确地实时逼近。最近的许多应用都要求研究球面上的微小扰动函数。例如，来自卫星的数据可能不是来自地球本身的表面(地球本身也不是完全球形的)，而是来自地表以上的不同高度。同样，在生物学应用中，最初被认为是球形的肿瘤在进化后不再是球形的。作者将利用小波分析和度量熵的思想，研究在近球形流形上的任意位置收集的数据的分析和建模。日期：2002年6月18日

DMS Award AbstractAward #: 0204704PI: Mhaskar, HrushikeshInstitution: California State University Los AngelesProgram: Applied MathematicsProgram Manager: Catherine MavriplisTitle: RUI: Modelling of Scattered Data on ManifoldsThe proposer will continue his research on the modelling and analysis of scattered data on the sphere and other manifolds. For the analysis of data, he will develop frames having small or compact supports, and having dual frames with other desirable properties. The proposer will study the optimal representation of functions on the manifolds using finitely many bits, rather than real coefficients, and develop algorithms for these representations based on scattered data. An important new mathematical tool in this study will be quadrature formulas with positive weights based on data near (rather than on) the manifolds, and local quadrature formulas based on data near parts of the manifolds. The research is expected to have applications in the areas of signal processing, analysis of satellite data, and the study of development of tumors.The problem of approximation of functions on the sphere is a very old one; among the early workers in this area was Gauss. It arises naturally in geodesic studies, for example, the study of the environment, and variations in the gravitational and electromagnetic fields around the earth. The data taken from a satellite is most naturally modelled as a function on the sphere. Additional applications arise in mathematical biology, where the growth of a tumor is modelled by a system of differential equations satisfied by functions on a sphere, which need to be approximated efficiently and accurately in real time. Many recent applications require a study of functions on slight perturbations of the sphere. For example, the data from the satellite may be taken not from the surface of the earth itself (which is not perfectly spherical either), but from different heights above the surface. Similarly, in biological applications, a tumor which may be initially assumed spherical, no longer remains so after its evolution. The proposer will study the analysis and modelling of data collected at arbitrary sites on a nearly spherical manifold, using ideas from wavelet analysis and metric entropy. Date: June 18, 2002