RUI: Modelling of Scattered Data on Manifolds

DMS Award AbstractAward #： 0204704 PI： Mhaskar，Hrushikesh机构： 加州州立大学洛杉矶分校课程： 应用数学项目经理：Catherine Mavriplis职务：- 我知道流形上散乱数据的建模提议者将继续研究球面和其他流形上散乱数据的建模和分析。为了分析数据，他将开发具有小型或紧凑支撑的框架，以及具有其他理想特性的双框架。提议者将研究流形上函数的最佳表示，使用多个位，而不是真实的系数，并开发基于分散数据的这些表示的算法。 在这项研究中，一个重要的新的数学工具将是正交公式与正权重的基础上的数据附近（而不是上）的流形，和局部求积公式的基础上的数据附近的部分流形。该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。问题的近似函数的领域是一个非常古老的，其中早期的工人在这方面是高斯。它自然地出现在测地线研究中，例如，环境研究，以及地球周围引力场和电磁场的变化。从卫星获取的数据最自然地被建模为球体上的函数。在数学生物学中出现了其他应用，其中肿瘤的生长由球上的函数所满足的微分方程系统建模，该系统需要在真实的时间内有效且准确地近似。最近的许多应用需要研究的功能轻微扰动的领域。例如，来自卫星的数据可以不是从地球本身的表面（其也不是完美的球形）获取，而是从表面上方的不同高度获取。类似地，在生物学应用中，最初可能被假定为球形的肿瘤在其演变之后不再保持球形。提议者将利用小波分析和度量熵的思想，研究在近球形流形上任意地点收集的数据的分析和建模。日期：二○ ○二年六月十八日

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