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Numerical harmonic analysis and its applications to image analysis and computational mathematics

数值调和分析及其在图像分析和计算数学中的应用

基本信息

批准号：
16340041
负责人：
OKADA Masami
金额：
$ 3.97万
依托单位：
TOKYO METROPOLITAN UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

项目摘要

Outline :We have constructed new modified spline functions in multi-dimensions and have applied them to numerical analysis of nonlinear partial differential equations and image analysis. Also, we invited from abroad totally 10 active researchers in three years to organize international meetings. The details are summarized as follows.1.Computational harmonic analysis :We succeeded in construction of basis functions with "minimal" supports which enables rigorous interpolation and good sampling approximation of functions. Basis functions defined in more general domains are also investigated, i.e. in the case of finite intervals, several typical domains in the plane, e.t.c. To our surprise, almost radial basis functions have been obtained in several dimensions.2.Application lo numerical analysis of PDE :As an application, we have found a new method of computiong a series of finite difference schemes of higher orders to partial differential oparators. Also, we computed stable numerical solution to the important KP equation whose analytical solutions are difficult to find. The study of multi-scale phenomena and image analysis is ongoing.3.Harmonic analysis of complex singularity :In a joint work with oversea researchers, we developed the theory of harmonic analysis on a complex variety with singularities. This work will hopefully inspire us in future works related to singularities.

我们构造了新的多维修正样条函数，并将其应用于非线性偏微分方程的数值分析和图像分析。同时，我们在三年内共邀请了10位国外活跃的研究人员组织国际会议。详细情况总结如下。计算调和分析：我们成功地构造了具有“最小”支持的基函数，这使得函数的严格插值和良好的抽样近似成为可能。我们还研究了在更一般的域上定义的基函数，即在有限区间的情况下，在平面上的几个典型域，等等。令我们惊讶的是，在几个维度上几乎得到了径向基函数。在偏微分算子数值分析中的应用：作为应用，我们找到了一种计算偏微分算子一系列高阶有限差分格式的新方法。此外，我们还计算了难以找到解析解的重要KP方程的稳定数值解。多尺度现象和图像分析的研究正在进行中。复奇异点的谐波分析：在与国外学者的合作中，我们发展了具有奇异点的复变的谐波分析理论。希望这一作品能对我们以后的奇点相关作品有所启发。