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

摘要：构造了新的多维修正的样条函数，并将其应用于非线性偏微分方程组的数值分析和图像分析。此外，我们还在三年内从国外邀请了10名活跃的研究人员来组织国际会议。具体内容如下：1.计算调和分析：我们成功地构造了具有“最小”支撑度的基函数，这使得我们能够对函数进行严格的内插和良好的采样逼近。还研究了定义在更一般区域上的基函数，即在有限区间的情况下，平面上的几个典型区域等。在偏微分方程数值分析中的应用：作为应用，我们找到了一种计算偏微分算子的一系列高阶有限差分格式的新方法。此外，我们还计算了重要的KP方程的稳定数值解，其解析解很难找到。多尺度现象和图像分析的研究正在进行中。3.复奇异的调和分析：在与国外研究人员的合作下，我们发展了具有奇异性的复变的调和分析理论。这项工作有望在以后与奇点相关的工作中启发我们。

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.