Multiscale Methods for the Rapid Solution of Boundary Integral Equations in Geometrically Complicated Domains

本文研究几何复杂曲面上边界积分算子的多尺度离散化问题。在参数空间中生成多尺度离散化并将其提升到曲面上的标准方法，当需要大量参数面片时，效率不高。该项目涉及多尺度离散化的另一种方法。基是在三空间的分层分解中产生的，并且随后被限制在边界表面上。这种构造导致积分运算符的稀疏表示，即使对于复杂的几何图形也是如此。计划将这种方法应用于一般椭圆型和含时抛物型问题的边界积分公式。所提出的研究可能影响的重要工程应用范围是相当不同的。例如对集成电路、互连和微机械系统的分析。在这些领域，为了在计算机上交互地生成原型，为复杂的三维结构找到计算高效的数值方法是很重要的。过去主要的计算工具是快速多极子方法(FMM)，因为它相对简单，对几何和积分算子也很灵活。我们利用三空间的层次分解来构造多尺度基。FMM也使用相同的立方体层次结构，因此所提出的方法能够结合FMM的灵活性和多尺度方法的优点。因此，计划中的研究可以导致更快和更健壮的算法，以及更好地理解这两种方法的理论和实践问题。

The proposed research is concerned with multiscale discretizations of boundary integral operators on geometrically complicated surfaces. The standard approach, generating multiscale discretizations in parameter spaces and lifting them on the surface, is not efficient when a large number of parameter patches is required. This project is concerned with an alternative approach to multiscale discretizations. The basis is generated in a hierarchical decomposition of the three-space and subsequently restricted on the boundary surface. This construction leads to a sparse representation of the integral operator even for complicated geometries. It is planned to apply this approach to boundary integral formulations of general elliptic as well as time dependent parabolic problems.The range of important engineering applications which the proposed research could impact is quite diverse. Examples include the analysis of integrated circuit interconnect and micromechanical systems. In these areas finding computationally efficient numerical methods for complex three dimensional structures is important in order to generate prototypes interactively on a computer. In the past the major computational tool has been the Fast Multipole Method (FMM), because of its relative simplicity and its flexibility to geometry and integral operator. We construct a multiscale basis by using a hierarchical decomposition of the three-space. The same hierarchy of cubes is also used by the FMM, and therefore the proposed approach is able to combine the flexibility of the FMM with the strengths of multiscale methods. Thus the planned research can lead to faster and more robust algorithms as well as to a better understanding of theoretical and practical issues of both approaches.