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Fast Implementation of the Boundary Element Method and its Application to Inverse Problems

边界元法的快速实现及其在反问题中的应用

基本信息

批准号：
12650060
负责人：
HAYAMI Ken
金额：
$ 1.73万
依托单位：
National Institute of Informatics (2001-2002)The University of Tokyo (2000)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12650060/
关键词：
Boundary Element Method inverse problem system of linear equations least-squares problem singular system Krylov subspace iterative method MEG source term identification 多重極展開反復解法特異クリロフ部分空間法一般化共役残差法固有値問題 Jacobi-Davidson法

项目摘要

1) Direct simulation of large amplitude standing waves using the boundary element methodWe developed a method using the two-dimensional boundary element method for the time dependent direct simulation of the free surface of liquids, and succeeded in the direct simulation of large amplitude standing waves, which was considered to be difficult.2) Inverse ProblemWe developed efficient direct methods for the inverse problem of source term identification of the three-dimensional Poisson equation, which is applied to MEG etc. Depending on whether the source is 1) relatively uniformly distributed in the domain, 2) concentrated near the boundary of the domain, infinite series of the 1) low-order moments, 2) high-order moments of the multipole expansion of the field are used in order to obtain reconstruction formulae concerning the position of the projection of the source on to the 1) xy-plane, 2) Riemann sphere.3) Iterative methods on singular systemsWe analyzed the behavior of Krylov subspace … More iterative methods on least-squares problems whose coefficient matrices are singular. Namely, we derived necessary and sufficient conditions for the CR (Conjugate Residual), GCR(k) (Generalized Conjugate Residual) and GMRES (Generalized Minimal Residual) methods to give a least-square solution without break-down. The methodology is to decompose the algorithms in to the range of the coefficient matrix and its orthogonal complement.We also developed a method of applying the GMRES method after preconditioning by incomplete QR decomposition, for (over-determined) least-squares problems.4) Iterative methods for eigenvalue problemsWe derived necessary and sufficient conditions for a solution to exist to the correction equation of the Jacobi-Davidson method, clarified the relation between the subspace expanded by such a solution and the invariant space of the coefficient matrix, and proposed a method for generating efficient starting vectors for the method. We also proposed methods for the validated numerics of the method. Less

1)用边界元法直接模拟大振幅驻波本文发展了一种用二维边界元法直接模拟与时间有关的液体自由表面的方法，并成功地直接模拟了大振幅驻波。这被认为是困难的。反问题我们发展了三维Poisson方程源项识别反问题的有效直接方法，其应用于MEG等。取决于源是否1）相对均匀地分布在域中，2）集中在域的边界附近，1）低阶矩的无穷级数，2）使用场的多极展开的高阶矩以获得关于源在xy平面上的投影位置的重建公式，2）Riemann球; 3）奇异系统的迭代方法我们分析了Krylov子空间的性质 ...更多信息系数矩阵奇异的最小二乘问题的迭代方法。即给出了CR（Conjugate Residual）、GCR（k）（Generalized Conjugate Residual）和GMRES（Generalized Minimum Residual）方法给出无故障最小二乘解的充要条件。该方法是将算法分解到系数矩阵及其正交补的范围内，并提出了一种在不完全QR分解预处理后应用GMRES方法的方法，（超定）最小二乘问题。4）特征值问题的迭代方法我们导出了Jacobi-Davidson方法的修正方程有解的充要条件，阐明了该解所展开的子空间与系数矩阵的不变空间之间的关系，并提出了一种有效的初始向量生成方法。我们还提出了方法的验证数值的方法。少