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）利用边界元法直接模拟大振幅驻波我们开发了一种利用二维边界元法对液体自由表面进行时间相关直接模拟的方法，并成功地直接模拟了大振幅驻波， 2）反问题我们针对三维泊松方程的源项识别反问题开发了高效的直接方法，该方法应用于MEG等。取决于源是否1）相对均匀分布域, 2)集中在域边界附近，使用场多极展开的 1）低阶矩，2）高阶矩的无限级数，以获得关于源在1) xy 平面，2) 黎曼球面。3) 奇异系统的迭代方法我们分析了 Krylov 子空间的行为...更多系数矩阵为奇异的最小二乘问题的迭代方法，即我们推导了必要的迭代方法。以及 CR（共轭残差）、GCR（k）（广义共轭残差）和 GMRES（广义最小残差）方法的充分条件，可以在不分解的情况下给出最小二乘解。该方法是将算法分解为系数矩阵的范围及其正交补集。我们还开发了一种在通过不完全 QR 分解进行预处理后应用 GMRES 方法的方法，用于（超定）最小二乘4）特征值问题的迭代方法推导了雅可比-戴维森法修正方程解存在的充要条件，阐明了该解展开的子空间与系数矩阵不变空间之间的关系，并提出了一种为该方法生成有效起始向量的方法，我们还提出了该方法的验证数值方法。