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

1)用边界元直接模拟大振幅驻波我们发展了一种用二维边界元方法直接模拟液体自由面随时间变化的方法，并成功地直接模拟了被认为是困难的大振幅驻波。2)逆问题我们发展了一种有效的直接方法来求解三维泊松方程的源项识别反问题，并将其应用于MEG等。根据源是1)在区域内相对均匀分布，2)集中在区域边界附近，1)低阶矩的无穷级数，2)利用场的多极展开的高阶矩，得到源在1)xy平面上，2)黎曼球上的投影位置的重构公式。3)奇异系统的迭代方法。我们分析了Krylov子空间…的行为系数矩阵奇异的最小二乘问题的更多迭代方法。也就是说，我们得到了CR(共轭残差)、GCR(K)(广义共轭残差)和GMRES(广义最小残差)方法在不崩溃的情况下给出最小二乘解的充要条件。方法是将算法分解到系数矩阵及其正交互补的范围内。我们还提出了一种将GMRES方法应用于(超定)最小二乘问题的方法。4)特征值问题的迭代方法我们推导了Jacobi-Davidson方法的修正方程存在解的充要条件，阐明了由这样的解展开的子空间与系数矩阵的不变空间之间的关系，并提出了一种生成方法的有效起始向量的方法。我们还提出了验证该方法的数值的方法。较少

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