Direct numerical solution to the inverse boundary-value problem of elliptic equations by using the adjoint variational method.

使用伴随变分法直接数值求解椭圆方程反边值问题。

基本信息

批准号：
14540099
负责人：
ONISHI Kazuei
金额：
$ 1.22万
依托单位：
Ibaraki University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540099/
关键词：
Partial differential equations Inverse problem Variational method Cauchy problem Coefficient identification Problem Adjoint problem High-order finite difference method Extended floating-point library ラプラス方程式数値解析境界値問題境界要素法

项目摘要

Inverse boundary-value problem of the Laplacs-Poissoon equation as tipical one of equations of the elliptic type is considered. A new technique for numerical solution of the equation. called high order finite difference method, is developed especially for approximate solution of the problem that is known to be ill-posed in the sense of Hadamard. This new numerical technique belongs to the class of quasi-spectral methods, allowing any number of grid points distributed arbitrarily over the domain of interest with its boundary included. The technique is based on the interpolation using exponential functions as approximating base functions. We can expect extremely high accuracy of the order of about several tens to the technique. However, as the order of accuracy increases, the conditioning of the linear system of equations to be solved gets worse. Therefore a special attention should be paid to numerical implementation of the technique on computers. We employ the extended floating point library, developed by Dr.Fujiwara of the Kyoto University, as a remedy against rounding errors in the numerical treatment of the ill-conditioned linear system. Two hundreds decimal digits are used in practice.The adjoint variational method is extensively applied to the coefficient identification problem of equations of the hyperbolic type. The scalar wave equation and the dynamic Navies equations in elasticity are considered. A reconstruction algorithm is developed for identification of Lame constants in linear elastic wave field from displacements and surface traction observed on the boundary. The algorithm has a nature of iterative procedure, in which the cost function is to be minimized by using the gradient of the sum of related functional and a penalty function. The method is shown to be more effective relative to the conventional identification methods through numerical experiments.

考虑到椭圆类型的方程之一，拉普拉克斯 - 散光方程的逆边界值问题。方程数值解的新技术。称为高阶有限差异方法，特别是为了近似于哈达玛德（Hadamard）意义上未知的问题的近似解决方案。这项新的数值技术属于准谱方法的类别，允许任意分布在带有其边界的感兴趣域上的任何数量的网格点。该技术基于使用指数函数作为近似基础函数的插值。我们可以期望该技术的数十个数十几个数量的准确性极高。但是，随着准确度的增加，要解决的方程式线性系统的调理变得更糟。因此，应特别注意计算机上该技术的数值实施。我们采用了由京都大学的Fujiwara博士开发的扩展浮点库，作为一种补救措施，反对在不良条件线性系统的数值处理中舍入错误。实践中使用了两百个十进制数字。伴随变异方法广泛应用于双曲线类型方程的系数识别问题。考虑弹性中的标量波方程和动态海军方程。开发了一种重建算法，用于鉴定线性弹性波场中的lame脚常数，从边界上观察到的位移和表面牵引力。该算法具有迭代过程的性质，在该过程中，使用相关功能和惩罚函数的总和的梯度将成本函数最小化。通过数值实验，该方法与常规识别方法相比更有效。