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

本文研究了一类椭圆型方程Laplacs-Poissoon方程的边值反问题。方程数值解的一种新方法。高阶有限差分法是专门为求解Hadamard意义下的不适定问题而发展起来的。这种新的数值技术属于类的准谱方法，允许任意数量的网格点分布在域的兴趣，其边界包括在内。该技术是基于使用指数函数作为近似基函数的插值。我们可以期望该技术具有极高的精度，约为几十级。然而，随着精度阶数的增加，待求解的线性方程组的条件变差。因此，应特别注意在计算机上数值实现的技术。我们采用扩展的浮点库，由京都大学的藤原博士开发的，作为对病态线性系统的数值处理中的舍入误差的补救措施。伴随变分法广泛应用于双曲型方程的系数识别问题。考虑了弹性力学中的标量波动方程和动力Navies方程。本文提出了一种由边界上的位移和表面张力识别线弹性波场中拉梅常数的重建算法。该算法具有迭代过程的性质，利用相关泛函之和的梯度与罚函数来最小化代价函数。数值实验表明，该方法相对于传统的辨识方法是更有效的。

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.