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

考虑了椭圆型方程中典型的Laplacs-Poison方程的逆边值问题。该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拟谱方法的一类，它允许任意数量的网格点在感兴趣的区域内任意分布，并包括其边界。该技术基于以指数函数作为基函数逼近的插值法。我们可以期待这项技术的极高精度，大约几十个数量级。然而，随着精度阶数的增加，要求解的线性方程组的条件变得更差。因此，应特别注意该技术在计算机上的数值实现。我们使用了京都大学藤原博士开发的扩展浮点库，作为对病态线性系统的数值处理中的舍入误差的补救。实践中使用了200个小数位，伴随变分方法被广泛地应用于双曲型方程的系数辨识问题。考虑了弹性力学中的标量波动方程和动力航海方程。提出了一种由边界上观测到的位移和表面牵引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，利用相关泛函和罚函数的和的梯度来最小化代价函数。数值实验表明，该方法比传统的辨识方法更有效。

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.