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

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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.