Abstract
The structure of the Tikhonov regularization for numerical analysis of an inverse boundary value problem is examined in this chapter. Singular value decomposition is applied to evaluate the amplitude of error magnification, which is called the condition number. It was found that the condition number in the Tikhonov regularization showed a V-shaped behavior with an increase in the regularization parameter α. For determining the optimum value of the Tikhonov regularization parameter α, the optimum condition number method is applied, in which the regularization parameter giving the optimum value of the condition number is employed as the effective one. Because of the V-shaped behavior of the condition number, the optimum condition number method has two or no candidates of the effective regularization parameter. To make the method applicable to the Tikhonov regularization, a new condition number is proposed. Numerical simulations are made for estimating boundary values in the Laplace filed from boundary values over described on a part of the boundary. It is found that the optimum condition number method with the new condition number is useful for estimating the effective regularization parameter. Good estimates of boundary values are obtained by applying the Tikhonov regularization with the estimated effective regularization parameter α.
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