On the Modified Tikhonov’s Regularization Method for the Cauchy Problem of the Laplace Equation

Abstract

Abstract:In this paper, an inverse problem of the Laplace equation with Cauchy data is examined. Due to the ill-posed behavior of this inverse problem, the Tikhonov’s regularization technique is employed and the L-curve concept is adopted to determine the optimal regularization parameter. Also, the singular value decomposition method is used in conjunction with the L-curve concept for the same problem. Numerical results show that neither the traditional Tikhonov’s regularization method nor the singular value decomposition method can yield acceptable results when the influence matrix is highly ill-posed. A modified regularization method, which combines the singular value decomposition method and regularization method, is thus proposed, and this new method shows that it is a better way to treat this kind of inverse problems comparing with the other two traditional methods. Numerical results also show that the inverse problem with Cauchy data is better to formulate by the singular integral equation than by the hypersingular integral equation for the constant element scheme. The inverted boundary data becomes closer to the exact solution when the number of elements increases, and numerical experiments show that the rate of convergence is higher for the formulation using the singular integral equation. Numerical experiments are made to examine how the boundary Cauchy data affect the inverted process. It is concluded that the inversion of unknown boundary data is more effective when the Cauchy data are given more precisely and are distributed on the whole boundary more diversely.