Regularization Theory And Algorithm For The Inverse Problems Of Elliptic Partial Differential Equations

It is well know that the Cauchy problem of elliptic partial equations are severely ill-posed in the sense of Hadamard:a small noise in the Cauchy data may cause dramatically large error in the solution. The Cauchy problem of elliptic equation arises from many science and engineering problems such as nondestructive testing techniques, geophysics and cardiology. The ill-posedness of Cauchy problem lead to enormous difficulty to research these problems, that is, it is difficult to construct stable, efficient algorithm. In general, the Cauchy problem of elliptic equation has not stability. Under an additional a-priori bound condition, a stable estimate can be obtained, such as Holder stability or logarithmic stability.In this thesis, we study three kinds of Cauchy problem of elliptic equa-tion:Cauchy problem of Laplace equation; Cauchy problem of Helmholtz equation and Cauchy problem of elliptic equation with variable coefficients.We analyze the ill-posedness of these inverse problems, and discuss their degree of ill-posedness. For stably computing these problems, we proposed several regularization methods.This thesis is divided into four parts. The first chapter is preface, the definition of inverse problem, the mathematical characteristic of inverse problem and the regularization method are reported. In the second chap-ter, we use line method and spectral method solve the Cauchy problem of Laplace equation, the stable error estimate is obtained. In the third chap-ter, we apply the quasi-reversibility method and the quasi-boundary value method to Helmholtz equation. Moreover, we get the optimal error bound for the Cauchy problem of Helmholtz equation at general source condition. The fourth chapter is devoted to Fourier method, revised Tikhonov method and Wavelet dual least squares method for the Cauchy problem of elliptic equation with variable coefficients, we obtain stable error estimate.In addition, we discuss the numerical implementation of all these meth-ods, and give large number of numerical examples to test various properties of the proposed regularization methods. These tests show that our methods are numerically feasible and effective.