Regularization Methods For Three Kinds Of Ill-posed Problems

This thesis investigates the regularization methods for three kinds of ill-posed problems, i.e., the unknown source identification problem for the diffusion equation, the Cauchy problem for the Laplace equation in strip domain, a class of the nonlinear backward heat equation. Although these problems has been discussed before, but most of the results belongs to the category of a-priori regularization, in which the nu-merical results are much affected by the unknown a-priori information. While using some new methods to study the aforementioned problems, we especially discuss and explore several non-classical a-posteriori regularization methods for related ill-posed problems, and construct relative rigorous and complete theory analysis, these results are completely new.In Chapter2, we mainly discuss the source identification problem for several kinds of diffusion equations. First, we consider identifying the unknown heat source which depends only on spatial variable. We use the mollification method with Gauss kernel to obtain the regularization solution and give the convergence error estimate between the regularization solution and the exact solution under an a-priori and an a-posteriori regularization parameter choice rule, respectively. Second, we consider identifying the unknown heat source which depends only on time variable. We adopt both the central difference regularization method and the mollification regularization method with Gauss kernel to obtain the regularization solution, respectively. For these regularization solutions, we obtain the convergence error estimates. Third, we propose to identify the unknown source which depends only on one variable for frac-tional diffusion equation. For the time-fractional diffusion equation, we analyze the ill-posedness of identifying the unknown source for time-fractional diffusion equation and give the optimal error bound. Then we take the quasi-reversibility regularization method and the Fourier regularization method to obtain the regularization solutions. For these methods, we obtain the error estimates between the regularization solu-tions and the exact solution. Fourth, we use the simplified Tikhonov regularization method to identify the unknown source which depends only on spatial variable for the spatial-fractional diffusion equation. Moreover, we give; the error estimates be-tween the regularization solutions and the exact solution under an a-priori and an a-posteriori parameter choice rules.In Chapter3, we study the Cauchy problem of Laplace equation in strip domain. This is a classical ill-posed problem, however, to the best of the authors’knowledge, there is very limited literature on the case that both given the nonhomogeneous Dirich-let boundary condition and the nonhomogeneous Neumann boundary condition. We propose to use the modified Tikhonov regularization method for obtaining the regu-larization solution under an a-priori and an a-posteriori parameter choice rules. We also obtain the error estimates between the regularization solutions and the exact solution, respectively.In Chapter4, we discuss a class of non-linear heat equation backward in time. Up to the present moment, the available methods and tools for dealing with this ill-posed problem are very limited, we use the Fourier regularization method to obtain the regularization solution and obtain the Holder type error estimate.