Regularied Methods For Ill-posed Problems

The dissertation investigates the regularizd methods for a few kinds of specifical inverse problems. We obtain the convergence and convergence rates of regular solution. At the same time, we extend Hilbert space in which Landweber iteration method are involved to Banach space. The generalized Landweber iteration method in Banach space is available.In chapter 2, we focus on an inverse coefficient problem of a semilinear parabolic equation. Under the weak source conditon:using least square method, we can get the convergence and convergence rate of approximate coefficent and approximate solution.In chapter 3, the problem related to controlled potential experiments in electrochemistry is studied. Modelling of the experiment leads to a problem for a nonlinear parabolic equation with additional condition. Driven by the needs of theoretical analysis, from the point of view an inverse coefficient problem, we analyze the monotonicity of input-output mappings in inverse coefficient and source problems for this parabolic equation. Additionally, we extend the nonlinear parabolic equation to a more general case. Under some proper conditions, we investigate the existence of quasisolution of the generalized nonlinear parabolic equation.In chapter 4, we devote to simplified Tikhonov regularization for a sideways parabolic equation, and a two-dimensional backward (inverse) heat conduction problem. We concentrate on the convergence rates of the simplified Tikhonov approximation of solutions of the sideways parabolic equations at 0≤x<1, and the two-dimensional backward (inverse) heat conduction problem at 0≤t