On Inverse Problems Of Fractional PDEs And Regularization Methods

In the thesis,some inverse problems and regularization methods are studied both theoretically and numerically.In Chapter 1,we give a brief review about the development and recent progress on inverse problems,especially the inverse problems related to fractional partial differential equations.The motivations and main contents of this thesis are also presented in this chapter.In Chapter 2,we investigate numerical methods for a backward problem of the timefractional wave equation.The main idea is first to transform the ill-posed backward problem into a weighted normal operator equation,then derive the regularization scheme through the classical Landweber iterative process.We propose two fractional regularization methods,which can be regarded as an extension of the classical Landweber regularization.Under the a priori and a posteriori regularization parameter choice rules,we obtain the convergence rates of the regularized solutions generated by the proposed methods.The theoretical estimate shows that the proposed fractional regularization efficiently overcome the well-known over-smoothing drawback caused by the classical regularization.Some numerical examples are provided to confirm the theoretical results.In Chapter 3,we consider an inverse source problem for the time-fractional diffusion equation.To deal with the ill-posedness of the problem,we propose to transform the problem into an regularized problem with L2 and total variational(TV)regularization terms.Differing from the classical Tikhonov regularization with L2 penalty terms,the TV regularization is beneficial for reconstructing discontinuous or piecewise constant solutions.The regularized problem is then approximated by a fully discrete scheme.The convergence of the discrete solution is established.Then based on a saddle-point reformulation of the regularized problem,an accelerated primal-dual iterative algorithm is proposed for the discrete problem.Finally,several numerical examples are provided to demonstrate the efficiency and the accuracy of the algorithm.In Chapter 4,we propose an inexact Newton regularization combined with twopoint gradient methods for nonlinear ill-posed problems.The basic idea of the proposed method is to linearize the equation around each outer iteration and subsequently apply a so-called two-point gradient method in the inner loop to accelerate the iterative process.Under suitable assumptions,we show that the iteration sequence generated by the proposed algorithm converges to a solution of the related problem in the noiseless situation.Furthermore,the stability and regularization properties of the proposed algorithm are analyzed in the noise-data case.Several numerical examples are provided to validate the theoretical results and to demonstrate the efficiency of the proposed method.