Deterministic Regularization Methods And Bayesian Approach To Ill-posed Problems

In this thesis,we study the deterministic regularization methods and Bayesian approach to some severely ill-posed inverse problems,i.e.,we will study the stabilized solution of these problems in the deterministic framework and stochastic framework,respectively.For the deterministic aspect,we take analytic continuation and some other severely ill-posed problems on regular rectangle domains as examples,and our emphasis here is on the a-posteriori regularization methods for solving these problems.First,we will discuss the problems of stable analytic continuation of analytic functions on infinite strip domain and general bounded domains,where for the first problem,we,respec-tively,use approximate inverse method and a-posteriori Fourier method to solve it,error estimates between the regularized solutions and the exact solutions are given,and the corresponding numerical tests are provided;for the problem of analytic contin-uation on general bounded domains,we study the numerical solution of this problem by the method of fundamental solutions combining with numerical differentiation,our results based on the theory of regularization about this problem is completely new.Second,based on the study of the a-posteriori truncation regularization method for solving the backward heat conduction problem with time variable coefficient and the Cauchy problems associated with Helmholtz-type equations on rectangle domains,we provide the general theory framework for solving such kind of ill-posed inverse problems with explicit expressions for the solutions,and the corresponding numerical experiments are given.For the stochastic aspect,we discuss the posterior contraction rates for a class of severely ill-posed inverse problems for which the forward mappings are diagonalizable,i.e.,under the framework of Bayesian,the solution of an inverse problem is no longer a single point estimate as that in the deterministic theory,but a conditional posterior probability distribution containing information about the relative probability of pos-sible states of the solution,and the so called contraction is a topic on the small noise limiting behavior of this conditional posterior probability distribution.Intuitively,we would hope the posterior mass concentrates on a small ball centered at the true solution.We assume that the forward operator and the prior and noise covariance op-erators commute with one another.We show how,for given smoothness assumptions on the truth,the scale parameter of the prior can be adjusted to optimize the rate of posterior contraction to the truth,and we explicitly compute the logarithmic rate.