Bayesian Method Of Parameter Inversion And Its Application

In recent years,the theoretical and applied research of inverse problem involves more and more fields.In terms of theoretical research,more and more researchers use different methods to overcome the uneasiness of inverse problem.Since the statistical method can quantify the uncertainty of the solution of the inverse problem,the research of the inverse problem is paid more and more attention.In addition,the inverse problem method is more and more widely used in the fields of petroleum exploration,hydrogeology and environmental science,signal and image processing,etc.,especially in the study of image denoising and recovery,in which the study of statistical method is of great significance.This paper mainly studies the Bayesian method of parameter inversion in time fractional diffusion equation and its application in image reconstruction.The first chapter introduces the research significance of the topic,and briefly discusses the research situation and development trend,then briefly introduces the relationship between the inverse problem and Bayesian method,and finally gives the main work of this thesis.The second chapter mainly introduces the basic concepts and theoretical knowledge of Bayesian method,as well as the application of Bayesian theory in image processing,so as to prepare for the research in the following chapters.In chapter three,the parameter inversion of one-dimensional fractional diffusion equation is studied by Bayesian method.Firstly,based on the difference scheme of the forward problem,the inversion of diffusion coefficient and the joint inversion of the fractional order and diffusion coefficient are studied respectively.By the additional observation data at the right end of the region,an inversion model is established based on Bayesian theory to solve the unknown parameters,the influence of likelihood variance on the inversion results and the influence of additional data dimensions on the inversion results are discussed,and the timeliness of the inversion is analyzed.For the inverse problem of determining the variable fractional order in the variable-time fractional diffusion equation,the Bayesian method is also used to carry out the statistical inversions,and the inversion results are given for the variable fractional order in different functionalforms.The fourth chapter mainly studies the Bayesian method in image reconstruction.Aiming at the image reconstruction of specific noise model,the corresponding algorithm and numerical experiment results are given on the basis of Bayesian theory.The experimental results show that the Bayesian method has some advantages in the field of image processing.The fifth chapter summarizes the content of the paper,and points out the existing problems in the research and the research directions in the future.