Study On Statistical Inversion Algorithm For Nonlinear Fredholm Integral Equations Of The First Kind

Inverse problems are widely used in geology,ocean,geophysics and other scientific fields.Usually,inverse problems are ill-posed and therefore it is quite difficult to solve this kind of problems numerically.This thesis focuses on solving the ill-posed nonlinear Fredholm integral equations which have wide application.We consider solving two practical physical problems,namely gravity measure and magnetic relief problems,and set out to solve the nonlinear Fredholm integral equations of the first kind generated by these two types of problems.The nonlinear Fredholm integral equations of the first kind belong to the category of operator equations of the first kind.For this kind of problems,even small disturbance of the right-hand data can cause enormous changes to the solution.In the framework of the classical method,the most popular approach is the regularization algorithms,the regularization method based on Newton iteration can obtain a stable numerical solution.However,for real-world problems,the data and noise are random in general.The classical regularization algorithms only provide a single estimation of the solution,and it is difficult to characterize the randomness of unknown parameter.Incorporating randomness into the research field,on the one hand,it is closer to the actual problem.On the other hand,it can not only give an approximate estimate of the solution,but also quantify the uncertainty of the solution.We devote to the study on the application of Bayesian statistical inversion algorithm to the solution of nonlinear Fredholm integral equation of the first kind.In the framework of Bayes inversion,instead of solving the original problem,we seek to quest the posterior distribution,the unknown on the data are given.The posterior distribution can provide us the sufficient statistical information of the unknown.In order to explore the posterior distribution,we mainly use the Markov chain Monte Carlo sampling algorithm,specifically using the preconditioned Crank-Nicolson(pCN)algorithm with random walk as the transfer kernel.In this process,we discuss the way of construction of prior distribution and likelihood function,and give the well-posedness analysis of the posterior distribution.Furthermore,the numerical simulations of gravity measurement and magnetic relief inverse problems are carried out.Numerical results show that the proposed method is effective.