The Bayesian Solution Of Inverse Problems

The inverse problem is ubiquitous in the technical neighborhood.The research on it has great development space and application prospect.In the actual mathe-matical and physical background,when the required parameters,operators,initial conditions,and boundary conditions are unknown,the unknown parameters in the model can be solved by the known observational data in the space.The inverse problem is often accompanied by the problems of nonlinearity and ill-posedness.The traditional algorithms for solving the inverse problem are usually carried out on the basis of the known characteristics of the measurement error or the small mea-surement error.However,in practical life,the measurement error exists and can not be neglected,so that the traditional algorithm can not get a good solution.This paper starts from the perspective of probability statistics.The bayesian approach is chosen to solve the inverse problem.Bayesian approach is briefly sum-marized as follows:set the priori distribution based on hypothesis and experience;set the likelihood function with error information;derive the posterior distribution from the optimization of the prior distribution.Because of the complexity of the posterior distribution,it is difficult to draw the target distribution directly.A numer-ical sampling method is used to approximate the posterior probability distribution.Then,the approximate values of the parameters are obtained.In this paper,The bayesian approach is briefly described from a mathematical perspective,including the setting and acquisition of prior information and likeli-hood functions,and the derivation of the posterior information.Based on the single-layer bayesian model,the distributions of hyper-parameters are set,then the hierar-chical bayesian model is deduced.Also,the well-posedness of problems is analyzed.The MCMC sampling algorithm is introduced.Combined with the Gibbs sam-pling method and the Metropolis-Hastings(MH)sampling method,the Metropolis-within-Gibbs sampling method can be deduced with the hierarchical bayesian mod-el.The innovation is that we combine the pCN method with the Metropolis-within-Gibbs method so that the acceptance function relies only on the likelihood function.Then,an improved pCN-Metropolis-within-Gibbs method can be derived.In addi-tion,the Bayesian model and the pCN-Metropolis-within-Gibbs method are used to solve the inverse source problem and the geometric inverse problem in the subsur-face flow.Good experimental results are obtained and the corresponding models and parameters are adjusted to solve the problem optimally.Bayesian algorithm combined with pCN-Metropolis-within-Gibbs algorithm to solve the inverse problem has several advantages as follows:(1)Better integration of prior information and error information to reduce the uncertainty of solving the problem;(2)The algorithm breaks the problem of ill-posedness and local con-vergence,converging to the global optimal solution;(3)The algorithm are suitable for the numerical calculation of the probability distribution density function with-out explicit expressions in high dimensional space;(4)The sampling algorithm con-structs the Markov chain by setting the acceptance criteria to complete the random simulation,which fits faster than the general Monte Carlo method.