The Study Of Bayesian Methods For Nonparametric Statistical Inverse Problems And Their Applications

To solve an inverse problem means to obtain information about the internal structure of a physical system form indirect noisy observations.The inverse problems have a wide range of applications in a variety of scientific fields,such as geology,atmospheric sciences,bioscience,epidemiology and battlefield information perception and processing,to name a few.Nonparametric statistical inverse problem means that the internal structure is modeled by some infinite-dimensional parameters and the noise is modeled by statistical methods.For given measurements,the only way to overcome the ill-posedness of ill-posed inverse problems is to use prior information.The Bayesian methods model prior information by prior distributions,and is one of the most effective ways for solving the ill-posed inverse problems.The thesis considers the Bayesian methods for nonparametric statistical inverse problems.The main contributions of the thesis include the following.1.To discuss the Bayesian method for nonparametric statistical inverse problems from prior construction,posterior computation and posterior contraction systematically.The ill-posedness of the inverse problem is measured by the modulus of continuities of the inverse operator,and a general posterior contraction theorem for nonlinear inverse problems is proved,which is the first innovation point of the thesis.Furthermore,we have discussed the influence of numeral errors to the convergence rates of various Bayesian estimators.The results indicate that as long as the numerical error converges to 0 fast enough,with the sample information increasing,the convergence rates of Bayesian estimators are exactly the posterior contraction rate.2.To present applications of the finite-dimensional sieve prior in different problems.Application of the sieve prior in regression with finite change-points is the second innovation point of the thesis: we present reversible jump MCMC sampler for the jump change-point model,and present posterior contraction rates for the structure change-point model which indicates that the sieve prior is rate adaptive for regression with finite changepoints.Application of the sieve prior in the nonlinear inverse problems is the third innovation point of the thesis: the modulus of continuities of the inverse operator is linearized by the so-called finite dimensional sieve,and thus the ill-posedness of linear inverse problems is generalized to nonlinear inverse problems;posterior contraction rates for mildly ill-posed and severally ill-posed inverse problems are both presented.3.To propose a multi-model adaptive trajectory inference method for space-based missile early-warning system,which is the fourth innovation point of the thesis.We model the boost-phase trajectory of different types of missiles by the method of sieves,adopt a multi-model Bayesian framework to realize the adaptation with respect to the types of the missiles.We divide the prior information into three different levels,type-free prior,typedependent prior and type-distribution prior,adopt a hierarchical Bayesian framework to realize adaptation with respect to the prior information.The performance of the method is validated by numerical experiments.