Bayesian Level Set Method For Boundary Recognition Of Unknown Sources

Inverse problems occur widely in mathematics,engineering fields.The core problems lie in the deep theories and complicated numerical computations.It plays an important role in the research of natural and social sciences.In recent years,it has gradually developed into a hot and difficult research topic at home and abroad.In the thesis,We consider a kind of geometric reconstruction problem,the inverse potential problem.For the inverse potential problem,the classical method is mainly based on the gradient descent iterative algorithm.In general,it is difficult to detect the location information of unknown sources with this method.In addition,the classical method only estimates the shape of the unknown geometry.However,in many cases,people not only need to consider its estimation,but also want to have a comprehensive understanding of its statistical information,that is,to make quantitative analysis of its uncertainty,which is difficult to achieve with the traditional parametric method.Therefore,we use the framework of Bayes inversion to describe the unknown information.The basic purpose of Bayes inversion is to discuss the posterior distribution of unknown variables when the observation data is known.There are many ways to describe the posterior distribution,the most common are:a)approximate the posterior distribution with the classical distribution;b)use the sample information of the posterior distribution to describe it.We use the method of sampling to estimate the statistical information of unknown variables,that is,we take a large number of samples from the posterior distribution and use these samples to estimate the statistics.The commonly used sampling algorithm is Markov chain Monte Carlo method,such as metropolis Hastings sampling algorithm.One advantage of the level set method is that it allows the topological information of the calculation area to change in the numerical calculation.The shape and position information of an unknown region are described by level set.Bayes inversion and level set method are combined to discuss the reconstruction of the support set of the source function for the boundary value problem of Poisson equation.By using the level set to parameterize the unknown area,we analyze the posterior distribution of the level set function when the observation data is known.Furthermore,the level set function is parameterized into the expansion form of radial basis function,so as to study the posteriori distribution property of expansion coefficient,i.e.the well posedness of posteriori distribution.Finally,through some numerical examples,we show the advantages and disadvantages of shape parameterization,level set parameterization and radial basis parameterization.These numerical results show that the proposed algorithm is feasible and competitive with the Matérn random field for the acoustic source problem.