Theory And Algorithms For Nonsmooth Discrete Hamilton-Jacobi-Bellman Equations And Inverse Scattering Problem

In this paper,we mainly study the theory and algorithms for the discrete Hamilton-Jacobi-Bellman equations and the inverse time-harmonic acoustic scattering prob-lem for an impenetrable scatter.Optimal value function for stochastic control prob-lem satisfies the Hamilton-Jacobi-Bellman(HJB)equations,which is a very impor-tant optimization problem and has many applications in engineering,management,economics and finance.Scattering is a common physical process and scattering the-ory is widely used in many scientific fields such as geophysical prospecting,medical imaging,non-destructive testing,underground petroleum resources,marine explo-ration,radar sensing and stealth technology,the mathematic model of scattering problems are Helmholtz equation and the corresponding boundary condition.The first chapter mainly introduces the related research background and sig-nificance of the researching.For discrete Hamilton-Jacobi-Bellman problem,we put forward the corresponding mathematical model,and then discrete continuous HJB equations to obtain the related discrete HJB equations.For the part of scat-tering problem,we mainly consider time-harmonic acoustic scattering problem for an impenetrable scatter,according to the wave equation,we derive that the time-harmonic acoustic scattering satisfies the Helmholtz equation and then combine the three boundary conditions to obtain three kinds of mathematical models of the scat-tering problem.Chapter two focuses on the discrete HJB problem,various numerical methods and algorithms has been developed for solving HJB equations effectively and ac-curately.In this chapter,we propose a Newton iteration algorithm to solve the nonsmooth discrete Hamilton-Jacobi-Bellman Equations.First,we introduce some auxiliary vectors to transform discrete HJB equations into an equivalent system of nonlinear non-differentiable equations.Then we use regularization method to ap-proximate the non-differentiable term by a related differentiable one.Finally,we solve the system by Newton iteration algorithm,whose convergence rate is super-linear.Some numerical examples are also presented to demonstrate the efficiency of this algorithm.The third chapter studies the inverse acoustic scattering problem of determin-ing the shape of an impenetrable scatter with Neumann boundary condition in two dimensions.Firstly,the existing algorithms for solving the backscattering problem as well as the advantages and disadvantages of these algorithms are summarized.Then we give the relevant reference knowledge which will be used in the study.The next section gives the mathematical model of the inverse problem.It is shown that under suitable hypothesis the boundary is uniquely determined form a knowledge of incident point sources and measurements on a curve inside the scatter.Then we put forward the modified linear sampling method to solve the inverse problem from measurements on the curve.By several preliminary examples of using the modified linear sampling method to reconstruct the boundary of scatter,we know that the method works effectively and is sensitive to noise.The last chapter summarizes the contents of the full text and puts forward the future research direction.