Multi-Scale Trust Region Inversion Methods For Inverse Problem Of Wave Equations

Inverse problem of wave equations is applied widely in many areas. It has nonlinear and ill-posed difficulties in essence and heavy computation in practice. Thus, the survey of inverse problem of wave equation and its numerical inversion method has meaningful significance in theory and value in practice.Facing the characters of inverse problem of wave equations and the difficulties of its numerical inversion method, the paper consider inverse problem of 2-D wave equations as concrete model. Combining trust region technique with homotopy method and introducing them to the course of numerical inversion of 2-D wave equations, it improves the performance of mono-scale inversion methods. Multi-scale inversion methods are constructed via multigrid which can decrease computation cost greatly.The inverse problem of 2-D wave equations can be transferred to a nonlinear optimization problem by employing Tikhonov regularization method for solving ill-posed problem. The regularization-Gauss-Newton method which solves it often can't converge in case of highly nonlinear and has local convergence.The method is improved by constructing corresponding quasi-Newton iterate formula and introducing control threshold which manages the choice of algorithm during the course of iterating inversion. The mono-scale method is thus derived with which we can choose iterate formula freely. The trust region technique is introduced to construct iterate formula so that regularization parameter can be chosen adaptively and mono-scale adaptive inversion methods are derived.In order to relax the limit of initial guess of solution, avoid numerous local minima in the inverse problem, ensure the convergence of the constructed algorithm, homotopy method is introduced to mono-scale adaptive inversion methods after brief introduction and arrive at widely convergent mono-scale adaptive inversion algorithms.The paper studies the multigrid so as to reduce the computation cost and enhance the capability for solving large-scale inverse problem of wave equations of numerical inversion methods. Based on trust region technique and information of gradient, widely convergent multi-scale inversion algorithms are formed separately...