A Homotopy-Projection Method For The Parameter Estimation Problems

The homotopy method is a widely convergent method for solving nonlinearsystem of equations, and also has important applications in many fields. Projec-tion method not only contains abundant mathematical theories, but also is a powerfulmethod and tool in engineering. This dissertation draws the homotopy method and theprojection method into the inversion process, acquires a homotopy-projection method, which has the abilities of noise suppression, global convergence and easy realizationof the procedure.At first, the widely convergent homotopy method is introduced to the param-eter estimation problem. The basic principle of homotopy inversion method is de-scribed. The regularizing numerical continuation method is constructed by combin-ing the Tikhonov regularization with the homotopy method, which mixed together theadvantages of the homotopy method and Tikhonov regularization. It theoretically en-sures that one can not only finally solve the solution of the inverse problem startingnearly from any point, but also effectively overcome the ill-posedness of the problem.In this paper, we consider the situation that closeness of some iterate aδ,m tothe exact solution is not sufficient for well-definedness of F(aδ,m), or that we searchfor a solution satisfying additional constrains. In this situation we eventually haveto use the metric projector PD(F) onto D(F) in order to carry out the next step or,more generally, stay in the feasible set. So, we combine the homotopy method and theprojection method to give a homotopy-projection method. This method is pervadedby the virtues of the two methods mentioned above, which not only overcome thelocal convergence, but also improves the solution quality. Numerical results show themethod's effectiveness.The proposed method to some extent overcomes some difficulties faced by theproblem of the parameter estimation problem. Besides, these method is very general,which can be used to solve other inverse problems.