Convergence Analysis Of Some Numerical Methods For Parameters Identification In Some Partial Differential Equations

Parameter identification problems arise widely from many areas of applications and have attracted much attention in both theoretical and applied sciences in recent years.In this thesis we present the quadratic convergence of Levenberg-Marquardt(L-M)method for solving general ill-posed nonlinear inverse problems including two parameters to be identified.The problem is formulated as follows:u(p,q)=z~δ,where p,q are unknown parameters,and zδ is the measured data of the exact solution u and the parameter δ is used here to emphasize the existence of the noise in the measured data.We first transform the problem into stabilized output least-square minimization with some Tikhonov regularizations.As the Tikhonov regularized minimization problem is non-convex due to the nonlinearity of the inverse problem,the L-M method is used to transform it into a convex minimization problem.The quadratic convergence of the L-M method is rigorously shown under some basic assumptions.We also study the application of the L-M method for the simultaneous identification of the Robin coefficient and heat flux in an elliptic system.The surrogate functional technique is introduced to solve the corresponding minimization problem which is still strongly ill-conditioned.The technique gives us an explicit minimizer of the minimization at each L-M iteration.Some numerical experiments are presented to illustrate the efficiency and accuracy of the proposed algorithm.We also study the parameter recovery for a dynamo equation.We formulated the ill-posed problem into an output least-squares nonlinear minimization by using a selected Tikhonov regularization and we demonstrate its regularizing effects.A fully discrete finite element method is used to approximate the nonlinear optimization problem and we established its convergence.The thesis consists of five chapters:In Chapter 1,we summarize the background of the related literature and we also give some preliminary results and notations.The general scheme of the L-M method is proposed and the quadratic convergence of the scheme is shown in Chapter 2.In Chapter 3,we present an application of the proposed L-M method and we give some numerical results.In Chapter 4,we present a parameter recovery for a dynamo equation.Chapter 5 is about the conclusion and some future work.