Coupled Complex Boundary Methods Of Cauchy Problem And Its Application In Nonlinear Inverse Robin Problem

In recent years,the study of elliptic partial differential equation and the related inverse problem has penetrated into all fields,with a wide range of applications in thermostatics,plasma physics,mechanical engineering,electrocardiogram and nondestructive evaluation of corrosion etc.,and thus draw more and more attention of mathematicians and engineers.In this thesis,we study the Cauchy problem governed by the elliptic equation,that is,the problem of recovering the Cauchy data on inaccessible boundary from the knowledge of the Cauchy data on the accessible boundary.It is well known that the Cauchy problem is a severely ill-posed inverse problem.Tikhonov regularization is a class of simple but effective methods for stably solving inverse problems.Different reconstruction frameworks use different data-fitting term or regularization norms.This thesis first studies a couple method of complex boundary(CCBM)-based functional which belongs to domain data-fitting functional.In CCBM,all Cauchy data are integrated into complex Robin boundary conditions.Then the inverse Cauchy problem is reduced to a complex one: finding the unknown data on the unmeasurable boundary so that the imaginary part of the solution of the forward problem vanishes in the problem domain.Tikhonov regularization is applied to the new model.Compared with previous work,the CCBM makes Dirichlet and Neumann data as parts of the Robin boundary conditions,and thus reduces the regularity requirements on the data.Moreover,the noisy data are used as boundary conditions,rather than in data-fitting term,and this make the problem become more stable.Due to large impact of regularization parameter on regularized solutions,its value should be selected properly.Therefore,in the second part of this thesis,we construct a parameter-dependent CCBM.The biggest advantage of the improved CCBM is: as long as the introduced parameter is chosen according to certain rule,the regularized solution is consistent with respect to the small values of the regularization parameter,and thus there is no need to consider the choice of regularization parameters: choose a small enough regularization parameter can readily give a reasonable approximation solution.This settles the regularization parameter selection problem of inverse Cauchy problem.In addition,this thesis studies a nonlinear inverse Robin problem through a linear Cauchy inverse problem,and thus avoid the nonlinearity.Adjoint techniques and finite element methods are applied to solve the proposed models for numerical solutions.The numerical examples show that the proposed methods in this thesis are feasible and effective.