Regularization Methods Of Inverse Problems Of Parameter Identification Of Partial Differential Equation

This paper presents some examples of problems that are inverse to each other, describes the general concepts of inverse problems, exactly defines the inverse problems of partial differential equations , and classifies them as four types. Due to the ill-posedness of inverse problems of partial differential equations, the causes of ill-posed problems are analyzed, and it is concluded that linear compact operators are always improperly-posed. The research of the ill-posedness is focused on the ill-posed problems without stability. For correct treatment of the ill-posedness of inverse problems of partial differential equations and obtaining stable approximate solutions depending continuously on data, some concepts related to regularization methods and the general theory are used, It is concluded that the regularization operators are not uniformly bounded and do not converge uniformly. Based on the spectral theory of self-adjoint compact operators and the singular value decomposition for compact operators, the paper, in detail, deduces the solvability conditions and solution expansion of compact operator equations, and illustrates that the ill-posedness of linear compact operators originates in the property of singular values that trends to zero.It followsthat it is possible to construct various kinds of regularization methods by combining the filter functions with the solution expression of compact operator equation. In the paper, there are three important filter functions to be offered, the regularization constructed by the third filter is called spectral cutoff, and the regularization by the first filter is just Tikhonov regularization. The basic idea of Tikhonov regularization is that linear compact operator equation of the first kind is replaced by the minimization problem of Tikhonov functional, the paper demonstrates the minimization of Tikhonov functional is an well-posed problem. In the end, the paper suggests a new type of iterative algorithm to solve inverse problems of parameter identification for one-dimension parabolic partial differential equation . the algorithm is based on approximation of function, Tikhonov regularization and optimal perturbation techniques to create an iterative process, It enlarges the applicable ranges of partial differential equations and initial-boundary values for these inverse problems. The results of numerical simulation illustrate that this numerical method has characteristics of higher numerical coputation precision, convergence rate and stability.