应用正则化子建立求解不适定问题的正则化方法的探讨

An Investigation on Constructing Regularization Strategy for Solving Ill-Posed Problems

根据紧算子的奇异系统理论，提出一种新的正则化子进而建立了一类新的求解不适定问题的正则化方法．分别通过正则参数的先验选取和后验确定方法，证明了正则解的收敛性并得到了其最优的渐近收敛阶；验证了应用Ｎｅｗｔｏｎ迭代法计算最佳参数的可行性，最后建立了当算子与右端均有扰动时相应的正则化求解策略．文中所述方法完善了一般优化正则化策略的构造理论．

According to the theory of singular system of compact operators, a new family of regularizing filters is set forth, and then a new class of reegularization strategies for solving ill-posed problems of the first kind equations is constructed. By a priori and a posteriori choice of the regularization parameter respectively, the optimum asymptotic convergence order of the regularized solution is obtained. And, using Newton's iteration, the feasibility of determining the optimal parameter is testified. Moreover, in the case that the operator and the right-hand side are both perturbed, an optimal regularization scheme is established applying the above methods. More optimal regularization can be constructed using the same methods, which improve and develop the constructing theory of general regularization strategy.