基于全变分正则化方法的不适定问题的研究及其应用

In practical applications, people often encounter this kind of inverse problem, the exact solution has a discontinuity or sharp corners. In this case, the classical Tikhonov regularization usually leads to oversmoothed or oscillatory. Total variation regularization is a standard method to solve this ill-posed prolem. But 1) this makes the algorithm and theoretical analysis of solution ignores the effects of parameters, because the parameter in substitution penalty of total variation is selected according to experience. 2) the nonlinear convex and non differentiability of total variational penalty makes the solution and its theoretical analysis of this class of problems encounter great difficulties. Therefore, the mathematical analysis in parameter of total variation penalty items, as well as the development of new efficient and stable algorithm is very necessary for the development and improvement of total variation regularization method.. The research content of this project is following as based on several categories of non smooth exact solution of ill posed problem : 1) the selection strategy of parameter in alternative penalty and regularization parameter; 2) developping some iterative total variation schemes; 3) considering a new regularization penalty combined TV with Tikhonov regularization. The aim of this part is to establish the corresponding theoretical results and to develop several techniques for ill-posed problems with nonsmooth solution based on total variation regularization. Through the study of these issues, the effective numerical algorithm and theoretical analysis of numerical solution with non smooth solution of the ill posed problem will be given, and the research method will be applied in the harmonic algorithm of magnetic resonance electrical impedance tomography.

具有非光滑真解的不适定问题，经典的Tihkonov正则化仅给出一个光滑的近似解，从而导致不光滑性的丢失。而全变分正则化法却能有效反演出真解的非光滑性。但是1) 由于全变分罚项的权重系数是按经验选取，这使得求解算法及理论分析忽略了该系数对解的影响。2)全变分罚项的非线性凸性和不可微性，使得这类问题的求解和理论分析遇到了很大困难。因此，对全变分罚项中的参数进行深入的数学理论分析，以及发展新的高效稳定算法是对全变分正则化法的发展与完善是非常有必要的。. 本项目围绕几类具有非光滑真解的不适定问题开展以下几个方面的研究：1)考虑全变分罚项的权重系数以及正则化参数的选取策略；2)发展快速稳定的算法，进一步提高该方法的有效性和实用性;3）将全变分正则化法和经典正则化法相结合的理论研究及数值求解。通过这些问题的研究，给出解决具有非光滑解的不适定问题的有效得数值计算方法和理论分析，并将研究方法应用到磁共振成像技术中。

本项目围绕几类具有非光滑真解的不适定问题展开以下几个方面的研究：1) 考虑全变分罚项的权重系数与输入数据误差之间的关系并建立相应的收敛性分析，结合数值实例验证了该方法的有限性以及用数值数据验证了理论给出的收敛率分析。2) 考虑全变分罚项的权重系数以及正则化参数的选取策略；3)发展快速稳定的算法，进一步提高该方法的有效性和实用性;4）将全变分正则化法和经典正则化法相结合的理论研究及数值求解。通过这些问题的研究，给出解决具有非光滑解的不适定问题的有效得数值计算方法和理论分析，并将研究的得到的方法应用到核磁共振成像的调和算法中。