非范数型多参数正则化方法的反问题理论与计算

In the recent decades，regularization with sparsity constraints has became the hot topic of inverse problems, because by changing the penalty term of regularization functionals, regularization with sparsity constraints allows reconstructed solutions approximating exact ones which are supposed to sparse. In this project, the research would contain theory, algorithm and application of multi-parameter regularization with non-norm l_0 + l_2. We would Focus on the nonlinear inverse problem and investigate the possibility that the multi-parameter regularization makes the solution sparse and guarantee the regularization properties. Our research would contain the following 3 aspects: 1)investigate the multi-parameter regularization with l_0 + l_2 penalty and its convergence property,regularization property and error estimates. 2) By combining Bayesian inference with MCMC methods, we would solve the difficulty in computing the minimizer of functionals caused by non-smooth of l_0 penalty and obtain the posterior distribution and other statistical information. 3)we would apply the result to the seismic full waveform inversion recovering the discontinuous media. We hope our research would throw some new light on the complicated geophysical exploration problems.

近十年来，人们发现稀疏约束正则化通过改变正则化泛函惩罚项，能更好地刻画反演解的稀疏性，因此该方法成为反问题正则化研究的热门课题。本项目将探讨l_0 + l_2非范数型多参数正则化的理论、算法与应用。我们将侧重非线性反问题，探索多参数正则化方法能否使反演结果尽可能稀疏又保持其正则性。我们的研究将包括如下三方面：1）研究以l_0 + l_2为惩罚项的多参数正则化方法，探讨其收敛性、正则性及误差估计；2）把Bayesian推断与MCMC（马尔科夫链蒙特卡罗）方法相结合，来克服由于l_0惩罚项的非光滑性带来的计算泛函极小值点这一困难，获得反演解的后验分布及诸多统计信息；3）将所获得的成果应用于与地质勘探有关的重构不连续介质等复杂参数的全波形反演问题，借此为复杂的地球物理勘探问题提供参考。

稀疏约束正则化通过改变正则化泛函惩罚项，能更好地刻画反演解的稀疏性，因此成为反问题研究的热门课题。本项目基于稀疏约束围绕多惩罚项正则化方法开展了三个方面的研究，1）含非光滑项的多参数正则化方法的算法研究；2）启发式参数选取准则的在迭代正则化方法中的应用；3）基于l_0惩罚项的正则化方法的理论与算法研究。围绕上述内容，进行算法理论与应用研究。获得如下成果：首先，针对非线性算子方程，引入p凸(p>=2) 惩罚项，构建全局收敛算法。这样的罚项可以包含L1、TV等非光滑项，从而对解的稀疏性或不连续性具有较好的刻画。其次，考虑无需噪音水平信息的启发式正则参数选取准则，探讨Gauss-Newton型迭代正则化方法的收敛性与收敛阶，并将该启发式参数选取方法与非光滑惩罚项的FISTA方法，分裂Bregman迭代相结合。最后探讨基于l_0为惩罚项的多参数正则化方法的收敛性、正则性及l_1范数意义下的误差估计，在贝叶斯框架下，应用MCMC方法来获得相应反演解的统计信息。期间将所提方法应用到地震勘探全波形反演问题、CT不完全数据图像重构等实际问题。