Banach空间基于非光滑惩罚项的迭代正则化算法及其应用

In the exploration of specific problems in engineering such us image denoising, geological exploration and biomedical imaging, the importance of non-smooth parameter reconstruction is becoming more and more important. The application of such problems in the reconstruction of medium interface is an important index to depict the typical characteristics of the medium. The mathematical difficulty of such reconstruction is how to keep the non-smoothness property of the solution, the Tikhonov regularization algorithm in the Hilbert space will over smooth the regularization solution. Therefore, it is a key method to deal with such problems by introducing the appropriate non-smooth penalty term and considering the corresponding optimization problem. This project research the theoretical analysis and numerical simulation of the iterative regularization algorithm in Banach space, including global convergent algorithm and Nesterov accelerating. Combining with Morozov discrepancy principle and Hanke-Raus heuristic stopping criterion, the project establishes the error estimates and discusses convergence speed. In addition, in order to reduce the computational cost, we provide the multi-level global convergence algorithm, the accuracy of regularized solution is improved by successive iteration of input data. Based on the rigorous mathematical theory, the results of this project provide efficient algorithms for non-smooth imaging application.

在图像恢复、地质勘探、生物医学成像等众多应用领域，非光滑参数的重建问题数学上的重要性日益显著。这类问题应用于介质和界面的恢复，是刻画介质典型特征的一个重要指标；数学上求解的难点在于利用正则化方法去噪的同时必须保持解的非光滑性，此时Hilbert空间中的经典的Tikhonov正则化算法化将会使反演解过度光滑。引入适当的非光滑约束作为惩罚项并考虑对应的优化问题是处理这类非光滑解恢复问题的一个关键方法。本项目在Banach空间中研究带有非光滑罚项的迭代正则化算法的相关理论和算法实现，包括全局收敛算法及加速迭代算法、结合广义残差准则或Hanke-Raus型启发式停止准则建立正则化解的误差估计及收敛速度。为了减少计算量，进而研究数据空间的多尺度全局收敛算法，通过对输入数据分层使用，逐次迭代提高近似解的精度。本项目的结果为应用领域的非光滑成像提供了具有严格数学理论基础的高效算法。

在图像恢复、地质勘探、生物医学成像等众多应用领域，非光滑参数的重建问题数学上的重要性日益显著。这类问题应用于介质和界面的恢复，是刻画介质典型特征的一个重要指标；数学上求解的难点在于利用正则化方法去噪的同时必须保持解的非光滑性，此时Hilbert空间中的经典的Tikhonov正则化算法化将会使反演解过度光滑。引入适当的非光滑约束作为惩罚项并考虑对应的优化问题是处理这类非光滑解恢复问题的一个关键方法。本项目研究了Banach空间中研究带有非光滑罚项的迭代正则化算法的相关理论和算法实现，完整讨论了全局收敛算法、多尺度算法、Bregman加速，两点梯度算法、渐进性分析、以及在地球科学中的应用。本项目的结果为应用领域的非光滑成像提供了具有严格数学理论基础的高效算法。