在Banach空间中迭代正则化的Gauss-Newton方法及其在参数识别问题中的应用
On the iteratively regularized Gauss--Newton method in Banach spaces with applications to parameter identification problems
AI总结：
本文将迭代正则化的Gauss-Newton方法扩展到Banach空间，通过凸优化问题定义迭代过程，提出后验停止规则并分析收敛性，允许非光滑惩罚项以恢复解的稀疏性和不连续性。
摘要：
本文提出将迭代正则化的Gauss-Newton方法扩展到Banach空间，通过定义凸优化问题来生成迭代过程。我们考虑了一些后验停止规则以终止迭代，并提供了详细的收敛分析。值得注意的是，在每个凸优化问题中，我们允许非光滑惩罚项，包括L1和总变差（TV）类惩罚泛函。这使我们能够恢复解的特殊特性，如稀疏性和不连续性。一些关于偏微分方程参数识别的数值实验被报告，以测试我们方法的性能。
英文摘要：
In this paper we propose an extension of the iteratively regularized Gauss--Newton method to the Banach space setting by defining the iterates via convex optimization problems. We consider some a posteriori stopping rules to terminate the iteration and present the detailed convergence analysis. The remarkable point is that in each convex optimization problem we allow non-smooth penalty terms including $L^1$ and total variation (TV) like penalty functionals. This enables us to reconstruct special features of solutions such as sparsity and discontinuities in practical applications. Some numerical experiments on parameter identification in partial differential equations are reported to test the performance of our method.
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