随机数据驱动Bouligand Landweber方法求解非光滑逆问题
Stochastic Data-Driven Bouligand Landweber Method for Solving Non-smooth Inverse Problems
AI总结：
本文提出随机数据驱动Bouligand Landweber方法，用于求解非光滑不适定逆问题系统，结合训练数据指导迭代，并证明其在精确和噪声数据下的收敛性与稳定性。
摘要：
在本研究中，我们提出并分析了一种新颖的随机梯度下降方法的变体，称为随机数据驱动Bouligand Landweber迭代，专门用于解决非光滑不适定逆问题系统。我们的方法结合了训练数据的使用，利用一个有界线性算子来指导迭代过程。在每一步迭代中，方法从带有数据驱动项的非线性系统中随机选择一个方程。在处理精确或准确数据时，已证明均方迭代误差收敛于零。然而，当面对噪声数据时，我们将我们的方法与预定义的停止准则结合使用，我们称之为先验停止规则。我们提供了全面的理论基础，在无限维希尔伯特空间领域内建立了该方案的收敛性和稳定性。这些理论基础通过讨论一个满足论文假设的示例得到进一步支持。
英文摘要：
In this study, we present and analyze a novel variant of the stochastic gradient descent method, referred as Stochastic data-driven Bouligand Landweber iteration tailored for addressing the system of non-smooth ill-posed inverse problems. Our method incorporates the utilization of training data, using a bounded linear operator, which guides the iterative procedure. At each iteration step, the method randomly chooses one equation from the nonlinear system with data-driven term. When dealing with the precise or exact data, it has been established that mean square iteration error converges to zero. However, when confronted with the noisy data, we employ our approach in conjunction with a predefined stopping criterion, which we refer to as an \textit{a-priori} stopping rule. We provide a comprehensive theoretical foundation, establishing convergence and stability for this scheme within the realm of infinite-dimensional Hilbert spaces. These theoretical underpinnings are further bolstered by discussing an example that fulfills assumptions of the paper.
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