Stochastic Data-Driven Bouligand Landweber Method for Solving Non-smooth Inverse Problems
Harshit Bajpai, Gaurav Mittal, Ankik Kumar Giri
TL;DR
This work develops a stochastic data-driven Bouligand Landweber iteration (SDBLI) to solve systems of nonlinear, potentially non-smooth ill-posed inverse problems in infinite-dimensional Hilbert spaces. By randomly selecting equations and using Bouligand subderivatives $G_i(u)$ together with data-driven operators $M_i$, the method achieves regularization without requiring Fréchet differentiability of $F_i$, and convergence is established for exact data while a-priori stopping yields stability under noise. The analysis hinges on a Lipschitz-type tangential cone condition and boundedness assumptions, culminating in asymptotic stability and convergence to a solution in $\mathcal{D}(u^{\dagger},\sigma)$; an example based on inverse source problems demonstrates applicability. The results offer a scalable, data-informed approach to non-smooth inverse problems with potential impact in tomography and related fields, and point to future work on discrepancy-based stopping and adaptive step-size strategies.
Abstract
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.