Stochastic gradient descent method with convex penalty for ill-posed problems in Banach spaces

作     者：Gu, Ruixue Fu, Zhenwu Han, Bo Fu, Hongsun 

作者机构：Dalian Maritime Univ Sch Sci Dalian 116026 Peoples R China Harbin Inst Technol Dept Math Harbin 150001 Heilongjiang Peoples R China 

出 版 物：《INVERSE PROBLEMS》 (Inverse Probl)

年 卷 期：2025年第41卷第5期

页      面：055003-055003页

核心收录：

学科分类：07[理学] 0701[理学-数学] 0702[理学-物理学] 

基　　金：National Natural Science Foundation of Chinahttp://dx.doi.org/10.13039/501100001809 National Natural Science Foundation of China [2022M710969] China Postdoctoral Science Foundation 

主　　题：stochastic gradient descent method linear and nonlinear inverse problems system of ill-posed equations convex penalty convergence analysis tomography 

摘      要：In this work, we investigate a stochastic gradient descent (SGD) method for solving inverse problems that can be written as systems of linear or nonlinear ill-posed equations in Banach spaces. The method uses only a randomly selected equation at each iteration and employs the convex function as the penalty term, and thus it is scalable to the problem size and has the ability to detect special features of solutions such as nonnegativity and piecewise constancy. To suppress the oscillation in iterates and reduce the semi-convergence of such methods, by incorporating the spirit of discrepancy principle, an adaptive strategy for choosing the step size is suggested. Under certain conditions, we establish the regularization results of the method under an a priori stopping rule. Further, we study an a posteriori stopping rule for SGD-theta method and show the finite iterations termination property. Several numerical simulations on computed tomography and schlieren imaging are provided to demonstrate the effectiveness of the method.