Stochastic gradient descent method with convex penalty for ill-posed problems in Banach spaces
<|reference_start|>Stochastic gradient descent method with convex penalty for ill-posed problems in Banach spaces: In this work, we investigate a stochastic gradient descent 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 {\it a priori} stopping rule. Several numerical simulations on computed tomography and schlieren imaging are provided to demonstrate the effectiveness of the method. Finally, we study an {\it a posteriori} stopping rule for SGD-$\theta$ method and show the finite iterations termination property.<|reference_end|>
@article{gu2024stochastic,
    title={Stochastic gradient descent method with convex penalty for ill-posed
  problems in Banach spaces},
    author={Ruixue Gu, Zhenwu Fu, Bo Han and Hongsun Fu},
    journal={arXiv preprint arXiv:2409.04973},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.04973},
    primaryClass={math.NA cs.NA}
}