arXiv subjects

Ruixue Gu

Publications and source records attributed to Ruixue Gu.

2 recordsLinked to original sources

arXiv2026

On a posteriori stopping rules of adaptive stochastic heavy ball method for ill-posed problems

In this paper we develop a stochastic heavy ball method for solving ill-posed inverse problems. The method updates the iterate using only a randomly selected equation at each iteration step while incorporating a momentum term into the process. To facilitate fast convergence, we propose an adaptive strategy for selecting the step size and the momentum coefficient. Inspired by the spirit of the discrepancy principle, we introduce an {\it a posteriori} stopping rule for our adaptive stochastic heavy ball method. This rule avoids the need to compute residuals of all equations in the system at every iteration or at fixed frequency intervals, thereby enhancing computational efficiency and practicality. Additionally, convex penalty functions are employed to capture the specific features of the desired solutions. Under suitable conditions, we establish almost sure convergence as well as convergence in expectation. Extensive numerical experiments are conducted to evaluate the performance of the proposed method, demonstrating its efficiency and promising potential for solving large-scale ill-posed problems.

math.NA↗

arXiv2024

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.

math.NA↗