一种自适应动量加速两点梯度法求解不适定反问题

数学物理学报 ›› 2026, Vol. 46 ›› Issue (1): 327-342.

一种自适应动量加速两点梯度法求解不适定反问题

肖显婷(), 何清龙^*()

贵州大学数学与统计学院贵阳 550025

收稿日期:2025-07-02 修回日期:2025-10-21 出版日期:2026-02-26 发布日期:2026-01-19
通讯作者: 何清龙 E-mail:2483186146@qq.com;qlhe@gzu.edu.cn
作者简介:肖显婷, Email: 2483186146@qq.com
基金资助:
国家自然科学基金(11801111);国家自然科学基金(12261021)

An Adaptive Momentum-Accelerated Two-Point Gradient Method for Solving Ill-Posed Inverse Problems

Xianting Xiao(), Qinglong He^*()

School of Mathematics and Statistics, Guizhou University, Guiyang 550025

Received:2025-07-02 Revised:2025-10-21 Online:2026-02-26 Published:2026-01-19
Contact: Qinglong He E-mail:2483186146@qq.com;qlhe@gzu.edu.cn
Supported by:
NSFC(11801111);NSFC(12261021)

摘要/Abstract

摘要：

Landweber 迭代正则化法是求解非线性不适定反问题的一种有效方法, 然而其收敛速度往往较慢, 极大限制了其在实际问题中的应用. 针对 Landweber 方法收敛速度慢的问题, 提出了自适应动量加速两点梯度法 (ATPGM). ATPGM 方法的主要思想是在两点梯度法的梯度方向更新中加入动量项, 充分利用历史梯度信息进行梯度修正, 从而实现加速收敛. 理论上证明了 ATPGM 方法的收敛性和正则性. 数值实验中, 基于一维和二维椭圆参数识别问题, 对 ATPGM 方法, Landweber 迭代正则化方法及两点梯度法 (TPG) 进行对比实验. 数值结果表明, ATPGM 方法在减少迭代次数和运行时间方面具有明显优势, 且其迭代步数对数据噪音不敏感.

关键词: 两点梯度, 非线性不适定反问题, 收敛性分析, 动量系数, 自适应

Abstract:

The Landweber iterative regularization method is an effective approach for solving nonlinear ill-posed inverse problems. However, its convergence speed is often slow, which greatly limits its practical applications. An adaptive two-point gradient method with momentum (ATPGM) is proposed to accelerate the classic Landweber method. The main idea of the ATPGM is that the momentum is introduced into the gradient direction of the two-point gradient method, thus it can take full advantage of the previous gradients information, resulting in a highly fast convergence.The convergence and regularity of ATPGM are given. In numerical experiments, we test the numerical performance of ATPGM, based on one-dimensional and two-dimensional elliptic parameter identification problems. A comprehensive comparision between the Landweber iterative regularization method, the two-point gradient method (TPG) and ATPGM is also presented. The numerical results show that ATPGM performs little better in terms of iterations and running time. Numerical results also show that ATPGM is not sensitive to noise in terms of iterations.

Key words: two-point gradient, nonlinear ill-posed inverse problems, convergence analysis, momentum coefficient, adaptive

中图分类号:

O241.82

肖显婷, 何清龙. 一种自适应动量加速两点梯度法求解不适定反问题[J]. 数学物理学报, 2026, 46(1): 327-342.

Xianting Xiao, Qinglong He. An Adaptive Momentum-Accelerated Two-Point Gradient Method for Solving Ill-Posed Inverse Problems[J]. Acta mathematica scientia,Series A, 2026, 46(1): 327-342.

图/表 4

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