Banach空间中求解线性反问题的对偶梯度流方法  

Dual gradient flow for solving linear inverse problems in Banach spaces

作　　者：金其年 王薇[2,3] Qinian Jin;Wei Wang

机构地区：[1]Mathematical Sciences Institute,Australian National University,Canberra ACT 2601,Australia [2]嘉兴学院数据科学学院,嘉兴314001 [3]浙江省医学电子与数字健康重点实验室,嘉兴314001

出　　处：《中国科学：数学》2023年第10期1377-1396,共20页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:12071184)资助项目。

摘　　要：本文考虑求解不适定问题Ax=y,并确定该问题的R-极小解,其中A:X→Y是Banach空间X到Hilbert空间Y的有界线性算子,R:X→(-∞,∞]为强凸函数.针对数据带噪声的情形,本文研究一种对偶梯度流方法.由于问题的不适定性会导致方法产生半收敛现象,需要选择合适的停止时间以保证重构解的正则性.本文讨论不同的选取方式(如先验选取、偏差原则和启发式偏差原则)下相应的收敛结果,并且基于解的变分源条件建立方法的收敛阶.数值实验结果展示了对偶梯度流方法在求解线性反问题中的有效性.We consider determining the R-minimizing solution of the ill-posed problem Ax=y for a bounded linear operator A:X→Y from a Banach space X to a Hilbert space Y,where R:X→(-∞,∞]is a strongly convex function.A dual gradient flow is proposed to approximate the sought solution by using noisy data.Due to the ill-posedness of the underlying problem,the flow demonstrates the semi-convergence phenomenon and a stopping time should be chosen carefully to find reasonable approximate solutions.We consider the choice of a proper stopping time by various rules such as the a priori rules,the discrepancy principle,and the heuristic discrepancy principle and establish the respective convergence results.Furthermore,convergence rates are derived under the variational source conditions on the sought solution.Numerical results are reported to test the performance of the dual gradient flow for linear inverse problems.

关 键 词：线性反问题 对偶梯度流 收敛阶 变分源条件 偏差原则 启发式偏差原则 

分 类 号：O177.2[理学—数学]