Solving Severely Ill⁃Posed Linear Systems with Time Discretization Based Iterative Regularization Methods  

求解严重病态线性方程组的基于时间离散的迭代正则化方法

作　　者：GONG Rongfang HUANG Qin 龚荣芳;黄沁(南京航空航天大学理学院,中国南京211106)

机构地区：[1]College of Science,Nanjing University of Aeronautics and Astronautics,Nanjing 211106,P.R.China

出　　处：《Transactions of Nanjing University of Aeronautics and Astronautics》2020年第6期979-994,共16页南京航空航天大学学报（英文版）

基　　金：supported by the Natural Science Foundation of China (Nos. 11971230, 12071215);the Fundamental Research Funds for the Central Universities(No. NS2018047);the 2019 Graduate Innovation Base(Laboratory)Open Fund of Jiangsu Province(No. Kfjj20190804)

摘　　要：Recently,inverse problems have attracted more and more attention in computational mathematics and become increasingly important in engineering applications.After the discretization,many of inverse problems are reduced to linear systems.Due to the typical ill-posedness of inverse problems,the reduced linear systems are often illposed,especially when their scales are large.This brings great computational difficulty.Particularly,a small perturbation in the right side of an ill-posed linear system may cause a dramatical change in the solution.Therefore,regularization methods should be adopted for stable solutions.In this paper,a new class of accelerated iterative regularization methods is applied to solve this kind of large-scale ill-posed linear systems.An iterative scheme becomes a regularization method only when the iteration is early terminated.And a Morozov’s discrepancy principle is applied for the stop criterion.Compared with the conventional Landweber iteration,the new methods have acceleration effect,and can be compared to the well-known acceleratedν-method and Nesterov method.From the numerical results,it is observed that using appropriate discretization schemes,the proposed methods even have better behavior when comparing withν-method and Nesterov method.近年来,反问题在计算数学中得到了越来越多的关注,在工程应用中也越来越重要。许多反问题在离散化后都退化为线性方程组。由于反问题的典型病态性,退化的线性方程组通常也是病态的,特别是当其规模很大时,这就给计算带来了很大的困难。特别地,病态线性方程组右端的一个小扰动可能会引起解的显著变化。因此需要采用正则化方法来获得稳定解。本文应用一类新的加速迭代正则化方法来求解这类大规模不适定线性方程组。一个迭代格式只有在迭代提前终止时才称为正则化方法。本文采用Morozov偏差原理作为终止准则。与传统的Landweber迭代法相比,新方法具有加速效果,可以与著名的ν方法和Nesterov方法相媲美。从数值结果可以看出,采用适当的离散化格式,本文的方法甚至比ν方法和Nesterov方法有更好的行为。

关 键 词：linear system ILL-POSEDNESS LARGE-SCALE iterative regularization methods ACCELERATION 

分 类 号：O241[理学—计算数学]