Some results on the regularization of LSQR for large-scale discrete ill-posed problems  

作　　者：HUANG Yi JIA ZhongXiao 

机构地区：[1]Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

出　　处：《Science China Mathematics》2017年第4期701-718,共18页中国科学：数学（英文版）

基　　金：supported by National Basic Research Program of China (Grant No. 2011CB302400);National Natural Science Foundation of China (Grant No. 11371219)

摘　　要：LSQR, a Lanczos bidiagonalization based Krylov subspace iterative method, and its mathematically equivalent conjugate gradient for least squares problems(CGLS) applied to normal equations system, are commonly used for large-scale discrete ill-posed problems. It is well known that LSQR and CGLS have regularizing effects, where the number of iterations plays the role of the regularization parameter. However, it has long been unknown whether the regularizing effects are good enough to find best possible regularized solutions. Here a best possible regularized solution means that it is at least as accurate as the best regularized solution obtained by the truncated singular value decomposition(TSVD) method. We establish bounds for the distance between the k-dimensional Krylov subspace and the k-dimensional dominant right singular space. They show that the Krylov subspace captures the dominant right singular space better for severely and moderately ill-posed problems than for mildly ill-posed problems. Our general conclusions are that LSQR has better regularizing effects for the first two kinds of problems than for the third kind, and a hybrid LSQR with additional regularization is generally needed for mildly ill-posed problems. Exploiting the established bounds, we derive an estimate for the accuracy of the rank k approximation generated by Lanczos bidiagonalization. Numerical experiments illustrate that the regularizing effects of LSQR are good enough to compute best possible regularized solutions for severely and moderately ill-posed problems, stronger than our theory, but they are not for mildly ill-posed problems and additional regularization is needed.LSQR, a Lanczos bidiagonalization based Krylov subspace iterative method, and its mathematically equivalent conjugate gradient for least squares problems （CGLS） applied to normal equations system, are com- monly used for large-scale discrete ill-posed problems. It is well known that LSQR and CGLS have regularizing effects, where the number of iterations plays the role of the regularization parameter. However, it has long been unknown whether the regularizing effects are good enough to find best possible regularized solutions. Here a best possible regularized solution means that it is at least as accurate as the best regularized solution obtained by the truncated singular value decomposition （TSVD） method. We establish bounds for the distance between the k-dimensional Krylov subspace and the k-dimensional dominant right singular space. They show that the Krylov subspace captures the dominant right singular space better for severely and moderately ill-posed prob- lems than for mildly ill-posed problems. Our general conclusions are that LSQR has better regularizing effects for the first two kinds of problems than for the third kind, and a hybrid LSQR with additional regularization is generally needed for mildly ill-posed problems. Exploiting the established bounds, we derive an estimate for the accuracy of the rank k approximation generated by Lanczos bidiagonalization. Numerical experiments illustrate that the regularizing effects of LSQR are good enough to compute best possible regularized solutions for severely and moderately ill-posed problems, stronger than our theory, but they are not for mildly ill-posed problems and additional regularization is needed.

关 键 词：ill-posed problem REGULARIZATION Lanczos bidiagonalization LSQR CGLS hybrid 

分 类 号：O241.6[理学—计算数学]