Krylov迭代求解器LSQR、CGLS、LSMR和CGME的正则化理论：线性离散病态问题
The Regularization Theory of the Krylov Iterative Solvers LSQR, CGLS, LSMR and CGME For Linear Discrete Ill-Posed Problems
AI总结：
本文分析了LSQR在严重、中等和轻微病态问题中的正则化效果，证明其在半收敛时能获得最佳正则化解，并对比了LSMR和CGME的正则化性能。
摘要：
对于大规模线性离散病态问题$\min\|Ax-b\|$或$Ax=b$（其中$b$受白噪声污染），基于Lanczos bidiagonalization的LSQR及其数学等价的CGLS是最常用的。它们具有内在的正则化效应，其中迭代次数$k$扮演正则化参数的角色。然而，迄今为止，关于Björck和Eldén在1979年提出的长期基本问题仍未有答案：LSQR和CGLS能为哪些问题找到最佳可能的正则化解？这里最佳可能的正则化解意味着其精度至少与通过截断奇异值分解（TSVD）方法或标准形式Tikhonov正则化得到的最佳正则化解相同，并且在某些条件下无法进一步改进。本文对LSQR在严重、中等和轻微病态问题中的正则化进行了详细分析。对于前两种问题，我们证明LSQR在半收敛时能找到最佳可能的解，并且在半收敛之前有以下结果成立：（i）$k$步Lanczos bidiagonalization总是生成$A$的近最佳秩$k$近似；（ii）$k$个Ritz值总是近似$A$的前$k$个大奇异值，按自然顺序排列；（iii）$k$步LSQR总是捕捉$A$的前$k$个主导SVD成分；（iv）Lanczos bidiagonalization生成的 bidiagonal 矩阵的对角线和次对角线衰减速度与$A$的奇异值衰减速度相同。然而，对于第三种问题，上述结果通常不成立。我们还分析了其他两个Krylov求解器LSMR和CGME的正则化，证明LSMR对每种问题都有与LSQR相似的正则化效应，且两者均优于CGME。数值实验验证了我们对LSQR的理论。
英文摘要：
For the large-scale linear discrete ill-posed problem $\min\|Ax-b\|$ or $Ax=b$ with $b$ contaminated by a white noise, Lanczos bidiagonalization based LSQR and its mathematically equivalent CGLS are most commonly used. They have intrinsic regularizing effects, where the number $k$ of iterations plays the role of regularization parameter. However, hitherto there has been no answer to the long-standing fundamental concern of Björck and Eldén in 1979: {\em for which kinds of problems LSQR and CGLS can 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 or by standard-form Tikhonov regularization and cannot be improved under certain conditions. In this paper we make a detailed analysis on the regularization of LSQR for severely, moderately and mildly ill-posed problems. For the first two kinds of problems, we prove that LSQR finds best possible solutions at semi-convergence and the following results hold until semi-convergence: (i) the $k$-step Lanczos bidiagonalization always generates a near best rank $k$ approximation to $A$; (ii) the $k$ Ritz values always approximate the first $k$ large singular values of $A$ in natural order; (iii) the $k$-step LSQR always captures the $k$ dominant SVD components of $A$; (iv) the diagonals and subdiagonals of the bidiagonal matrices generated by Lanczos bidiagonalization decay as fast as the singular values of $A$. However, for the third kind of problem, the above results do not hold generally. We also analyze the regularization of the other two Krylov solvers LSMR and CGME, proving that LSMR has similar regularizing effects to LSQR for each kind of problem and both are superior to CGME. Numerical experiments confirm our theory on LSQR.
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