Noise representation in residuals of LSQR, LSMR, and CRAIG regularization
Noise representation in residuals of LSQR, LSMR, and CRAIG regularization
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DOI：
10.1016/j.laa.2017.07.031
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发表时间：
2016-12
影响因子：
1.1
通讯作者：
Iveta Hnvetynkov'a;Marie Kub'inov'a;Martin Plevsinger
中科院分区：
文献类型：
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作者：
Iveta Hnvetynkov'a;Marie Kub'inov'a;Martin Plevsinger
Golub-Kahan迭代双对角化是求解大型线性噪声污染不适定问题的几种正则化方法中的核心算法。我们考虑一个一般的噪声设置，并推导出（噪声污染）双对角化向量和基于双对角化的正则化方法LSQR，LSMR和克雷格的残差之间的显式关系。对于LSQR和LSMR残差，我们证明了计算的双对角化向量的线性组合的系数反映了这些向量中的每个向量中的传播噪声的量。对于克雷格，残差只是特定双对角化向量的倍数。我们展示了它的大小如何表示在每次迭代中的正则化效果，通过表示克雷格解决方案作为修改后的兼容问题的精确解。较大的二维问题的结果的有效性和正交性的损失的影响进行了讨论。
Golub–Kahan iterative bidiagonalization represents the core algorithm in several regularization methods for solving large linear noise-polluted ill-posed problems. We consider a general noise setting and derive explicit relations between (noise contaminated) bidiagonalization vectors and the residuals of bidiagonalization-based regularization methods LSQR, LSMR, and CRAIG. For LSQR and LSMR residuals we prove that the coefficients of the linear combination of the computed bidiagonalization vectors reflect the amount of propagated noise in each of these vectors. For CRAIG the residual is only a multiple of a particular bidiagonalization vector. We show how its size indicates the regularization effect in each iteration by expressing the CRAIG solution as the exact solution to a modified compatible problem. Validity of the results for larger two-dimensional problems and influence of the loss of orthogonality is also discussed.