THE USE OF THE L-CURVE IN THE REGULARIZATION OF DISCRETE III-POSED PROBLEMS
THE USE OF THE L-CURVE IN THE REGULARIZATION OF DISCRETE III-POSED PROBLEMS
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DOI：
10.1137/0914086
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发表时间：
1993-11-01
影响因子：
3.1
通讯作者：
OLEARY, DP
中科院分区：
文献类型：
--
作者：
HANSEN, PC;OLEARY, DP
正则化算法通常用于对病态问题产生合理的解。对于所有有效的正则化参数，l曲线是正则化解的大小与相应残差大小的关系图。得出了两个主要结果。首先给出了各种正则化方法的统一特征，并表明“尺寸”的测量依赖于所选择的特定正则化方法。例如，2范数适用于Tikhonov正则化，但奇异值分解（SVD）坐标系中的1范数适用于截断的SVD正则化。其次，提出了一种基于l曲线选择正则化参数的新方法，并说明了该方法的有效性。将该方法与广义交叉验证方法进行了比较，结果表明该方法在存在相关误差时具有更强的鲁棒性。
Regularization algorithms are often used to produce reasonable solutions to ill-posed problems. The L-curve is a plot-for all valid regularization parameters-of the size of the regularized solution versus the size of the corresponding residual. Two main results are established. First a unifying characterization of various regularization methods is given and it is shown that the measurement of ''size'' is dependent on the particular regularization method chosen. For example, the 2-norm is appropriate for Tikhonov regularization, but a 1-norm in the coordinate system of the singular value decomposition (SVD) is relevant to truncated SVD regularization. Second, a new method is proposed for choosing the regularization parameter based on the L-curve, and it is shown how this method can be implemented efficiently. The method is compared to generalized cross validation and this new method is shown to be more robust in the presence of correlated errors.