The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems

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. 1. Introduction. In many applications such as spectroscopy [1], seismography [13], and medical imaging [11], data are gathered by convolution of a noisy signal with a detector. A linear model of this process leads to an integral equation of the first kind: