Abstract
The L-curve criterion is often applied to determine a suitable value of the regularization parameter when solving ill-conditioned linear systems of equations with a right-hand side contaminated by errors of unknown norm. However, the computation of the L-curve is quite costly for large problems; the determination of a point on the L-curve requires that both the norm of the regularized approximate solution and the norm of the corresponding residual vector be available. Therefore, usually only a few points on the L-curve are computed and these values, rather than the L-curve, are used to determine a value of the regularization parameter. We propose a new approach to determine a value of the regularization parameter based on computing an L-ribbon that contains the L-curve in its interior. An L-ribbon can be computed fairly inexpensively by partial Lanczos bidiagonalization of the matrix of the given linear system of equations. A suitable value of the regularization parameter is then determined from the L-ribbon, and we show that an associated approximate solution of the linear system can be computed with little additional work.
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Calvetti, D., Golub, G.H. & Reichel, L. Estimation of the L-Curve via Lanczos Bidiagonalization. BIT Numerical Mathematics 39, 603–619 (1999). https://doi.org/10.1023/A:1022383005969
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DOI: https://doi.org/10.1023/A:1022383005969