L-curve

Last updated July 01, 2025

L-curve is a visualization method used in the field of regularization in numerical analysis and mathematical optimization.^[1] It represents a logarithmic plot where the norm of a regularized solution is plotted against the norm of the corresponding residual norm. It is useful for picking an appropriate regularization parameter for the given data.^[2]

This method can be applied on methods of regularization of least-square problems, such as Tikhonov regularization and the Truncated SVD,^[2] and iterative methods of solving ill-posed inverse problems, such as the Landweber algorithm, Modified Richardson iteration and Conjugate gradient method.

References

↑ "L-Curve and Curvature Bounds for Tikhonov Regulairzation" (PDF). math.kent.edu. Retrieved June 15, 2025.
1 2 Hansen, P. C. (2001). "The L-curve and its use in the numerical treatment of inverse problems". In Johnston, P. R. (ed.). Computational Inverse Problems in Electrocardiography (PDF). WIT Press. pp. 119–142. ISBN 978-1-85312-614-7.

Hanke, Martin. "Limitations of the L-curve method in ill-posed problems." BIT Numerical Mathematics 36.2 (1996): 287-301.
Engl, Heinz W., and Wilhelm Grever. "Using the L--curve for determining optimal regularization parameters." Numerische Mathematik 69.1 (1994): 25-31.

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[1] "L-Curve and Curvature Bounds for Tikhonov Regulairzation" (PDF). math.kent.edu. Retrieved June 15, 2025.

[Hansen-2] 1 2 Hansen, P. C. (2001). "The L-curve and its use in the numerical treatment of inverse problems". In Johnston, P. R. (ed.). Computational Inverse Problems in Electrocardiography (PDF). WIT Press. pp. 119–142. ISBN 978-1-85312-614-7.

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