# Non-convergence of the L-curve regularization parameter selection method

@article{Vogel1996NonconvergenceOT,
title={Non-convergence of the L-curve regularization parameter selection method},
author={Curtis R. Vogel},
journal={Inverse Problems},
year={1996},
volume={12},
pages={535-547}
}
• C. Vogel
• Published 1 August 1996
• Mathematics
• Inverse Problems
The L-curve method was developed for the selection of regularization parameters in the solution of discrete systems obtained from ill-posed problems. An analysis of this method is given for selecting a parameter for Tikhonov regularization. This analysis, which is carried out in a semi-discrete, semi-stochastic setting, shows that the L-curve approach yields regularized solutions which fail to converge for a certain class of problems. A numerical example is also presented which indicates that…
275 Citations

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## References

SHOWING 1-10 OF 17 REFERENCES

### Limitations of the L-curve method in ill-posed problems

This paper considers the Tikhonov regularization method with the regularization parameter chosen by the so-called L-curve criterion. An infinite dimensional example is constructed for which the

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

• Mathematics
SIAM J. Sci. Comput.
• 1993
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, and a new method is proposed for choosing the regularization parameter based on the L-curve.

### Using the L--curve for determining optimal regularization parameters

• Mathematics
• 1994
Summary. The L--curve'' is a plot (in ordinary or doubly--logarithmic scale) of the norm of (Tikhonov--) regularized solutions of an ill--posed problem versus the norm of the residuals. We show

### A Regularization Parameter in Discrete Ill-Posed Problems

An analysis of the shape of this plot is given and a theoretical justification for choosing the regularization parameter so it is related to the "L-corner" of the plot considered in the logarithmic scale is given.

### Analysis of Discrete Ill-Posed Problems by Means of the L-Curve

The main purpose of this paper is to advocate the use of the graph associated with Tikhonov regularization in the numerical treatment of discrete ill-posed problems, and to demonstrate several important relations between regularized solutions and the graph.

### Convergence rates of approximate least squares solutions of linear integral and operator equations of the first kind

• Mathematics
• 1974
We consider approximations {xn } obtained by moment discretization to (i) the minimal ?2-norm solution of XCx = y where XC is a Hilbert-Schmidt integral operator on ?2, and to (ii) the least squares

### Optimal choice of a truncation level for the truncated SVD solution of linear first kind integral equations when data are noisy

Given error contaminated discrete data $z_i = \int_0^1 {k(s_i ,t)f(t)dt + \varepsilon _i }$, $i = 1, \cdots ,n$, we apply the truncated singular value decomposition to find an approximate solution

### Spline Models for Observational Data

Foreword 1. Background 2. More splines 3. Equivalence and perpendicularity, or, what's so special about splines? 4. Estimating the smoothing parameter 5. 'Confidence intervals' 6. Partial spline

### Inverse Problems in the Mathematical Sciences

Contents: Introduction - Inverse problems modeled by integral equations of the first kind: Causation - Parameter estimation in differential equations: Model identification - Mathematical background

### Practical Approximate Solutions to Linear Operator Equations When the Data are Noisy

• G. Wahba
• Mathematics, Computer Science
• 1977
It is shown that the weighted cross-validation estimate of $\hat \lambda$ estimates the value of $\lambda$ which minimizes $({1 / n) E\sum\nolimits_{j = 1}^n {[(\mathcal{K}f_{n,\lambda } )(t_j ) - (\mathcal(K)f)(t-j )]} ^2$ .