Cited by 3,903Numerical methods in inverse problemsMatrix Theory and AlgorithmsAdvanced Statistical Methods and Models

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

Hansen, Per Christian·1992·SIAM Review

Abstract

When discrete ill-posed problems are analyzed and solved by various numerical regularization techniques, a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector. In particular, the graph associated with Tikhonov regularization plays a central role. The main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill-posed problems. The graph is characterized quantitatively, and several important relations between regularized solutions and the graph are derived. It is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic L-shaped “corner” of the graph.

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Crossref · 10.1137/1034115

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Authors
Hansen, P. C.
Corresponding author
Hansen, Per Christian
Venue
SIAM Review
Published
1992-12
Volume
34
Issue
4
Pages
561-580
ISSN
  • 0036-1445 (ISSN-L)
  • 1095-7200
OpenAlex
W2031604650
Cited by
3,903
References
47
Citations by year
2026: 169 · 2025: 147 · 2024: 144 · 2023: 149 · 2022: 178 · 2021: 232 · 2020: 191 · 2019: 185 · 2018: 220 · 2017: 189 · 2016: 201 · 2015: 168 · 2014: 139 · 2013: 154 · 2012: 151
FWCI
21.40
Type
journal article
Language
en
Open access
closed
Locations
SIAM Review (publishedVersion)
Topics
  • Physical Sciences › Mathematics › Mathematical Physics › Numerical methods in inverse problems
  • Physical Sciences › Computer Science › Computational Theory and Mathematics › Matrix Theory and Algorithms
  • Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
Keywords
discrete ill-posed problems, L-curve, Tikhonov regularization, regularization parameter selection, regularized solutions, residual norm, ill-posed inverse problems

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