Analysis of Discrete Ill-Posed Problems by Means of the L-Curve
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.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Regularization helps stabilize the electrophysiological source imaging solution against noise.
“Regularization also helps to stabilize the solution against noise (17).”
From Electrophysiological source imaging: A noninvasive window to brain dynamics (2018) · cites this paper as [17]
Cites
Solutions of Ill-Posed Problems.
Bell, J. B., Tikhonov, A. N., & Arsenin, V. Y. (1978) · Mathematics of Computation · doi:10.2307/2006360 · Cited by 8,096 · see citing works
Solving Least Squares Problems
Lawson, C. L., & Hanson, R. J. (1995) · doi:10.1137/1.9781611971217 · Cited by 7,103 · see citing works
Spline Models for Observational Data.
Müller, H., & Wahba, G. (1991) · Journal of the American Statistical Association · doi:10.2307/2290434 · Cited by 5,016 · see citing works
LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares
Paige, C. C., & Saunders, M. A. (1982) · ACM Transactions on Mathematical Software · doi:10.1145/355984.355989 · Cited by 4,455 · see citing works
Spline Models for Observational Data
Wahba, G. (1990) · doi:10.1137/1.9781611970128 · Cited by 4,184 · see citing works
Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter
Golub, G. H., Heath, M. T., & Wahba, G. (1979) · Technometrics · doi:10.1080/00401706.1979.10489751 · Cited by 3,940 · see citing works
The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems
Hansen, P. C., & O’Leary, D. P. (1993) · SIAM Journal on Scientific Computing · doi:10.1137/0914086 · Cited by 3,158 · see citing works
Self-citationThe Mathematics of Computerized Tomography
Natterer, F., & Wang, G. (2002) · Medical Physics · doi:10.1118/1.1429631 · Cited by 2,570 · see citing works
(1925) · Nature · doi:10.1038/115492c0 · Cited by 1,953 · see citing works
The Mathematics of Computerized Tomography
Natterer, F. (1986) · Vieweg+Teubner eBooks · doi:10.1007/978-3-663-01409-6 · Cited by 1,774 · see citing works
Methods for Solving Incorrectly Posed Problems
Morozov, V. A. (1984) · Springer eBooks · doi:10.1007/978-1-4612-5280-1 · Cited by 1,366 · see citing works
Eigenvalues and Condition Numbers of Random Matrices
Edelman, A. S. (1988) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/0609045 · Cited by 1,331 · see citing works
The theory of Tikhonov regularization for Fredholm equations of the first kind
Groetsch, C. W. (1984) · Medical Entomology and Zoology · Cited by 1,183 · see citing works
Kreß, R. (1989) · Applied mathematical sciences · doi:10.1007/978-3-642-97146-4 · Cited by 839 · see citing works
The truncatedSVD as a method for regularization
Hansen, P. C. (1987) · BIT Numerical Mathematics · doi:10.1007/bf01937276 · Cited by 830 · see citing works
Self-citationIll-posed problems in early vision
Bertero, M., Poggio, T., & Torre, V. (1988) · Proceedings of the IEEE · doi:10.1109/5.5962 · Cited by 724 · see citing works
Skilling, J., & Gull, S. F. (1985) · Springer eBooks · doi:10.1007/978-94-017-2221-6_5 · Cited by 565 · see citing works
Hansen, P. C. (1990) · SIAM Journal on Scientific and Statistical Computing · doi:10.1137/0911028 · Cited by 515 · see citing works
Self-citationLeast Squares Methods for Ill-Posed Problems with a Prescribed Bound
Miller, K. (1970) · SIAM Journal on Mathematical Analysis · doi:10.1137/0501006 · Cited by 510 · see citing works
Inverse Problems of Mathematical Physics
Romanov, V. G. (1986) · De Gruyter eBooks · doi:10.1515/9783110926019 · Cited by 489 · see citing works
The discrete picard condition for discrete ill-posed problems
Hansen, P. C. (1990) · BIT Numerical Mathematics · doi:10.1007/bf01933214 · Cited by 325 · see citing works
Self-citationBjörck, Å. (1990) · Handbook of numerical analysis · doi:10.1016/s1570-8659(05)80036-5 · Cited by 314 · see citing works
Inverse and Ill-Posed Problems
Ryabov, V. M., Lebedeva, A. V., & Pakulina, A. N. (1987) · Elsevier eBooks · doi:10.1016/c2013-0-10621-6 · Cited by 235 · see citing works
Ill-Posed Problems in Natural Sciences
Tikhonov, A. N. (1992) · De Gruyter eBooks · doi:10.1515/9783112313930 · Cited by 193 · see citing works
Regularization,GSVD and truncatedGSVD
Hansen, P. C. (1989) · BIT Numerical Mathematics · doi:10.1007/bf02219234 · Cited by 179 · see citing works
Self-citationRamos, A. A., & Arregui, I. (2018) · Cambridge University Press eBooks · doi:10.1017/9781316182406.003 · Cited by 160 · see citing works
A Practical Examination of Some Numerical Methods for Linear Discrete Ill-Posed Problems
Varah, J. M. (1979) · SIAM Review · doi:10.1137/1021007 · Cited by 154 · see citing works
Pitfalls in the Numerical Solution of Linear Ill-Posed Problems
Varah, J. M. (1983) · SIAM Journal on Scientific and Statistical Computing · doi:10.1137/0904012 · Cited by 141 · see citing works
Regularization for Applied Inverse and III-Posed Problems
(1986) · Vieweg+Teubner eBooks · doi:10.1007/978-3-322-93034-7 · Cited by 129 · see citing works
Bates, R. H. T. (1986) · Physics Bulletin · doi:10.1088/0031-9112/37/4/035 · Cited by 127 · see citing works
Engl, H. W., & Gfrerer, H. (1988) · Applied Numerical Mathematics · doi:10.1016/0168-9274(88)90017-7 · Cited by 103 · see citing works
SIRT- and CG-type methods for the iterative solution of sparse linear least-squares problems
van der Sluis, A., & van der Vorst, H. A. (1990) · Linear Algebra and its Applications · doi:10.1016/0024-3795(90)90215-x · Cited by 103 · see citing works
Improvement of the resolution of an instrument by numerical solution of an integral equation
Shaw, C. B. (1972) · Journal of Mathematical Analysis and Applications · doi:10.1016/0022-247x(72)90259-4 · Cited by 91 · see citing works
Mathematical Programming and the Numerical Solution of Linear Equations.
Relles, D. A., Rust, B. W., & Burrus, W. R. (1976) · Journal of the American Statistical Association · doi:10.2307/2285796 · Cited by 89 · see citing works
Inverse problems in mathematical physics
Prilepko, A. I., Орловский, Д. Г., & Vasin, I. A. (1992) · De Gruyter eBooks · doi:10.1515/9783112313930-045 · Cited by 74 · see citing works
An Augmented Galerkin Method for First Kind Fredholm Equations
Babolian, E., & Delves, L. M. (1979) · IMA Journal of Applied Mathematics · doi:10.1093/imamat/24.2.157 · Cited by 59 · see citing works
THREE TOPICS IN ILL-POSED PROBLEMS
Wahba, G. (1987) · Elsevier eBooks · doi:10.1016/b978-0-12-239040-1.50008-9 · Cited by 49 · see citing works
Inverse and Ill-Posed Problems
Engl, H. W., Groetsch, C. W., & Problems, I.-P. (1987) · Medical Entomology and Zoology · Cited by 43 · see citing works
Calculating the Isochrones of Ventricular Depolarization
Cuppen, J. J. M. (1984) · SIAM Journal on Scientific and Statistical Computing · doi:10.1137/0905008 · Cited by 39 · see citing works
Perturbation bounds for discrete Tikhonov regularisation
Hansen, P. C. (1989) · Inverse Problems · doi:10.1088/0266-5611/5/4/002 · Cited by 26 · see citing works
Self-citationHansen, P. C. (1988) · Journal of Computational and Applied Mathematics · doi:10.1016/0377-0427(88)90336-6 · Cited by 22 · see citing works
Self-citationNumerical treatment of ill-posed problems
Natterer, F. (1986) · Lecture notes in mathematics · doi:10.1007/bfb0072662 · Cited by 21 · see citing works
Relations between SVD and GSVD of Discrete regularization problems in standard and general form
Hansen, P. C. (1990) · Linear Algebra and its Applications · doi:10.1016/0024-3795(90)90315-4 · Cited by 20 · see citing works
Self-citationAsymptotic theory of filtering for linear operator equations with discrete noisy data
Groetsch, C. W., & Vogel, C. R. (1987) · Mathematics of Computation · doi:10.1090/s0025-5718-1987-0906184-2 · Cited by 16 · see citing works
Regularization and the general Gauss-Markov linear model
Zha, H. Y., & Hansen, P. C. (1990) · Mathematics of Computation · doi:10.1090/s0025-5718-1990-1035946-9 · Cited by 15 · see citing works
Self-citationA deterministic approach to optimal regularization —The finite dimensional case—
Kitagawa, T. (1987) · Japan Journal of Applied Mathematics · doi:10.1007/bf03167812 · Cited by 12 · see citing works
Solution of Ill-Posed Problems by Means of Truncated SVD
Hansen, P. C. (1988) · International series of numerical mathematics · doi:10.1007/978-3-0348-6303-2_15 · Cited by 10 · see citing works
Self-citation
Flagged first, then by citations
Cited by
Sorted by citations
Topological defects in epithelia govern cell death and extrusion
Saw, T. B., Doostmohammadi, A., Nier, V., Kocgozlu, L., Thampi, S., Toyama, Y., Marcq, P., Lim, C. T., Yeomans, J. M., & Ladoux, B. (2017) · Nature · doi:10.1038/nature21718 · Cited by 831 · see citing works
Hou, R., & Xia, Y. (2021) · Journal of Sound and Vibration · doi:10.1016/j.jsv.2020.115741 · Cited by 674 · see citing works
Babiloni, F., Cincotti, F., Babiloni, C., Carducci, F., Mattia, D., Astolfi, L., Basilisco, A., Rossini, P. M., Ding, L., Ni, Y., et al. (2005) · NeuroImage · doi:10.1016/j.neuroimage.2004.09.036 · Cited by 412 · see citing works
Electrophysiological source imaging: A noninvasive window to brain dynamics
He, B., Sohrabpour, A., Brown, E., & Liu, Z. (2018) · Annual Review of Biomedical Engineering · doi:10.1146/annurev-bioeng-062117-120853 · Cited by 297 · see citing works
Motor imagery classification by means of source analysis for brain–computer interface applications
Qin, L., Ding, L., & He, B. (2004) · Journal of Neural Engineering · doi:10.1088/1741-2560/1/3/002 · Cited by 239 · see citing works
Noninvasive Brain-Computer Interfaces Based on Sensorimotor Rhythms
He, B., Baxter, B., Edelman, B. J., Cline, C. C., & Ye, W. W. (2015) · Proceedings of the IEEE · doi:10.1109/jproc.2015.2407272 · Cited by 233 · see citing works
Deep neural network processing of DEER data
Worswick, S. G., Spencer, J. A., Jeschke, G., & Kuprov, I. (2018) · Science Advances · doi:10.1126/sciadv.aat5218 · Cited by 212 · see citing works
High-resolution EEG techniques for brain-computer interface applications
Cincotti, F., Mattia, D., Aloise, F., Bufalari, S., Astolfi, L., De Vico Fallani, F., Tocci, A., Bianchi, L., Marciani, M. G., Gao, S., et al. (2008) · Journal of Neuroscience Methods · doi:10.1016/j.jneumeth.2007.06.031 · Cited by 115 · see citing works
Zhang, C. D., & Xu, Y. L. (2016) · Structural Control and Health Monitoring · doi:10.1002/stc.1785 · Cited by 106 · see citing works
fMRI–EEG integrated cortical source imaging by use of time-variant spatial constraints
Liu, Z., & He, B. (2008) · NeuroImage · doi:10.1016/j.neuroimage.2007.10.003 · Cited by 103 · see citing works
Lessons Learned During the Conduct of Clinical Studies in The Dental PBRN
Gilbert, G. H., Richman, J. S., Gordan, V. V., Rindal, D. B., Fellows, J. L., Benjamin, P. L., Wallace‐Dawson, M., Williams, O. D., & for The DPBRN Collaborative Group (2011) · Journal of Dental Education · doi:10.1002/j.0022-0337.2011.75.4.tb05069.x · Cited by 59 · see citing works
An inverse problem approach to optical homodyne tomography
Tan, S. M. (1997) · Journal of Modern Optics · doi:10.1080/09500349708231881 · Cited by 40 · see citing works
Electrophysiological Mapping and Neuroimaging
He, B., & Ding, L. (2013) · Neural Engineering · doi:10.1007/978-1-4614-5227-0_12 · Cited by 26 · see citing works
- 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
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review