A Regularization Parameter in Discrete Ill-Posed Problems
- 1 May 1996
- journal article
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Scientific Computing
- Vol. 17 (3) , 740-749
- https://doi.org/10.1137/s1064827593252672
Abstract
The Tikhonov regularization method for discrete ill-posed problems is considered. For the practical choice of the regularization parameter $\alpha$, some authors use a plot of the norm of the regularized solution versus the norm of the residual vector for all $\alpha$ considered. This paper contains an analysis of the shape of this plot and gives a theoretical justification for choosing the regularization parameter so it is related to the "L-corner" of the plot considered in the logarithmic scale. Moreover, a new criterion for choosing $\alpha$ is introduced (independent of the shape of the plot) which gives a new interpretation of the "corner criterion" mentioned above. The existence of "L-corner" is discussed.
Keywords
This publication has 10 references indexed in Scilit:
- Solving Least Squares ProblemsPublished by Society for Industrial & Applied Mathematics (SIAM) ,1995
- Analysis of Discrete Ill-Posed Problems by Means of the L-CurveSIAM Review, 1992
- A class of regularization methods for ill-posed problems with nonexact dataNumerical Functional Analysis and Optimization, 1992
- Algorithms for the computation of functionals defined on the solution of a discrete ill-posed problemBIT Numerical Mathematics, 1990
- Truncated Singular Value Decomposition Solutions to Discrete Ill-Posed Problems with Ill-Determined Numerical RankSIAM Journal on Scientific and Statistical Computing, 1990
- Least squares methodsPublished by Elsevier ,1990
- Methods for Solving Incorrectly Posed ProblemsPublished by Springer Nature ,1984
- Generalized Cross-Validation as a Method for Choosing a Good Ridge ParameterTechnometrics, 1979
- Practical Approximate Solutions to Linear Operator Equations When the Data are NoisySIAM Journal on Numerical Analysis, 1977
- Approximate regularized solutions to improperly posed linear integral and operator equationsPublished by Springer Nature ,1974