On convergence rates for asymptotic discrepancy principle
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Anatoly Bakushinsky
Abstract
A series of recent numerical experiments for parameter estimation inverse problems in epidemiology [7, 6] have indicated that applicability of the discrepancy principle (DP) does not depend on the structure of a particular regularizing operator. This observation confirmed the original theoretical analysis by Bakushinsky [1, 2] on the construction of fairly general stabilizing algorithms in Banach and Hilbert spaces. In [7, 6], a unified approach to the implementation of the DP for linear ill-posed problems, the Abstract Discrepancy Principle, has been proposed and justified. The current paper investigates the convergence rates of the ADP. Special cases of sectorial and self-adjoint operators are studied.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1112897
Funding statement: This work is supported by NSF under grant (DMS-1112897).
References
[1] Bakushinsky A., Principle of the residual in the case of a perturbed operator for general regularizing algorithms (in Russian), Zh. Vychisl. Mat. i Mat. Fiz. 22 (1982), 989–993. 10.1016/0041-5553(82)90024-6Search in Google Scholar
[2] Bakushinsky A., Regularizing algorithms in a Banach space that are based on the generalized residual principle (in Russian), Ill-Posed Problems of Mathematical Physics and Analysis, Nauka, Novosibirsk (1984), 18–21. Search in Google Scholar
[3] Bakushinsky A. B. and Kokurin M. Y., Iterative Methods for Ill-Posed Operator Equations with Smooth Operators, Springer, Dordrecht, 2004. Search in Google Scholar
[4] Haase M., Functional Calculus for Sectorial Operators, Birkhäuser, Basel, 2006. 10.1007/3-7643-7698-8Search in Google Scholar
[5] Morozov V. A., Methods for Solving Incorrectly Posed Problems, Springer, Berlin, 1984. 10.1007/978-1-4612-5280-1Search in Google Scholar
[6] Smirnova A., Bakushinsky A. and DeCamp L., On stable parameter estimation in epidemiology using abstract discrepancy principle, Proceedings of the Inverse Problems from Theory to Applications Conference (IPTA 2014), IOP Publishing, Bristol (2014), 63–67. Search in Google Scholar
[7] Smirnova A., Bakushinsky A. and DeCamp L., On application of asymptotic generalized discrepancy principle to the analysis of epidemiology models, Appl. Anal. 94 (2015), no. 4, 672–693. 10.1080/00036811.2014.898272Search in Google Scholar
[8] Tikhonov A., Leonov A. and Yagola A., Nonlinear Ill-Posed Problems. Vol. 1 and 2, Chapman and Hall, London, 1998. 10.1007/978-94-017-5167-4_1Search in Google Scholar
[9] Vainikko G. and Veretennikov A., Iterative Procedures in Ill-Posed Problems, Nauka, Moscow, 1986. Search in Google Scholar
© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- Inversions of the windowed ray transform
- Inverse problems of demand analysis and their applications to computation of positively-homogeneous Konüs–Divisia indices and forecasting
- Some generalizations for Landweber iteration for nonlinear ill-posed problems in Hilbert scales
- An inverse problem for Sturm–Liouville operators with non-separated boundary conditions containing the spectral parameter
- Use of difference-based methods to explore statistical and mathematical model discrepancy in inverse problems
- Computing quasisolutions of nonlinear inverse problems via efficient minimization of trust region problems
- Accuracy estimates of Gauss–Newton-type iterative regularization methods for nonlinear equations with operators having normally solvable derivative at the solution
- On convergence rates for asymptotic discrepancy principle
- Identification of an unknown coefficient in KdV equation from final time measurement
- Improved asymptotic analysis for dynamical probe method
Articles in the same Issue
- Frontmatter
- Inversions of the windowed ray transform
- Inverse problems of demand analysis and their applications to computation of positively-homogeneous Konüs–Divisia indices and forecasting
- Some generalizations for Landweber iteration for nonlinear ill-posed problems in Hilbert scales
- An inverse problem for Sturm–Liouville operators with non-separated boundary conditions containing the spectral parameter
- Use of difference-based methods to explore statistical and mathematical model discrepancy in inverse problems
- Computing quasisolutions of nonlinear inverse problems via efficient minimization of trust region problems
- Accuracy estimates of Gauss–Newton-type iterative regularization methods for nonlinear equations with operators having normally solvable derivative at the solution
- On convergence rates for asymptotic discrepancy principle
- Identification of an unknown coefficient in KdV equation from final time measurement
- Improved asymptotic analysis for dynamical probe method
