Computing quasisolutions of nonlinear inverse problems via efficient minimization of trust region problems
-
Barbara Kaltenbacher
,
Franz Rendl
Abstract
In this paper we present a method for the regularized solution of nonlinear inverse problems, based on Ivanov regularization (also called method of quasi solutions or constrained least squares regularization). This leads to the minimization of a nonconvex cost function under a norm constraint, where nonconvexity is caused by nonlinearity of the inverse problem. Minimization is done by iterative approximation, using (nonconvex) quadratic Taylor expansions of the cost function. This leads to repeated solution of quadratic trust region subproblems with possibly indefinite Hessian. Thus, the key step of the method consists in application of an efficient method for solving such quadratic subproblems, developed by Rendl and Wolkowicz [10]. We here present a convergence analysis of the overall method as well as numerical experiments.
Funding statement: This work is supported by the Initial Training Network Mixed Integer Nonlinear Optimization (MINO) of the European Union and the ICT COST action TD1207, as well as by Karl Popper Kolleg Modeling-Simulation-Optimization funded by the Alpen-Adria-Universität Klagenfurt and by the Carinthian Economic Promotion Fund (KWF).
We wish to thank both reviewers for fruitful comments leading to an improved version of the manuscript.
References
[1] Clason C., Kaltenbacher B. and Klassen A., On convergence and convergence rates for Ivanov and Morozov regularization, in preparation. Suche in Google Scholar
[2] Grodzevich O. and Wolkowicz H., Regularization using a parameterized trust region subproblem, Math. Program. Ser. B 116 (2009), 193–220. 10.1007/s10107-007-0126-4Suche in Google Scholar
[3] Hanke M., A regularization Levenberg–Marquardt scheme, with applications to inverse groundwater filtration problems, Inverse Problems 13 (1997), 79–95. 10.1088/0266-5611/13/1/007Suche in Google Scholar
[4] Ivanov V. K., On linear problems which are not well-posed, Dokl. Akad. Nauk SSSR 145 (1962), 270–272. Suche in Google Scholar
[5] Ivanov V. K., On ill-posed problems, Mat. Sb. (N.S.) 61(103) (1963), 211–223. Suche in Google Scholar
[6] Ivanov V. K., Vasin V. V. and Tanana V. P., Theory of Linear Ill-posed Problems and Its Applications, Inverse Ill-Posed Problems Ser., VSP, Utrecht, 2002. 10.1515/9783110944822Suche in Google Scholar
[7] Kaltenbacher B., Kirchner A. and Vexler B., Adaptive discretizations for the choice of a Tikhonov regularization parameter in nonlinear inverse problems, Inverse Problems 27 (2011), Article ID 125008. 10.1088/0266-5611/27/12/125008Suche in Google Scholar
[8] Lorenz D. and Worliczek N., Necessary conditions for variational regularization schemes, Inverse Problems 29 (2013), Article ID 075016. 10.1088/0266-5611/29/7/075016Suche in Google Scholar
[9] Neubauer A. and Ramlau R., On convergence rates for quasi-solutions of ill-posed problems, Electron. Trans. Numer. Anal. 41 (2014), 81–92. Suche in Google Scholar
[10] Rendl F. and Wolkowicz H., A semidefinite framework for trust region subproblems with applications to large scale minimization, Math. Program. 77 (1997), 273–299. 10.1007/BF02614438Suche in Google Scholar
[11] Seidman T. I. and Vogel C. R., Well posedness and convergence of some regularisation methods for nonlinear ill-posed problems, Inverse Problems 5 (1989), 227–238. 10.1088/0266-5611/5/2/008Suche in Google Scholar
[12] Sorensen D., Newton’s method with a model trust region modification, SIAM J. Numer. Anal. 19 (1982), 409–426. 10.2172/6836252Suche in Google Scholar
[13] Vogel C. R., A constrained least squares regularization method for nonlinear iii-posed problems, SIAM J. Control Optim. 28 (1990), 34–49. 10.1137/0328002Suche in Google Scholar
© 2016 by De Gruyter
Artikel in diesem Heft
- 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
Artikel in diesem Heft
- 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
