# An analysis of a multi-level projected steepest descent iteration for nonlinear inverse problems in Banach spaces subject to stability constraints

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## Abstract

We consider nonlinear inverse problems described by operator equations in Banach spaces. Assuming conditional stability of the inverse problem, that is, assuming that stability holds on a compact, convex subset of the domain of the operator, we introduce a novel nonlinear projected steepest descent iteration and analyze its convergence to an approximate solution given limited accuracy data. We proceed with developing a multi-level algorithm based on a nested family of compact, convex subsets on which stability holds and the stability constants are ordered. Growth of the stability constants is coupled to the increase in accuracy of approximation between neighboring levels to ensure that the algorithm can continue from level to level until the iterate satisfies a desired discrepancy criterion, after a finite number of steps.

## Mathematics Subject Classification

35R30 65J22 47J25## Notes

### Acknowledgments

The research was initiated at the Isaac Newton Institute for Mathematical Sciences (Cambridge, England) during a programme on Inverse Problems in Fall 2011. The authors would like to thank the anonymous referees for their valuable comments and suggestions to improve the quality of the paper.

## References

- 1.Alber, Y.I., Butnariu, D.: Convergence of Bregman projection methods for solving consistent convex feasibility problems in reflexive Banach spaces. J. Optim. Theory Appl.
**92**, 33–61 (1997)CrossRefMATHMathSciNetGoogle Scholar - 2.Alber, Y.I., Kartsatos, A.G., Litsyn, E.: Iterative solution of unstable variational inequalities on approximately given sets. Abstr. Appl. Anal.
**1**(1), 45–64 (1996)CrossRefMATHMathSciNetGoogle Scholar - 3.Alessandrini, G., Vessella, S.: Lipschitz stability for the inverse conductivity problem. Adv. Appl. Math.
**35**(2), 207–241 (2005)CrossRefMATHMathSciNetGoogle Scholar - 4.Ammari, H., Bahouri, H., Dos Santos Ferreira, D., Gallagher, I.: Stability estimates for an inverse scattering problem at high frequencies. ArXiv e-prints (2012)Google Scholar
- 5.Beretta, E., de Hoop, M.V., Qiu, L.: Lipschitz stability of an inverse boundary value problem for a Schrödinger-type equation. SIAM J. Math. Anal.
**45**(2), 679–699 (2013)CrossRefMATHMathSciNetGoogle Scholar - 6.Beretta, E., Francini, E.: Lipschitz stability for the electrical impedance tomography problem: the complex case. Commun. Partial Differ. Equ.
**36**(10), 1723–1749 (2011)CrossRefMATHMathSciNetGoogle Scholar - 7.Bregman, L.M.: The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys.
**7**(3), 200–217 (1967)CrossRefGoogle Scholar - 8.Butnariu, D., Iusem, A.N., Resmerita, E.: Total convexity for powers of the norm in uniformly convex Banach spaces. J. Convex Anal.
**7**(2), 319–334 (2000)MATHMathSciNetGoogle Scholar - 9.Chavent, G., Kunisch, K.: On weakly nonlinear inverse problems. SIAM. J. Appl. Math.
**56**(2), 542–572 (1996)CrossRefMATHMathSciNetGoogle Scholar - 10.Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Mathematics and Its Applications, vol. 62. Kluwer Academic Publishers Group, Dordrecht (1990)Google Scholar
- 11.Daubechies, I., Fornasier, M., Loris, I.: Accelerated projected gradient method for linear inverse problems with sparsity constraints. J. Fourier Anal. Appl.
**14**(5–6), 764–792 (2008)CrossRefMATHMathSciNetGoogle Scholar - 12.de Hoop, M.V., Qiu, L., Scherzer, O.: Local analysis of inverse problems: Hölder stability and iterative reconstruction. Inverse Probl.
**28**(4), 045001 (2012)CrossRefGoogle Scholar - 13.Eicke, B.: Iteration methods for convexly constrained ill-posed problems in Hilbert space. Numer. Funct. Anal. Optim.
**13**(5–6), 413–429 (1992)CrossRefMATHMathSciNetGoogle Scholar - 14.Gilyazov, S.F.: Iterative solution methods for inconsistent linear equations with nonself-adjoint operator. Moscow Univ. Comput. Math. Cybernet.
**13**, 8–13 (1977)Google Scholar - 15.Hanke, M.: Accelerated Landweber iterations for the solution of ill-posed equations. Numer. Math.
**60**(3), 341–373 (1991)CrossRefMATHMathSciNetGoogle Scholar - 16.Kaltenbacher, B.: Toward global convergence for strongly nonlinear ill-posed problems via a regularizing multilevel approach. Numer. Funct. Anal. Optim.
**27**(5–6), 637–665 (2006)CrossRefMATHMathSciNetGoogle Scholar - 17.Kaltenbacher, B.: Convergence rates of a multilevel method for the regularization of nonlinear ill-posed problems. J. Integral Equ. Appl.
**20**(2), 201–228 (2008)CrossRefMATHMathSciNetGoogle Scholar - 18.Kaltenbacher, B., Neubauer, A., Scherzer, O.: Iterative Regularization Methods for Nonlinear Ill-Posed Problems. Radon Series on Computational and Applied Mathematics, vol. 6. Walter de Gruyter GmbH & Co. KG, Berlin (2008)Google Scholar
- 19.Mandache, N.: Exponential instability in an inverse problem for the Schrödinger equation. Inverse Probl.
**17**(5), 1435–1444 (2001)CrossRefMATHMathSciNetGoogle Scholar - 20.Neubauer, A., Scherzer, O.: A convergence rate result for a steepest descent method and a minimal error method for the solution of nonlinear ill-posed problems. Z. Anal. Anwend.
**14**(2), 369–377 (1995)CrossRefMATHMathSciNetGoogle Scholar - 21.Scherzer, O.: A convergence analysis of a method of steepest descent and a two-step algorithm for nonlinear ill-posed problems. Numer. Funct. Anal. Optim.
**17**(1–2), 197–214 (1996)MATHMathSciNetGoogle Scholar - 22.Scherzer, O.: An iterative multi-level algorithm for solving nonlinear ill-posed problems. Numer. Math.
**80**(4), 579–600 (1998)CrossRefMATHMathSciNetGoogle Scholar - 23.Schuster, T., Kaltenbacher, B., Hofmann, B., Kazimierski, K.S.: Regularization Methods in Banach Spaces. Radon Series on Computational and Applied Mathematics, vol. 10. Walter de Gruyter GmbH & Co. KG, Berlin (2012)Google Scholar
- 24.Teschke, G., Borries, C.: Accelerated projected steepest descent method for nonlinear inverse problems with sparsity constraints. Inverse Probl.
**26**(2), 025007 (23 pp) (2010)Google Scholar - 25.Xu, Z.B., Roach, G.F.: Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces. J. Math. Anal. Appl.
**157**(1), 189–210 (1991)CrossRefMATHMathSciNetGoogle Scholar