Numerische Mathematik

, Volume 129, Issue 1, pp 127–148 | Cite as

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

  • Maarten V. de Hoop
  • Lingyun Qiu
  • Otmar Scherzer


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 



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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Maarten V. de Hoop
    • 1
  • Lingyun Qiu
    • 2
  • Otmar Scherzer
    • 3
  1. 1.Center for Computational and Applied MathemematicsPurdue UniversityWest LafayetteUSA
  2. 2.Institute for Mathematics and its ApplicationsUniversity of MinnesotaMinneapolisUSA
  3. 3.Computational Science CenterUniversity of ViennaViennaAustria

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