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Variational regularisation for inverse problems with imperfect forward operators and general noise models.

Leon Bungert1,2, Martin Burger1, Yury Korolev3,4

  • 1Department Mathematik, University of Erlangen-Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany.

Inverse Problems
|June 21, 2021
PubMed
Summary

This study introduces variational regularization for inverse problems with uncertain forward operators. It establishes convergence rates for solutions using Bregman distances, applicable to various data fidelity terms.

Keywords:
Banach latticesBregman distancesKullback–Leibler divergenceWasserstein distancesdiscrepancy principlef-divergencesimperfect forward models

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Area of Science:

  • Inverse Problems
  • Functional Analysis
  • Optimization

Background:

  • Inverse problems require regularization due to ill-posedness.
  • Forward operator errors can be modeled using interval arithmetic within Banach lattices.
  • Variational methods offer a robust framework for addressing these challenges.

Purpose of the Study:

  • To develop and analyze variational regularization methods for inverse problems with imperfect forward operators.
  • To investigate existence and convex duality properties for general data fidelity and regularization functionals.
  • To derive convergence rates for regularized solutions under both a priori and a posteriori parameter choices.

Main Methods:

  • Analysis of existence and convex duality for variational regularization.
  • Application of interval arithmetic to model forward operator errors.
  • Derivation of convergence rates using Bregman distances.

Main Results:

  • Established convergence rates for regularized solutions in terms of Bregman distances.
  • Demonstrated applicability to a wide range of data fidelity terms including Wasserstein distances, φ-divergences, and norms.
  • Provided theoretical guarantees for parameter choice rules (a priori and a posteriori).

Conclusions:

  • The proposed variational regularization methods are effective for inverse problems with interval-bounded forward operator errors.
  • The theoretical framework supports the use of diverse data fidelity measures.
  • The results offer a rigorous foundation for practical applications in scientific imaging and data analysis.