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Iterative regularization for constrained minimization formulations of nonlinear inverse problems.

Barbara Kaltenbacher1, Kha Van Huynh1

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Computational Optimization and Applications
|February 28, 2022
PubMed
Summary

This study presents iterative methods for solving inverse problems, focusing on identifying spatially varying diffusivity in elliptic partial differential equations (PDEs). Numerical experiments demonstrate the effectiveness of these regularization techniques, including for impedance acoustic tomography.

Keywords:
Coefficient identification in elliptic PDEsImpedance acoustic tomographyInverse problemsIterative regularization

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Area of Science:

  • Applied Mathematics
  • Computational Science
  • Imaging Science

Background:

  • Inverse problems are crucial in science and engineering for inferring model parameters from observed data.
  • Traditional methods often face challenges with ill-posedness and require robust solution techniques.
  • Identifying spatially varying properties, like diffusivity, is a key challenge in many inverse problems.

Purpose of the Study:

  • To formulate inverse problems as constrained minimization problems.
  • To develop and analyze iterative solution methods, specifically gradient and Newton-type approaches.
  • To investigate the application of these methods to identify spatially varying diffusivity in elliptic PDEs.

Main Methods:

  • Formulation of inverse problems as constrained minimization problems.
  • Iterative solution using gradient and Newton-type methods.
  • Convergence analysis based on regularization theory.

Main Results:

  • Demonstrated convergence properties of the proposed iterative methods.
  • Successful application to identifying spatially varying diffusivity in elliptic PDEs.
  • Numerical validation using impedance acoustic tomography data.

Conclusions:

  • The proposed constrained minimization framework effectively addresses inverse problems.
  • Iterative regularization methods provide a robust approach for parameter identification.
  • The study validates the utility of impedance acoustic tomography for diffusivity imaging.