Iterative regularization for constrained minimization formulations of nonlinear inverse problems

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.