Regularization of Ill-Posed Problems by Envelope Guided Conjugate Gradients

Abstract

We propose a new way to iteratively solve large scale ill-posed problems by exploiting the relation between Tikhonov regularization and multiobjective optimization to obtain, iteratively, approximations to the Tikhonov L-curve and its corner. Monitoring the change of the approximate L-curves allows us to adjust the regularization parameter adaptively during a preconditioned conjugate gradient iteration, so that the desired solution can be reconstructed with a low number of iterations. We apply the technique to an idealized image reconstruction problem in positron emission tomography.

Key Words:

• Envelope
• Ill-posed
• L-curve
• Multiobjective optimization
• Preconditioned conjugate gradients
• Tikhonov regularization