Abstract
We consider an ill-posed problem Ta = f* in Hilbert spaces and suppose that the linear bounded operator T is approximately available, with a known estimate for the operator perturbation at the solution. As a numerical scheme the CGNR-method is considered, that is, the classical method of conjugate gradients by Hestenes and Stiefel applied to the associated normal equations. Two a posteriori stopping rules are introduced, and convergence results are provided for the corresponding approximations, respectively. As a specific application, a parameter estimation problem is considered.
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Plato, R. The conjugate gradient method for linear ill-posed problems with operator perturbations. Numerical Algorithms 20, 1–22 (1999). https://doi.org/10.1023/A:1019139414435
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DOI: https://doi.org/10.1023/A:1019139414435