Regularization of Discrete Ill-posed Problems
Marielba RojasPresented at the 1997 CRPC Annual Meeting Poster Session
Large-scale discrete ill-posed problems arise naturally from inverse problems in a variety of fields like medical imaging, geophysics and satellite imaging.
The fact that the coefficient matrices arising in these problems are very ill-conditioned and that the data usually contain a high level of noise, precludes the use of standard methods for linear systems of equations or for least squares problems.
Regularization methods attempt to overcome this difficulty by replacing the original ill-conditioned problem with a better conditioned related problem. The solution of the latter is called a regularized solution for the original problem.
We pose the regularization problem as a quadratically constrained least squares problem, or trust region subproblem. We then apply a recent method for the large scale trust region subproblem that recasts the problem as a parameterized eigenvalue problem. The method relies on the Implicitly Restarted Lanczos Method for solving a sequence of eigenproblems.
We will present the method and some preliminary numerical results, and discuss the difficulties that arise in the ill-conditioned case.
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