We are interested in solving large-scale least squares problems in which the coefficient matrices come from the discretization of continuous ill-posed operators, and instead of the exact right-hand side we have a vector contaminated by noise. This kind of problems are known as discrete ill-posed problems and usually arise in the numerical treatment of inverse problems.
When we apply standard methods to this kind of problems we usually obtain meaningless solutions and therefore we must use different techniques to solve them. These techniques are known as regularization methods. We pose the regularization problem as a quadratically constrained least squares problem which is a special case of an optimization problem known as the trust-region subproblem. This discrete ill-posed trust-region subproblem presents a high degree of singularities, which makes its numerical solution a very challenging task.
We have developed a method for the large-scale trust-region subproblem that can handle the singularities associated with discrete ill-posed problem, uses the coefficient matrix only in matrix-vector products and has low and fixed storage requirements.
In this talk we will discuss the regularization problem and present a brief description of our method and numerical results on discrete ill-posed problems from inverse problems.
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