A Hybrid Multilevel-active Set Method for Large Box-constrained Discrete Ill-posed Inverse Problems

Many questions in science and engineering give rise to ill-posed inverse problems whose solution is known to satisfy box constrains, such as nonnegativity. The solution of discretized versions of these problems is highly sensitive to perturbations in the data, discretization errors, and round-off errors introduced during the computations. It is therefore often beneficial to impose known constraints during the solution process. This paper paper describes a two-phase algorithm for the solution of large-scale box-constrained discrete ill-posed problems. The first phase applies a cascadic multilevel method and imposes the constraints on each level by orthogonal projection. The second phase improves the computed approximate solution on the finest level by an active set method. The latter allows several indices of the active set to be updated simultaneously. This reduces the computational effort significantly, when compared to standard active set methods that update one index at a time. Applications to image restoration are presented.