Discrepancy curves for multi-parameter regularization

For solving linear ill-posed problems regularization methods are required when the right-hand side is with some noise. In the present paper regularized solutions are obtained by multi-parameter regularization and the regularization parameters are chosen by a multi-parameter discrepancy principle. Under certain smoothness assumptions we provide order optimal error bounds that characterize the accuracy of the regularized solutions. For the computation of the regularization parameters fast algorithms of Newton type are applied which are based on special transformations. These algorithms are globally and monotonically convergent. Some of our theoretical results are illustrated by numerical experiments. We also show how the proposed approach may be employed for multi-task approximation.