Multi-parameter regularization and its numerical realization

In this paper we propose and analyse a choice of parameters in the multi-penalty regularization. A modified discrepancy principle is presented within the multi-parameter regularization framework. An order optimal error bound is obtained under standard smoothness assumptions. We also propose a numerical realization of the multi-parameter discrepancy principle based on the model function approximation. Numerical experiments on a series of test problems support theoretical results. Finally we show how proposed approach can be successfully implemented in Laplacian Regularized Least Squares for learning from labeled and unlabeled examples.