Multi-penalty regularization with a component-wise penalization

We discuss a new regularization scheme for reconstructing the solution of a linear ill-posed operator equation from given noisy data in the Hilbert space setting. In this new scheme, the regularized approximation is decomposed into several components, which are defined by minimizing a multi-penalty functional. We show theoretically and numerically that under a proper choice of the regularization parameters, the regularized approximation exhibits the so-called compensatory property, in the sense that it performs similar to the best of the single-penalty regularization with the same penalizing operator.