Data Based Regularization Matrices for the Tikhonov‐Phillips Regularization

In Tikhonov‐Phillips regularization of general form the given ill‐posed linear system is replaced by a Least Squares problem including a minimization of the solution vector x , relative to a seminorm $||L_X||_2$ with some regularization matrix L . Based on the finite difference matrix L k , given by a discretization of the first or second derivative, we introduce the seminorm $||L_k D_{\bar{x}}^{-1} x||_2$ where the diagonal matrix $D_{\bar{x}} := {\rm diag}\, (|\tilde{x}_1|,\ldots,|\tilde{x}_n|)$ and $\tilde{x}$ is the best available approximate solution to x . (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)