Premium
Data Based Regularization Matrices for the Tikhonov‐Phillips Regularization
Author(s) -
Huckle Thomas,
Sedlacek Matous
Publication year - 2012
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201210310
Subject(s) - tikhonov regularization , regularization (linguistics) , backus–gilbert method , discretization , mathematics , regularization perspectives on support vector machines , diagonal , matrix (chemical analysis) , block matrix , diagonal matrix , mathematical analysis , computer science , inverse problem , physics , chemistry , eigenvalues and eigenvectors , artificial intelligence , geometry , chromatography , quantum mechanics
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)