Parameter Choice Strategies for Multipenalty Regularization

The widespread applicability of the multipenalty regularization is limited by the fact that theoretically optimal rate of reconstruction for a given problem can be realized by a one-parameter counterpart, provided that relevant information on the problem is available and taken into account in the regularization. In this paper, we explore the situation where no such information is given, but still accuracy of optimal order can be guaranteed by employing multipenalty regularization. Our focus is on the analysis and the justification of an a posteriori parameter choice rule for such a regularization scheme. First we present a modified version of the discrepancy principle within the multipenalty regularization framework. As a consequence we provide a theoretical justification to the multipenalty regularization scheme equipped with the a posteriori parameter choice rule. We then establish a fast numerical realization of the proposed discrepancy principle based on a model function approximation. Finally, we pro...

[1]  Kazufumi Ito,et al.  On the Choice of the Regularization Parameter in Nonlinear Inverse Problems , 1992, SIAM J. Optim..

[2]  R. DeVore.  Optimal computation , 2006 .

[3]  Jun Zou,et al.  An improved model function method for choosing regularization parameters in linear inverse problems , 2002 .

[4]  Sergei V. Pereverzev,et al.  Regularization Theory for Ill-Posed Problems: Selected Topics , 2013 .

[5]  Bernd Hofmann,et al.  A multi-parameter regularization approach for estimating parameters in jump diffusion processes , 2006 .

[6]  Mark A. Lukas,et al.  Comparing parameter choice methods for regularization of ill-posed problems , 2011, Math. Comput. Simul..

[7]  U. TAUTENHAHN,et al.  Dual Regularized Total Least Squares And Multi-Parameter Regularization , 2008 .

[8]  Lixin Shen,et al.  Multi-Parameter Regularization Methods for High-Resolution Image Reconstruction With Displacement Errors , 2007, IEEE Transactions on Circuits and Systems I: Regular Papers.

[9]  Peter Mathé,et al.  The discretized discrepancy principle under general source conditions , 2006, J. Complex..

[10]  A. Tikhonov,et al.  Use of the regularization method in non-linear problems , 1965 .

[11]  V. Morozov.  On the solution of functional equations by the method of regularization , 1966 .

[12]  Bangti Jin,et al.  Multi-Parameter Tikhonov Regularization , 2011, ArXiv.

[13]  F. Natterer.  Error bounds for tikhonov regularization in hilbert scales , 1984 .

[14]  Bernd Hofmann,et al.  How general are general source conditions? , 2008 .

[15]  Markus Grasmair,et al.  Multi-parameter Tikhonov Regularisation in Topological Spaces , 2011, 1109.0364.

[16]  Kalyanmoy Deb,et al.  Multi-objective Optimization , 2014 .

[17]  A. N. Tikhonov,et al.  The regularization of ill-posed problems , 1963 .

[18]  Zhongying,et al.  MULTI-PARAMETER TIKHONOV REGULARIZATION FOR LINEAR ILL-POSED OPERATOR EQUATIONS , 2008 .

[19]  E. Miller,et al.  Efficient determination of multiple regularization parameters in a generalized L-curve framework , 2002 .

[20]  Lea Fleischer,et al.  Regularization of Inverse Problems , 1996 .

[21]  Peiliang Xu,et al.  Multiple Parameter Regularization: Numerical Solutions and Applications to the Determination of Geopotential from Precise Satellite Orbits , 2006 .

[22]  Sergei V. Pereverzev,et al.  Regularization in Hilbert scales under general smoothing conditions , 2005 .

[23]  Shuai Lu,et al.  Multi-parameter regularization and its numerical realization , 2011, Numerische Mathematik.

[24]  P. Mathé,et al.  Geometry of linear ill-posed problems in variable Hilbert scales Inverse Problems 19 789-803 , 2003 .

[25]  Stefan Kindermann,et al.  On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization , 2008 .

[26]  K. Kunisch,et al.  Iterative choices of regularization parameters in linear inverse problems , 1998 .

[27]  K. Ilk.  On the Regularization of Ill-Posed Problems , 1987 .

[28]  Bangti Jin,et al.  Multi-parameter Tikhonov regularization — An augmented approach , 2013 .

[29]  Claude Brezinski,et al.  Multi-parameter regularization techniques for ill-conditioned linear systems , 2003, Numerische Mathematik.

[30]  Valeriya Naumova,et al.  Multi-penalty regularization with a component-wise penalization , 2013 .

[31]  Per Christian Hansen,et al.  REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems , 1994, Numerical Algorithms.