# American Institute of Mathematical Sciences

• Previous Article
Anisotropic total variation regularized $L^1$ approximation and denoising/deblurring of 2D bar codes
• IPI Home
• This Issue
• Next Article
A fast algorithm for global minimization of maximum likelihood based on ultrasound image segmentation
August  2011, 5(3): 619-643. doi: 10.3934/ipi.2011.5.619

## Errors of regularisation under range inclusions using variable Hilbert scales

 1 Centre for Mathematics and its Applications, The Australian National University, Canberra ACT, 0200, Australia 2 Department of Mathematics, Chemnitz University of Technology, 09107 Chemnitz, Germany

Received  May 2010 Revised  June 2011 Published  August 2011

Based on the variable Hilbert scale interpolation inequality, bounds for the error of regularisation methods are derived under range inclusions. In this context, new formulae for the modulus of continuity of the inverse of bounded operators with non-closed range are given. Even if one can show the equivalence of this approach to the version used previously in the literature, the new formulae and corresponding conditions are simpler than the former ones. Several examples from image processing and spectral enhancement illustrate how the new error bounds can be applied.
Citation: Markus Hegland, Bernd Hofmann. Errors of regularisation under range inclusions using variable Hilbert scales. Inverse Problems & Imaging, 2011, 5 (3) : 619-643. doi: 10.3934/ipi.2011.5.619
##### References:

show all references

##### References:
 [1] Haixia Yu. Hilbert transforms along double variable fractional monomials. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1433-1446. doi: 10.3934/cpaa.2019069 [2] Thorsten Hohage, Mihaela Pricop. Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise. Inverse Problems & Imaging, 2008, 2 (2) : 271-290. doi: 10.3934/ipi.2008.2.271 [3] G. M. de Araújo, S. B. de Menezes. On a variational inequality for the Navier-Stokes operator with variable viscosity. Communications on Pure & Applied Analysis, 2006, 5 (3) : 583-596. doi: 10.3934/cpaa.2006.5.583 [4] Jean Dolbeault, Maria J. Esteban, Michał Kowalczyk, Michael Loss. Improved interpolation inequalities on the sphere. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 695-724. doi: 10.3934/dcdss.2014.7.695 [5] Charles Fefferman. Interpolation by linear programming I. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 477-492. doi: 10.3934/dcds.2011.30.477 [6] Hong-Gunn Chew, Cheng-Chew Lim. On regularisation parameter transformation of support vector machines. Journal of Industrial & Management Optimization, 2009, 5 (2) : 403-415. doi: 10.3934/jimo.2009.5.403 [7] Konstantinos Papafitsoros, Kristian Bredies. A study of the one dimensional total generalised variation regularisation problem. Inverse Problems & Imaging, 2015, 9 (2) : 511-550. doi: 10.3934/ipi.2015.9.511 [8] Yvon Maday, Ngoc Cuong Nguyen, Anthony T. Patera, S. H. Pau. A general multipurpose interpolation procedure: the magic points. Communications on Pure & Applied Analysis, 2009, 8 (1) : 383-404. doi: 10.3934/cpaa.2009.8.383 [9] Irene Benedetti, Luisa Malaguti, Valentina Taddei. Nonlocal problems in Hilbert spaces. Conference Publications, 2015, 2015 (special) : 103-111. doi: 10.3934/proc.2015.0103 [10] Fritz Gesztesy, Rudi Weikard, Maxim Zinchenko. On a class of model Hilbert spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5067-5088. doi: 10.3934/dcds.2013.33.5067 [11] Doǧan Çömez. The modulated ergodic Hilbert transform. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 325-336. doi: 10.3934/dcdss.2009.2.325 [12] Ilesanmi Adeboye, Harrison Bray, David Constantine. Entropy rigidity and Hilbert volume. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1731-1744. doi: 10.3934/dcds.2019075 [13] Małgorzata Wyrwas, Dorota Mozyrska, Ewa Girejko. Subdifferentials of convex functions on time scales. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 671-691. doi: 10.3934/dcds.2011.29.671 [14] John Sylvester. An estimate for the free Helmholtz equation that scales. Inverse Problems & Imaging, 2009, 3 (2) : 333-351. doi: 10.3934/ipi.2009.3.333 [15] Maxime Breden, Jean-Philippe Lessard. Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2825-2858. doi: 10.3934/dcdsb.2018164 [16] Anita Mayo. Accurate two and three dimensional interpolation for particle mesh calculations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1205-1228. doi: 10.3934/dcdsb.2012.17.1205 [17] V. Rehbock, K.L. Teo, L.S. Jennings. Suboptimal feedback control for a class of nonlinear systems using spline interpolation. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 223-236. doi: 10.3934/dcds.1995.1.223 [18] Silvia Allavena, Michele Piana, Federico Benvenuto, Anna Maria Massone. An interpolation/extrapolation approach to X-ray imaging of solar flares. Inverse Problems & Imaging, 2012, 6 (2) : 147-162. doi: 10.3934/ipi.2012.6.147 [19] Lucio Boccardo, Daniela Giachetti. A nonlinear interpolation result with application to the summability of minima of some integral functionals. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 31-42. doi: 10.3934/dcdsb.2009.11.31 [20] Rolando Mosquera, Aziz Hamdouni, Abdallah El Hamidi, Cyrille Allery. POD basis interpolation via Inverse Distance Weighting on Grassmann manifolds. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1743-1759. doi: 10.3934/dcdss.2019115

2018 Impact Factor: 1.469