February  2016, 10(1): 1-25. doi: 10.3934/ipi.2016.10.1

On the choice of the Tikhonov regularization parameter and the discretization level: A discrepancy-based strategy

1. 

Computational Science Center, University of Vienna, Oskar Morgenstern-Platz 1, 1090 Vienna, Austria

2. 

Institute of Mathematics, Statistics and Physics, Federal University of Rio Grande, Av. Italia km 8, 96201-900 Rio Grande, Brazil

3. 

Instituto Nacional de Matemática Pura e Aplicada, Rio do Janeiro, RJ 22460-320

Received  October 2014 Revised  September 2015 Published  February 2016

We address the classical issue of appropriate choice of the regularization and discretization level for the Tikhonov regularization of an inverse problem with imperfectly measured data. We focus on the fact that the proper choice of the discretization level in the domain together with the regularization parameter is a key feature in adequate regularization. We propose a discrepancy-based choice for these quantities by applying a relaxed version of Morozov's discrepancy principle. Indeed, we prove the existence of the discretization level and the regularization parameter satisfying such discrepancy. We also prove associated regularizing properties concerning the Tikhonov minimizers. We conclude by presenting some numerical examples of interest.
Citation: Vinicius Albani, Adriano De Cezaro, Jorge P. Zubelli. On the choice of the Tikhonov regularization parameter and the discretization level: A discrepancy-based strategy. Inverse Problems & Imaging, 2016, 10 (1) : 1-25. doi: 10.3934/ipi.2016.10.1
References:
[1]

V. Albani, A. De Cezaro and J. Zubelli, Convex regularization of local volatility estimation in a discrete setting,, Submitted (SSRN ID: 2308138), (2308). doi: 10.2139/ssrn.2308138. Google Scholar

[2]

V. Albani and J. P. Zubelli, Online local volatility calibration by convex regularization,, Appl. Anal. Discrete Math., 8 (2014), 243. doi: 10.2298/AADM140811012A. Google Scholar

[3]

S. Anzengruber, B. Hofmann and P. Mathé, Regularization properties of the sequential discrepancy principle for Tikhonov regularization in Banach spaces,, Appl. Anal., 93 (2014), 1382. doi: 10.1080/00036811.2013.833326. Google Scholar

[4]

S. Anzengruber and R. Ramlau, Morozov's discrepancy principle for Tikhonov-type functionals with nonlinear operators,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/2/025001. Google Scholar

[5]

S. Anzengruber and R. Ramlau, Convergence rates for Morozov's discrepancy principle using variational inequalities,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/10/105007. Google Scholar

[6]

T. Bonesky, Morozov's discrepancy principle and Tikhonov-type functionals,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/1/015015. Google Scholar

[7]

S. Crepey, Calibration of the local volatility in a generalized Black-Scholes model using Tikhonov regularization,, SIAM J. Math. Anal., 34 (2003), 1183. doi: 10.1137/S0036141001400202. Google Scholar

[8]

A. De Cezaro, O. Scherzer and J. Zubelli, Convex regularization of local volatility models from option prices: Convergence analysis and rates,, Nonlinear Anal., 75 (2012), 2398. doi: 10.1016/j.na.2011.10.037. Google Scholar

[9]

H. Egger and H. Engl, Tikhonov regularization applied to the inverse problem of option pricing: Convergence analysis and rates,, Inverse Problems, 21 (2005), 1027. doi: 10.1088/0266-5611/21/3/014. Google Scholar

[10]

I. Ekland and R. Teman, Convex Analysis and Variational Problems,, North Holland, (1976). Google Scholar

[11]

H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems,, Mathematics and its Applications, (1996). Google Scholar

[12]

C. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of First Kind,, 1st edition, (1984). Google Scholar

[13]

C. Groetsch and A. Neubauer, Regularization of ill-posed problems: Optimal parameter choice in finite dimensions,, J. Approx. Theory, 58 (1989), 184. doi: 10.1016/0021-9045(89)90019-1. Google Scholar

[14]

B. Hofmann and P. Mathé, Parameter choice in Banach space regularization under variational inequalities,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/10/104006. Google Scholar

[15]

B. Hofmann and O. Scherzer, Local ill-posedness and source conditions of operator equations in Hilbert spaces,, Inverse Problems, 14 (1998), 1189. doi: 10.1088/0266-5611/14/5/007. Google Scholar

[16]

B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear ill-Posed Problems,, Walter de Gruyter, (2008). doi: 10.1515/9783110208276. Google Scholar

[17]

A. Kirsch and A. Rieder, Seismic tomography is lolocal ill-posed,, Inverse Problems, 30 (2014). doi: 10.1088/0266-5611/30/12/125001. Google Scholar

[18]

J. Lindenstrauss, On nonlinear projections in Banach spaces,, Michigan Math. J., 11 (1964), 263. doi: 10.1307/mmj/1028999141. Google Scholar

[19]

V. Morozov, On the solution of functional equations by the method of regularization,, Dokl. Math., 7 (1966), 414. Google Scholar

[20]

A. Neubauer and O. Scherzer, Finite-dimensional approximation of tikhonov regularized solutions of non-linear ill-posed problems,, Numer. Funct. Anal. Optim., 11 (1990), 85. doi: 10.1080/01630569008816362. Google Scholar

[21]

J. Nocedal and S. Wright, Numerical Optimization,, Springer Series in Operations Research and Financial Engineering, (2006). Google Scholar

[22]

C. Pöschl, E. Resmerita and O. Scherzer, Discretization of variational regularization in Banach spaces,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/10/105017. Google Scholar

[23]

E. Resmerita and R. Anderssen, Joint additive Kullback-Leibler residual minimization and regularization for linear inverse problems,, Math. Methods Appl. Sci., 30 (2007), 1527. doi: 10.1002/mma.855. Google Scholar

[24]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging,, Applied Mathematical Sciences, (2008). Google Scholar

[25]

T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods in Banach Spaces,, Walter de Gruyter, (2012). doi: 10.1515/9783110255720. Google Scholar

[26]

E. Somersalo and J. Kapio, Statistical and Computational Inverse Problems,, Applied Mathematical Sciences, (2004). doi: 10.1007/978-3-662-08966-8. Google Scholar

[27]

A. Tikhonov and V. Arsenin, Nonlinear Ill-posed Problems,, Chapman and Hall, (1998). Google Scholar

show all references

References:
[1]

V. Albani, A. De Cezaro and J. Zubelli, Convex regularization of local volatility estimation in a discrete setting,, Submitted (SSRN ID: 2308138), (2308). doi: 10.2139/ssrn.2308138. Google Scholar

[2]

V. Albani and J. P. Zubelli, Online local volatility calibration by convex regularization,, Appl. Anal. Discrete Math., 8 (2014), 243. doi: 10.2298/AADM140811012A. Google Scholar

[3]

S. Anzengruber, B. Hofmann and P. Mathé, Regularization properties of the sequential discrepancy principle for Tikhonov regularization in Banach spaces,, Appl. Anal., 93 (2014), 1382. doi: 10.1080/00036811.2013.833326. Google Scholar

[4]

S. Anzengruber and R. Ramlau, Morozov's discrepancy principle for Tikhonov-type functionals with nonlinear operators,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/2/025001. Google Scholar

[5]

S. Anzengruber and R. Ramlau, Convergence rates for Morozov's discrepancy principle using variational inequalities,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/10/105007. Google Scholar

[6]

T. Bonesky, Morozov's discrepancy principle and Tikhonov-type functionals,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/1/015015. Google Scholar

[7]

S. Crepey, Calibration of the local volatility in a generalized Black-Scholes model using Tikhonov regularization,, SIAM J. Math. Anal., 34 (2003), 1183. doi: 10.1137/S0036141001400202. Google Scholar

[8]

A. De Cezaro, O. Scherzer and J. Zubelli, Convex regularization of local volatility models from option prices: Convergence analysis and rates,, Nonlinear Anal., 75 (2012), 2398. doi: 10.1016/j.na.2011.10.037. Google Scholar

[9]

H. Egger and H. Engl, Tikhonov regularization applied to the inverse problem of option pricing: Convergence analysis and rates,, Inverse Problems, 21 (2005), 1027. doi: 10.1088/0266-5611/21/3/014. Google Scholar

[10]

I. Ekland and R. Teman, Convex Analysis and Variational Problems,, North Holland, (1976). Google Scholar

[11]

H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems,, Mathematics and its Applications, (1996). Google Scholar

[12]

C. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of First Kind,, 1st edition, (1984). Google Scholar

[13]

C. Groetsch and A. Neubauer, Regularization of ill-posed problems: Optimal parameter choice in finite dimensions,, J. Approx. Theory, 58 (1989), 184. doi: 10.1016/0021-9045(89)90019-1. Google Scholar

[14]

B. Hofmann and P. Mathé, Parameter choice in Banach space regularization under variational inequalities,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/10/104006. Google Scholar

[15]

B. Hofmann and O. Scherzer, Local ill-posedness and source conditions of operator equations in Hilbert spaces,, Inverse Problems, 14 (1998), 1189. doi: 10.1088/0266-5611/14/5/007. Google Scholar

[16]

B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear ill-Posed Problems,, Walter de Gruyter, (2008). doi: 10.1515/9783110208276. Google Scholar

[17]

A. Kirsch and A. Rieder, Seismic tomography is lolocal ill-posed,, Inverse Problems, 30 (2014). doi: 10.1088/0266-5611/30/12/125001. Google Scholar

[18]

J. Lindenstrauss, On nonlinear projections in Banach spaces,, Michigan Math. J., 11 (1964), 263. doi: 10.1307/mmj/1028999141. Google Scholar

[19]

V. Morozov, On the solution of functional equations by the method of regularization,, Dokl. Math., 7 (1966), 414. Google Scholar

[20]

A. Neubauer and O. Scherzer, Finite-dimensional approximation of tikhonov regularized solutions of non-linear ill-posed problems,, Numer. Funct. Anal. Optim., 11 (1990), 85. doi: 10.1080/01630569008816362. Google Scholar

[21]

J. Nocedal and S. Wright, Numerical Optimization,, Springer Series in Operations Research and Financial Engineering, (2006). Google Scholar

[22]

C. Pöschl, E. Resmerita and O. Scherzer, Discretization of variational regularization in Banach spaces,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/10/105017. Google Scholar

[23]

E. Resmerita and R. Anderssen, Joint additive Kullback-Leibler residual minimization and regularization for linear inverse problems,, Math. Methods Appl. Sci., 30 (2007), 1527. doi: 10.1002/mma.855. Google Scholar

[24]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging,, Applied Mathematical Sciences, (2008). Google Scholar

[25]

T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods in Banach Spaces,, Walter de Gruyter, (2012). doi: 10.1515/9783110255720. Google Scholar

[26]

E. Somersalo and J. Kapio, Statistical and Computational Inverse Problems,, Applied Mathematical Sciences, (2004). doi: 10.1007/978-3-662-08966-8. Google Scholar

[27]

A. Tikhonov and V. Arsenin, Nonlinear Ill-posed Problems,, Chapman and Hall, (1998). Google Scholar

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