# American Institute of Mathematical Sciences

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 and 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), 2014. doi: 10.2139/ssrn.2308138. [2] V. Albani and J. P. Zubelli, Online local volatility calibration by convex regularization, Appl. Anal. Discrete Math., 8 (2014), 243-268. doi: 10.2298/AADM140811012A. [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), 1382-1400. doi: 10.1080/00036811.2013.833326. [4] S. Anzengruber and R. Ramlau, Morozov's discrepancy principle for Tikhonov-type functionals with nonlinear operators, Inverse Problems, 26 (2010), 025001. doi: 10.1088/0266-5611/26/2/025001. [5] S. Anzengruber and R. Ramlau, Convergence rates for Morozov's discrepancy principle using variational inequalities, Inverse Problems, 27 (2011), 105007. doi: 10.1088/0266-5611/27/10/105007. [6] T. Bonesky, Morozov's discrepancy principle and Tikhonov-type functionals, Inverse Problems, 25 (2009), 015015. doi: 10.1088/0266-5611/25/1/015015. [7] S. Crepey, Calibration of the local volatility in a generalized Black-Scholes model using Tikhonov regularization, SIAM J. Math. Anal., 34 (2003), 1183-1206. doi: 10.1137/S0036141001400202. [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), 2398-2415. doi: 10.1016/j.na.2011.10.037. [9] H. Egger and H. Engl, Tikhonov regularization applied to the inverse problem of option pricing: Convergence analysis and rates, Inverse Problems, 21 (2005), 1027-1045. doi: 10.1088/0266-5611/21/3/014. [10] I. Ekland and R. Teman, Convex Analysis and Variational Problems, North Holland, Amsterdan, 1976. [11] H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996. [12] C. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of First Kind, 1st edition, Research Notes in Mathematics, 105, Pitman Advanced Publishing Program, 1984. [13] C. Groetsch and A. Neubauer, Regularization of ill-posed problems: Optimal parameter choice in finite dimensions, J. Approx. Theory, 58 (1989), 184-200. doi: 10.1016/0021-9045(89)90019-1. [14] B. Hofmann and P. Mathé, Parameter choice in Banach space regularization under variational inequalities, Inverse Problems, 28 (2012), 104006. doi: 10.1088/0266-5611/28/10/104006. [15] B. Hofmann and O. Scherzer, Local ill-posedness and source conditions of operator equations in Hilbert spaces, Inverse Problems, 14 (1998), 1189-1206. doi: 10.1088/0266-5611/14/5/007. [16] B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear ill-Posed Problems, Walter de Gruyter, 2008. doi: 10.1515/9783110208276. [17] A. Kirsch and A. Rieder, Seismic tomography is lolocal ill-posed, Inverse Problems, 30 (2014), 125001. doi: 10.1088/0266-5611/30/12/125001. [18] J. Lindenstrauss, On nonlinear projections in Banach spaces, Michigan Math. J., 11 (1964), 263-287. doi: 10.1307/mmj/1028999141. [19] V. Morozov, On the solution of functional equations by the method of regularization, Dokl. Math., 7 (1966), 414-417. [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), 85-99. doi: 10.1080/01630569008816362. [21] J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operations Research and Financial Engineering, Springer, 2006. [22] C. Pöschl, E. Resmerita and O. Scherzer, Discretization of variational regularization in Banach spaces, Inverse Problems, 26 (2010), 105017. doi: 10.1088/0266-5611/26/10/105017. [23] E. Resmerita and R. Anderssen, Joint additive Kullback-Leibler residual minimization and regularization for linear inverse problems, Math. Methods Appl. Sci., 30 (2007), 1527-1544. doi: 10.1002/mma.855. [24] O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, Applied Mathematical Sciences, 167, Springer, New York, 2008. [25] T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods in Banach Spaces, Walter de Gruyter, 2012. doi: 10.1515/9783110255720. [26] E. Somersalo and J. Kapio, Statistical and Computational Inverse Problems, Applied Mathematical Sciences, 160, Springer, 2004. doi: 10.1007/978-3-662-08966-8. [27] A. Tikhonov and V. Arsenin, Nonlinear Ill-posed Problems, Chapman and Hall, 1998.

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##### References:
 [1] V. Albani, A. De Cezaro and J. Zubelli, Convex regularization of local volatility estimation in a discrete setting, Submitted (SSRN ID: 2308138), 2014. doi: 10.2139/ssrn.2308138. [2] V. Albani and J. P. Zubelli, Online local volatility calibration by convex regularization, Appl. Anal. Discrete Math., 8 (2014), 243-268. doi: 10.2298/AADM140811012A. [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), 1382-1400. doi: 10.1080/00036811.2013.833326. [4] S. Anzengruber and R. Ramlau, Morozov's discrepancy principle for Tikhonov-type functionals with nonlinear operators, Inverse Problems, 26 (2010), 025001. doi: 10.1088/0266-5611/26/2/025001. [5] S. Anzengruber and R. Ramlau, Convergence rates for Morozov's discrepancy principle using variational inequalities, Inverse Problems, 27 (2011), 105007. doi: 10.1088/0266-5611/27/10/105007. [6] T. Bonesky, Morozov's discrepancy principle and Tikhonov-type functionals, Inverse Problems, 25 (2009), 015015. doi: 10.1088/0266-5611/25/1/015015. [7] S. Crepey, Calibration of the local volatility in a generalized Black-Scholes model using Tikhonov regularization, SIAM J. Math. Anal., 34 (2003), 1183-1206. doi: 10.1137/S0036141001400202. [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), 2398-2415. doi: 10.1016/j.na.2011.10.037. [9] H. Egger and H. Engl, Tikhonov regularization applied to the inverse problem of option pricing: Convergence analysis and rates, Inverse Problems, 21 (2005), 1027-1045. doi: 10.1088/0266-5611/21/3/014. [10] I. Ekland and R. Teman, Convex Analysis and Variational Problems, North Holland, Amsterdan, 1976. [11] H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996. [12] C. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of First Kind, 1st edition, Research Notes in Mathematics, 105, Pitman Advanced Publishing Program, 1984. [13] C. Groetsch and A. Neubauer, Regularization of ill-posed problems: Optimal parameter choice in finite dimensions, J. Approx. Theory, 58 (1989), 184-200. doi: 10.1016/0021-9045(89)90019-1. [14] B. Hofmann and P. Mathé, Parameter choice in Banach space regularization under variational inequalities, Inverse Problems, 28 (2012), 104006. doi: 10.1088/0266-5611/28/10/104006. [15] B. Hofmann and O. Scherzer, Local ill-posedness and source conditions of operator equations in Hilbert spaces, Inverse Problems, 14 (1998), 1189-1206. doi: 10.1088/0266-5611/14/5/007. [16] B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear ill-Posed Problems, Walter de Gruyter, 2008. doi: 10.1515/9783110208276. [17] A. Kirsch and A. Rieder, Seismic tomography is lolocal ill-posed, Inverse Problems, 30 (2014), 125001. doi: 10.1088/0266-5611/30/12/125001. [18] J. Lindenstrauss, On nonlinear projections in Banach spaces, Michigan Math. J., 11 (1964), 263-287. doi: 10.1307/mmj/1028999141. [19] V. Morozov, On the solution of functional equations by the method of regularization, Dokl. Math., 7 (1966), 414-417. [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), 85-99. doi: 10.1080/01630569008816362. [21] J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operations Research and Financial Engineering, Springer, 2006. [22] C. Pöschl, E. Resmerita and O. Scherzer, Discretization of variational regularization in Banach spaces, Inverse Problems, 26 (2010), 105017. doi: 10.1088/0266-5611/26/10/105017. [23] E. Resmerita and R. Anderssen, Joint additive Kullback-Leibler residual minimization and regularization for linear inverse problems, Math. Methods Appl. Sci., 30 (2007), 1527-1544. doi: 10.1002/mma.855. [24] O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, Applied Mathematical Sciences, 167, Springer, New York, 2008. [25] T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods in Banach Spaces, Walter de Gruyter, 2012. doi: 10.1515/9783110255720. [26] E. Somersalo and J. Kapio, Statistical and Computational Inverse Problems, Applied Mathematical Sciences, 160, Springer, 2004. doi: 10.1007/978-3-662-08966-8. [27] A. Tikhonov and V. Arsenin, Nonlinear Ill-posed Problems, Chapman and Hall, 1998.
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