# American Institute of Mathematical Sciences

April 2019, 13(2): 401-430. doi: 10.3934/ipi.2019020

## Regularization of a backwards parabolic equation by fractional operators

 1 Department of Mathematics, Alpen-Adria-Universität Klagenfurt, 9020 Klagenfurt, Austria 2 Department of Mathematics, Texas A&M University, College Station, Texas 77843, USA

* corresponding author

Received  June 2018 Revised  September 2018 Published  January 2019

Fund Project: The work of the first author was supported by the Austrian Science Fund FWF under the grants I2271 and P30054 as well as partially by the Karl Popper Kolleg "Modeling-Simulation-Optimization", funded by the Alpen-Adria-Universität Klagenfurt and by the Carinthian Economic Promotion Fund (KWF)
The work of the second author was supported in part by the National Science Foundation through award DMS-1620138

The backwards diffusion equation is one of the classical ill-posed inverse problems, related to a wide range of applications, and has been extensively studied over the last 50 years. One of the first methods was that of quasireversibility whereby the parabolic operator is replaced by a differential operator for which the backwards problem in time is well posed. This is in fact the direction we will take but will do so with a nonlocal operator; an equation of fractional order in time for which the backwards problem is known to be "almost well posed."

We shall look at various possible options and strategies but our conclusion for the best of these will exploit the linearity of the problem to break the inversion into distinct frequency bands and to use a different fractional order for each. The fractional exponents will be chosen using the discrepancy principle under the assumption we have an estimate of the noise level in the data. An analysis of the method is provided as are some illustrative numerical examples.

Citation: Barbara Kaltenbacher, William Rundell. Regularization of a backwards parabolic equation by fractional operators. Inverse Problems & Imaging, 2019, 13 (2) : 401-430. doi: 10.3934/ipi.2019020
##### References:
 [1] V. Akcelik, G. Biros, A. Draganescu, O. Ghattas, J. Hill and B. G. van Bloemen Waanders, Inversion of airborne contaminants in a regional model, In Computational Science - ICCS 2006, 6th International Conference, Reading, UK, May 28-31, 2006, Proceedings, Part III, (2006), 481–488. doi: 10.1007/11758532_64. [2] F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, Comptes Rendus Mathematique, 347 (2009), 867-872. doi: 10.1016/j.crma.2009.05.011. [3] K. A. Ames, G. W. Clark, J. F. Epperson and S. F. Oppenheimer, A comparison of regularizations for an ill-posed problem, Math. Comp., 67 (1998), 1451-1471. doi: 10.1090/S0025-5718-98-01014-X. [4] G. I. Barenblatt, P. Zheltov and I. N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks (strata), PMM24, Transl. of Priklad. Mat. Mekh.24, 24 (1960), 1286-1303. doi: 10.1016/0021-8928(60)90107-6. [5] A. Bonito, W. Lei and J. E. Pasciak, Numerical approximation of space-time fractional parabolic equations, Comput. Methods Appl. Math., 17 (2017), 679-705. doi: 10.1515/cmam-2017-0032. [6] M. Caputo, Linear models of dissipation whose Q is almost frequency independent – Ⅱ, Geophys. J. Int., 13 (1967), 529-539. doi: 10.1111/j.1365-246X.1967.tb02303.x. [7] M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure Appl. Geophys., 91 (1971), 134-147. [8] P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Zeitschrift für angewandte Mathematik und Physik ZAMP, 19 (1968), 614-627. doi: 10.1007/BF01594969. [9] S. P. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55. doi: 10.2140/pjm.1989.136.15. [10] G. W. Clark and S. F. Oppenheimer, Quasireversibility methods for non-well-posed problems, Electron. J. Differential Equations, (1994), pages No. 08, approx. 9pp. [11] B. D. Coleman, R. J. Duffin and V. J. Mizel, Instability, uniqueness and nonexistence theorems for the equation ut = uxx-uxtx on a strip, Arch. Rational Mech. Anal., 19 (1965), 100-116. doi: 10.1007/BF00282277. [12] M. M. Djrbashian, Harmonic Analysis and Boundary Value Problems in the Complex Domain, Birkhäuser, Basel, 1993. doi: 10.1007/978-3-0348-8549-2. [13] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996. [14] R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics. Springer, Heidelberg, 2014. doi: 10.1007/978-3-662-43930-2. [15] R. Gorenflo and F. Mainardi, Fractional calculus, In Fractals and Fractional Calculus in Continuum Mechanics, Springer Vienna, 378 (1997), 223–276. [16] P. C. Hansen, Rank-deficient and Discrete Ill-posed Problems: Numerical Aspects of Linear Inversion, SIAM monographs on mathematical modeling and computation, Philadelphia, PA, 1998. doi: 10.1137/1.9780898719697. [17] T. Hohage, Logarithmic convergence rates of the iteratively regularized Gauẞ-Newton method for an inverse potential and an inverse scattering problem, Inverse Problems, 13 (1997), 1279-1299. doi: 10.1088/0266-5611/13/5/012. [18] V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences. Springer New York, 2006. [19] B. Jin, R. Lazarov, Y. Liu and Z. Zhou, The Galerkin finite element method for a multi-term time-fractional diffusion equation, J. Comput. Phys., 281 (2015), 825-843. doi: 10.1016/j.jcp.2014.10.051. [20] B. Jin and W. Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse Problems, 31 (2015), 035003, 40pp. doi: 10.1088/0266-5611/31/3/035003. [21] T. Kato, Frational powers of dissipative operators. Ⅱ, J. Math. Soc. Japan, 14 (1962), 242-248. doi: 10.2969/jmsj/01420242. [22] R. Kowar and O. Scherzer, Attenuation models in photoacoustics, In H. Ammari, editor, Mathematical Modeling in Biomedical Imaging II: Optical, Ultrasound, and Opto-Acoustic Tomographies, volume 2035 of Lecture Notes in Mathematics, pages 85–130. Springer Verlag, Berlin Heidelberg, 2012. doi: 10.1007/978-3-642-22990-9_4. [23] I. Lasiecka, S. A. Messaoudi and M. I. Mustafa, Note on intrinsic decay rates for abstract wave equations with memory, Journal of Mathematical Physics, 54 (2013), 031504, 18pp. doi: 10.1063/1.4793988. [24] R. Lattès and J.-L. Lions, The Method of Quasi-Reversibility. Applications to Partial Differential Equations, Translated from the French edition and edited by Richard Bellman. Modern Analytic and Computational Methods in Science and Mathematics, No. 18. American Elsevier Publishing Co., Inc., New York, 1969. [25] Z. Li, Y. Liu and M. Yamamoto, Initial-boundary value problems for multi-term timefractional diffusion equations with positive constant coefficients, Applied Mathematics and Computation, 257 (2015), 381-397. Recent Advances in Fractional Differential Equations. doi: 10.1016/j.amc.2014.11.073. [26] S. Lu and S. V. Pereverzev, Regularization Theory for Ill-posed Problems: Selected Topics, Inverse and ill-posed problems series. Walter de Gruyter GmbH & Company KG, 2013. doi: 10.1515/9783110286496. [27] Y. Luchko and R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives, Acta Math. Vietnam., 24 (1999), 207-233. [28] F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett., 9 (1996), 23-28. doi: 10.1016/0893-9659(96)00089-4. [29] E. W Montroll and G. H. Weiss, Random walks on lattices. Ⅱ, J. Math. Phys., 6 (1965), 167-181. doi: 10.1063/1.1704269. [30] V. A. Morozov, Choice of parameter for the solution of functional equations by the regularization method, Dokl. Akad. Nauk SSSR, 175 (1967), 1225-1228. [31] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447. doi: 10.1016/j.jmaa.2011.04.058. [32] R. E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl., 47 (1974), 563-572. doi: 10.1016/0022-247X(74)90008-0. [33] R. E. Showalter, Quasi-reversibility of first and second order parabolic evolution equations, In Improperly Posed Boundary Value Problems (Conf., Univ. New Mexico, Albuquerque, N. M., 1974), Res. Notes in Math., No. 1. Pitman, London, 1975, 76–84. [34] R. E. Showalter, Regularization and approximation of second order evolution equations, SIAM J. Math. Anal., 7 (1976), 461-472. doi: 10.1137/0507037. [35] I. M Sokolov, J. Klafter and A. Blumen, Fractional kinetics, Physics Today, 55 (2002), 48-54. doi: 10.1063/1.1535007. [36] E. M. Wright, On the coefficients of power series having exponential singularities, J. London Math. Soc, 8 (1933), 71-79. doi: 10.1112/jlms/s1-8.1.71.

show all references

##### References:
 [1] V. Akcelik, G. Biros, A. Draganescu, O. Ghattas, J. Hill and B. G. van Bloemen Waanders, Inversion of airborne contaminants in a regional model, In Computational Science - ICCS 2006, 6th International Conference, Reading, UK, May 28-31, 2006, Proceedings, Part III, (2006), 481–488. doi: 10.1007/11758532_64. [2] F. Alabau-Boussouira and P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, Comptes Rendus Mathematique, 347 (2009), 867-872. doi: 10.1016/j.crma.2009.05.011. [3] K. A. Ames, G. W. Clark, J. F. Epperson and S. F. Oppenheimer, A comparison of regularizations for an ill-posed problem, Math. Comp., 67 (1998), 1451-1471. doi: 10.1090/S0025-5718-98-01014-X. [4] G. I. Barenblatt, P. Zheltov and I. N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks (strata), PMM24, Transl. of Priklad. Mat. Mekh.24, 24 (1960), 1286-1303. doi: 10.1016/0021-8928(60)90107-6. [5] A. Bonito, W. Lei and J. E. Pasciak, Numerical approximation of space-time fractional parabolic equations, Comput. Methods Appl. Math., 17 (2017), 679-705. doi: 10.1515/cmam-2017-0032. [6] M. Caputo, Linear models of dissipation whose Q is almost frequency independent – Ⅱ, Geophys. J. Int., 13 (1967), 529-539. doi: 10.1111/j.1365-246X.1967.tb02303.x. [7] M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure Appl. Geophys., 91 (1971), 134-147. [8] P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Zeitschrift für angewandte Mathematik und Physik ZAMP, 19 (1968), 614-627. doi: 10.1007/BF01594969. [9] S. P. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55. doi: 10.2140/pjm.1989.136.15. [10] G. W. Clark and S. F. Oppenheimer, Quasireversibility methods for non-well-posed problems, Electron. J. Differential Equations, (1994), pages No. 08, approx. 9pp. [11] B. D. Coleman, R. J. Duffin and V. J. Mizel, Instability, uniqueness and nonexistence theorems for the equation ut = uxx-uxtx on a strip, Arch. Rational Mech. Anal., 19 (1965), 100-116. doi: 10.1007/BF00282277. [12] M. M. Djrbashian, Harmonic Analysis and Boundary Value Problems in the Complex Domain, Birkhäuser, Basel, 1993. doi: 10.1007/978-3-0348-8549-2. [13] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996. [14] R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics. Springer, Heidelberg, 2014. doi: 10.1007/978-3-662-43930-2. [15] R. Gorenflo and F. Mainardi, Fractional calculus, In Fractals and Fractional Calculus in Continuum Mechanics, Springer Vienna, 378 (1997), 223–276. [16] P. C. Hansen, Rank-deficient and Discrete Ill-posed Problems: Numerical Aspects of Linear Inversion, SIAM monographs on mathematical modeling and computation, Philadelphia, PA, 1998. doi: 10.1137/1.9780898719697. [17] T. Hohage, Logarithmic convergence rates of the iteratively regularized Gauẞ-Newton method for an inverse potential and an inverse scattering problem, Inverse Problems, 13 (1997), 1279-1299. doi: 10.1088/0266-5611/13/5/012. [18] V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences. Springer New York, 2006. [19] B. Jin, R. Lazarov, Y. Liu and Z. Zhou, The Galerkin finite element method for a multi-term time-fractional diffusion equation, J. Comput. Phys., 281 (2015), 825-843. doi: 10.1016/j.jcp.2014.10.051. [20] B. Jin and W. Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse Problems, 31 (2015), 035003, 40pp. doi: 10.1088/0266-5611/31/3/035003. [21] T. Kato, Frational powers of dissipative operators. Ⅱ, J. Math. Soc. Japan, 14 (1962), 242-248. doi: 10.2969/jmsj/01420242. [22] R. Kowar and O. Scherzer, Attenuation models in photoacoustics, In H. Ammari, editor, Mathematical Modeling in Biomedical Imaging II: Optical, Ultrasound, and Opto-Acoustic Tomographies, volume 2035 of Lecture Notes in Mathematics, pages 85–130. Springer Verlag, Berlin Heidelberg, 2012. doi: 10.1007/978-3-642-22990-9_4. [23] I. Lasiecka, S. A. Messaoudi and M. I. Mustafa, Note on intrinsic decay rates for abstract wave equations with memory, Journal of Mathematical Physics, 54 (2013), 031504, 18pp. doi: 10.1063/1.4793988. [24] R. Lattès and J.-L. Lions, The Method of Quasi-Reversibility. Applications to Partial Differential Equations, Translated from the French edition and edited by Richard Bellman. Modern Analytic and Computational Methods in Science and Mathematics, No. 18. American Elsevier Publishing Co., Inc., New York, 1969. [25] Z. Li, Y. Liu and M. Yamamoto, Initial-boundary value problems for multi-term timefractional diffusion equations with positive constant coefficients, Applied Mathematics and Computation, 257 (2015), 381-397. Recent Advances in Fractional Differential Equations. doi: 10.1016/j.amc.2014.11.073. [26] S. Lu and S. V. Pereverzev, Regularization Theory for Ill-posed Problems: Selected Topics, Inverse and ill-posed problems series. Walter de Gruyter GmbH & Company KG, 2013. doi: 10.1515/9783110286496. [27] Y. Luchko and R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives, Acta Math. Vietnam., 24 (1999), 207-233. [28] F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett., 9 (1996), 23-28. doi: 10.1016/0893-9659(96)00089-4. [29] E. W Montroll and G. H. Weiss, Random walks on lattices. Ⅱ, J. Math. Phys., 6 (1965), 167-181. doi: 10.1063/1.1704269. [30] V. A. Morozov, Choice of parameter for the solution of functional equations by the regularization method, Dokl. Akad. Nauk SSSR, 175 (1967), 1225-1228. [31] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447. doi: 10.1016/j.jmaa.2011.04.058. [32] R. E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl., 47 (1974), 563-572. doi: 10.1016/0022-247X(74)90008-0. [33] R. E. Showalter, Quasi-reversibility of first and second order parabolic evolution equations, In Improperly Posed Boundary Value Problems (Conf., Univ. New Mexico, Albuquerque, N. M., 1974), Res. Notes in Math., No. 1. Pitman, London, 1975, 76–84. [34] R. E. Showalter, Regularization and approximation of second order evolution equations, SIAM J. Math. Anal., 7 (1976), 461-472. doi: 10.1137/0507037. [35] I. M Sokolov, J. Klafter and A. Blumen, Fractional kinetics, Physics Today, 55 (2002), 48-54. doi: 10.1063/1.1535007. [36] E. M. Wright, On the coefficients of power series having exponential singularities, J. London Math. Soc, 8 (1933), 71-79. doi: 10.1112/jlms/s1-8.1.71.
Amplification factor $A(\lambda_k, \alpha)$
Reconstructions from single and double split frequency method
Reconstructions from SVD and double split frequency method
Reconstructions from SVD as well as double and triple split frequency method
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