Fast matrix exponential-based quasi-boundary value methods for inverse space-dependent source problems

Fermín S. V. Bazán; Luciano Bedin; Koung Hee Leem; Jun Liu; George Pelekanos; Fermín S. V. Bazán; Luciano Bedin; Koung Hee Leem; Jun Liu; George Pelekanos

doi:10.3934/nhm.2023026

Networks and Heterogeneous Media

2023, Volume 18, Issue 2: 601-621. doi: 10.3934/nhm.2023026

Previous Article Next Article

Research article Special Issues

Fast matrix exponential-based quasi-boundary value methods for inverse space-dependent source problems

1.
Department of Mathematics, Federal University of Santa Catarina, Florianopolis SC, Santa Catarina, Brazil
2.
Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026, USA

Received: 26 October 2022 Revised: 26 December 2022 Accepted: 22 January 2023 Published: 06 February 2023

In this paper, we study the well-established quasi-boundary value methods for regularizing inverse state-dependent source problems, where the convergence analysis of three typical cases is presented in the framework of filtering regularization method under suitable source conditions. Interestingly, the quasi-boundary value methods can be interpreted as certain Lavrentiev-type regularization, which was not known in literature. As another major contribution, efficient numerical implementation based on matrix exponential in time is developed, which shows much improved computational efficiency than MATLAB's backslash solver based on the all-at-once space-time discretization scheme. Numerical examples are reported to illustrate the promising computational performance of our proposed algorithms based on matrix exponential techniques.
- ill-posed problem,
- inverse source problem,
- quasi-boundary value method,
- filtering regularization,
- Lavrentiev regularization,
- matrix exponential
Citation: Fermín S. V. Bazán, Luciano Bedin, Koung Hee Leem, Jun Liu, George Pelekanos. Fast matrix exponential-based quasi-boundary value methods for inverse space-dependent source problems[J]. Networks and Heterogeneous Media, 2023, 18(2): 601-621. doi: 10.3934/nhm.2023026

Related Papers:

Abstract

In this paper, we study the well-established quasi-boundary value methods for regularizing inverse state-dependent source problems, where the convergence analysis of three typical cases is presented in the framework of filtering regularization method under suitable source conditions. Interestingly, the quasi-boundary value methods can be interpreted as certain Lavrentiev-type regularization, which was not known in literature. As another major contribution, efficient numerical implementation based on matrix exponential in time is developed, which shows much improved computational efficiency than MATLAB's backslash solver based on the all-at-once space-time discretization scheme. Numerical examples are reported to illustrate the promising computational performance of our proposed algorithms based on matrix exponential techniques.

References

[1]	M. N. Ahmadabadi, M. Arab, F. M. Ghaini, The method of fundamental solutions for the inverse space-dependent heat source problem, Eng Anal Bound Elem, 33 (2009), 1231–1235. https://doi.org/10.1016/j.enganabound.2009.05.001 doi: 10.1016/j.enganabound.2009.05.001
[2]	A. H. Al-Mohy, N. J. Higham, Computing the action of the matrix exponential, with an application to exponential integrators, SIAM J Sci Comput, 33 (2011), 488–511. https://doi.org/10.1137/100788860 doi: 10.1137/100788860
[3]	M. Ali, S. Aziz, S. A. Malik, Inverse source problems for a space–time fractional differential equation, Inverse Probl Sci Eng, 28 (2020), 47–68. https://doi.org/10.1080/17415977.2019.1597079 doi: 10.1080/17415977.2019.1597079
[4]	F. S. Bazán, J. B. Francisco, K. H. Leem, G. Pelekanos, A maximum product criterion as a tikhonov parameter choice rule for kirsch's factorization method, J. Comput. Appl. Math., 236 (2012), 4264–4275. https://doi.org/10.1016/j.cam.2012.05.008 doi: 10.1016/j.cam.2012.05.008
[5]	F. S. Bazán, J. B. Francisco, K. H. Leem, G. Pelekanos, Using the linear sampling method and an improved maximum product criterion for the solution of the electromagnetic inverse medium problem, J. Comput. Appl. Math., 273 (2015), 61–75. https://doi.org/10.1016/j.cam.2014.06.003 doi: 10.1016/j.cam.2014.06.003
[6]	J. R. Cannon, P. DuChateau, Structural identification of an unknown source term in a heat equation, Inverse Probl, 14 (1998), 535–551. https://doi.org/10.1088/0266-5611/14/3/010 doi: 10.1088/0266-5611/14/3/010
[7]	A. Chambolle, An algorithm for total variation minimization and applications, J Math Imaging Vis, 20 (2004), 89–97. https://doi.org/10.1023/B:JMIV.0000011321.19549.88 doi: 10.1023/B:JMIV.0000011321.19549.88
[8]	N. M. Dien, D. N. D. Hai, T. Q. Viet, D. D. Trong, On tikhonov's method and optimal error bound for inverse source problem for a time-fractional diffusion equation, Comput. Math. with Appl., 80 (2020), 61–81. https://doi.org/10.1016/j.camwa.2020.02.024 doi: 10.1016/j.camwa.2020.02.024
[9]	A. Doicu, T. Trautmann, F. Schreier, Numerical Regularization for Atmospheric Inverse Problems, Heidelberg: Springer Berlin, 2010,
[10]	F. F. Dou, C. L. Fu, F. Yang, Identifying an unknown source term in a heat equation, Inverse Probl Sci Eng, 17 (2009), 901–913. https://doi.org/10.1080/17415970902916870 doi: 10.1080/17415970902916870
[11]	F. F. Dou, C. L. Fu, F. L. Yang, Optimal error bound and Fourier regularization for identifying an unknown source in the heat equation, J. Comput. Appl. Math., 230 (2009), 728–737. https://doi.org/10.1016/j.cam.2009.01.008 doi: 10.1016/j.cam.2009.01.008
[12]	H. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, Netherlands: Springer, 2000.
[13]	L. Evans, Partial Differential Equations, Providence: American Mathematical Society, 2016.
[14]	A. Farcas, D. Lesnic, The boundary-element method for the determination of a heat source dependent on one variable, J Eng Math, 54 (2006), 375–388. https://doi.org/10.1007/s10665-005-9023-0 doi: 10.1007/s10665-005-9023-0
[15]	A. Fatullayev, Numerical solution of the inverse problem of determining an unknown source term in a heat equation, Math Comput Simul, 58 (2002), 247–253. https://doi.org/10.1016/S0378-4754(01)00365-2 doi: 10.1016/S0378-4754(01)00365-2
[16]	A. Fatullayev, Numerical solution of the inverse problem of determining an unknown source term in a two-dimensional heat equation, Appl. Math. Comput., 152 (2004), 659–666. https://doi.org/10.1016/S0096-3003(03)00582-4 doi: 10.1016/S0096-3003(03)00582-4
[17]	D. N. Hào, J. Liu, N. V. Duc, N. V. Thang, Stability results for backward time-fractional parabolic equations, Inverse Probl, 35 (2019), 125006. https://doi.org/10.1088/1361-6420/ab45d3 doi: 10.1088/1361-6420/ab45d3
[18]	M. Hochbruck, C. Lubich, H. Selhofer, Exponential integrators for large systems of differential equations, SIAM J Sci Comput, 19 (1998), 1552–1574. https://doi.org/10.1137/S1064827595295337 doi: 10.1137/S1064827595295337
[19]	M. Hochbruck, A. Ostermann, Exponential integrators, Acta Numer., 19 (2010), 209–286. https://doi.org/10.1017/S0962492910000048 doi: 10.1017/S0962492910000048
[20]	Y. Jiang, J. Liu, X. S. Wang, A direct parallel-in-time quasi-boundary value method for inverse space-dependent source problems, J. Comput. Appl. Math., 423 (2023), 114958. https://doi.org/10.1016/j.cam.2022.114958 doi: 10.1016/j.cam.2022.114958
[21]	B. Jin, W. Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse Probl, 31 (2015), 035003. https://doi.org/10.1088/0266-5611/31/3/035003 doi: 10.1088/0266-5611/31/3/035003
[22]	B. T. Johansson, D. Lesnic, A variational method for identifying a spacewise-dependent heat source, IMA J Appl Math, 72 (2007), 748–760. https://doi.org/10.1093/imamat/hxm024 doi: 10.1093/imamat/hxm024
[23]	T. Johansson, D. Lesnic, Determination of a spacewise dependent heat source, J. Comput. Appl. Math., 209 (2007), 66–80. https://doi.org/10.1016/j.cam.2006.10.026 doi: 10.1016/j.cam.2006.10.026
[24]	S. I. Kabanikhin, Inverse and ill-posed problems: Theory and Applications, Berlin: De Gruyter, 2011.
[25]	R. Ke, M. K. Ng, T. Wei, Efficient preconditioning for time fractional diffusion inverse source problems, SIAM J. Matrix Anal. Appl., 41 (2020), 1857–1888. https://doi.org/10.1137/20M1320304 doi: 10.1137/20M1320304
[26]	A. Kirsch, An introduction to the mathematical theory of inverse problems, New York: Springer, 2011.
[27]	D. Kressner, R. Luce, Fast computation of the matrix exponential for a Toeplitz matrix, SIAM J. Matrix Anal. Appl., 39 (2018), 23–47. https://doi.org/10.1137/16M1083633 doi: 10.1137/16M1083633
[28]	M. M. Lavrentiev, Some Improperly Posed Problems of Mathematical Physics, Berlin: Springer, 1967,
[29]	S. T. Lee, H. K. Pang, H. W. Sun, Shift-invert Arnoldi approximation to the Toeplitz matrix exponential, SIAM J Sci Comput, 32 (2010), 774–792. https://doi.org/10.1137/090758064 doi: 10.1137/090758064
[30]	D. Lesnic, Inverse Problems with Applications in Science and Engineering, Boca Raton: CRC Press, 2021,
[31]	J. Lohwater, O. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, New York: Springer, 2013.
[32]	Y. Ma, C. Fu, Y. Zhang, Identification of an unknown source depending on both time and space variables by a variational method, Appl. Math. Model., 36 (2012), 5080–5090. https://doi.org/10.1016/j.apm.2011.12.046 doi: 10.1016/j.apm.2011.12.046
[33]	M. T. Nair, U. Tautenhahn, Lavrentiev regularization for linear ill-posed problems under general source conditions, Zeitschrift für Anal. und ihre Anwendung, 23 (2004), 167–185.
[34]	M. Nair, Regularization of ill-posed operator equations: an overview, The Journal of Analysis, 29 (2021), 519–541. https://doi.org/10.1007/s41478-020-00263-9 doi: 10.1007/s41478-020-00263-9
[35]	H. T. Nguyen, D. L. Le, V. T Nguyen, Regularized solution of an inverse source problem for a time fractional diffusion equation, Appl. Math. Model., 40 (2016), 8244–8264. https://doi.org/10.1016/j.apm.2016.04.009 doi: 10.1016/j.apm.2016.04.009
[36]	L. I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259–268.
[37]	E. G. Savateev, On problems of determining the source function in a parabolic equation, J Inverse Ill Posed Probl, 3 (1995), 83–102. https://doi.org/10.1515/jiip.1995.3.1.83 doi: 10.1515/jiip.1995.3.1.83
[38]	U. Tautenhahn, On the method of Lavrentiev regularization for nonlinear ill-posed problems, Inverse Probl, 18 (2002), 191. https://doi.org/10.1088/0266-5611/18/1/313 doi: 10.1088/0266-5611/18/1/313
[39]	D. Trong, N. Long, P. Alain, Nonhomogeneous heat equation: Identification and regularization for the inhomogeneous term, J. Math. Anal. Appl., 312 (2005), 93–104. https://doi.org/10.1016/j.jmaa.2005.03.037 doi: 10.1016/j.jmaa.2005.03.037
[40]	D. Trong, P. Quan, P. Alain, Determination of a two-dimensional heat source: uniqueness, regularization and error estimate, J. Comput. Appl. Math., 191 (2006), 50–67. https://doi.org/10.1016/j.cam.2005.04.022 doi: 10.1016/j.cam.2005.04.022
[41]	Z. Wang, W. Zhang, B. Wu, Regularized optimization method for determining the space-dependent source in a parabolic equation without iteration, J. Comput. Anal. Appl., 20 (2016), 1107–1126.
[42]	T. Wei, X. Li, Y. Li, An inverse time-dependent source problem for a time-fractional diffusion equation, Inverse Probl, 32 (2016), 085003. https://doi.org/10.1088/0266-5611/32/8/085003 doi: 10.1088/0266-5611/32/8/085003
[43]	T. Wei, J. G. Wang, A modified quasi-boundary value method for the backward time-fractional diffusion problem, Esaim Math Model Numer Anal, 48 (2014), 603–621. https://doi.org/10.1051/m2an/2013107 doi: 10.1051/m2an/2013107
[44]	T. Wei, J. Wang, A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation, Appl Numer Math, 78 (2014), 95–111. https://doi.org/10.1016/j.apnum.2013.12.002 doi: 10.1016/j.apnum.2013.12.002
[45]	T. Wei, J. Wang, A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation, Appl Numer Math, 78 (2014), 95–111. https://doi.org/10.1016/j.apnum.2013.12.002 doi: 10.1016/j.apnum.2013.12.002
[46]	L. Yan, C. L. Fu, F. L. Yang, The method of fundamental solutions for the inverse heat source problem, Eng Anal Bound Elem, 32 (2008), 216–222. https://doi.org/10.1016/j.enganabound.2007.08.002 doi: 10.1016/j.enganabound.2007.08.002
[47]	L. Yan, F. L. Yang, C. L. Fu, A meshless method for solving an inverse spacewise-dependent heat source problem, J. Comput. Phys., 228 (2009), 123–136. https://doi.org/10.1016/j.jcp.2008.09.001 doi: 10.1016/j.jcp.2008.09.001
[48]	F. Yang, C. L. Fu, A simplified Tikhonov regularization method for determining the heat source, Appl. Math. Model., 34 (2010), 3286–3299. https://doi.org/10.1016/j.apm.2010.02.020 doi: 10.1016/j.apm.2010.02.020
[49]	F. Yang, C. L. Fu, X. X. Li, A quasi-boundary value regularization method for determining the heat source, Math. Methods Appl. Sci., 37 (2013), 3026–3035. https://doi.org/10.1002/mma.3040 doi: 10.1002/mma.3040
[50]	F. Yang, C. L. Fu, X. X. Li, The inverse source problem for time-fractional diffusion equation: stability analysis and regularization, Inverse Probl Sci Eng, 23 (2015), 969–996. https://doi.org/10.1080/17415977.2014.968148 doi: 10.1080/17415977.2014.968148
[51]	L. Yang, M. Dehghan, J. Yu, G. Luo, Inverse problem of time-dependent heat sources numerical reconstruction, Math Comput Simul, 81 (2011), 1656–1672. https://doi.org/10.1016/j.matcom.2011.01.001 doi: 10.1016/j.matcom.2011.01.001
[52]	L. Yang, J. Yu, G. Luo, Z. Deng, Numerical identification of source terms for a two dimensional heat conduction problem in polar coordinate system, Appl. Math. Model., 37 (2013), 939–957. https://doi.org/10.1016/j.apm.2012.03.024 doi: 10.1016/j.apm.2012.03.024
[53]	Z. Yi, D. Murio, Source term identification in 1D IHCP, Comput. Math. with Appl., 47 (2004), 1921–1933. https://doi.org/10.1016/j.camwa.2002.11.025 doi: 10.1016/j.camwa.2002.11.025

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)