An extension of the landweber regularization for a backward time fractional wave problem

Bin Fan; Mejdi Azaïez; Chuanju Xu

doi:10.3934/dcdss.2020409

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August 2021, 14(8): 2893-2916. doi: 10.3934/dcdss.2020409

An extension of the landweber regularization for a backward time fractional wave problem

Bin Fan ^1, , Mejdi Azaïez ^2, and Chuanju Xu ^2,3,,

1.	School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
2.	School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical, Modeling and High Performance Scientific Computing, Xiamen University, Xiamen 361005, China
3.	Bordeaux INP, Laboratoire I2M UMR 5295, Pessac 33607, France

^*Corresponding author: Chuanju Xu

Received February 2020 Revised May 2020 Published August 2021 Early access July 2020

Fund Project: This research is partially supported by NSFC grant 11971408 and NSFC/ANR joint program 51661135011/ANR-16-CE40-0026-01. The first author gratefully acknowledges the financial support from China Scholarship Council and the laboratory I2M UMR 5295 for hosting. The second author has received financial support from the French State in the frame of the "Investments for the future" Programme Idex Bordeaux, reference ANR-10-IDEX-03-02

In this paper, we investigate numerical methods for a backward problem of the time-fractional wave equation in bounded domains. We propose two fractional filter regularization methods, which can be regarded as an extension of the classical Landweber iteration for the time-fractional wave backward problem. The idea is first to transform the ill-posed backward problem into a weighted normal operator equation, then construct the regularization methods for the operator equation by introducing suitable fractional filters. Both a priori and a posteriori regularization parameter choice rules are investigated, together with an estimate for the smallest regularization parameter according to a discrepancy principle. Furthermore, an error analysis is carried out to derive the convergence rates of the regularized solutions generated by the proposed methods. The theoretical estimate shows that the proposed fractional regularizations efficiently overcome the well-known over-smoothing drawback caused by the classical regularizations. Some numerical examples are provided to confirm the theoretical results. In particular, our numerical tests demonstrate that the fractional regularization is actually more efficient than the classical methods for problems having low regularity.

Keywords: Backward problem, Time-fractional wave equation, Fractional filter regularization, Convergence.

Mathematics Subject Classification: Primary: 65M32, 35R11; Secondary: 47A52, 35L05.

Citation: Bin Fan, Mejdi Azaïez, Chuanju Xu. An extension of the landweber regularization for a backward time fractional wave problem. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2893-2916. doi: 10.3934/dcdss.2020409

References:

[1]	O. P. Agrawal, Formulation of euler-lagrange equations for fractional variational problems, Journal of Mathematical Analysis and Applications, 272 (2002), 368-379. doi: 10.1016/S0022-247X(02)00180-4.
[2]	O. P. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynamics, 29 (2002), 145-155. doi: 10.1023/A:1016539022492.
[3]	D. A. Benson, S. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resources Research, 36 (2000), 1403-1412. doi: 10.1029/2000WR900031.
[4]	D. Bianchi, A. Buccini, M. Donatelli and S. Serra-Capizzano, Iterated fractional tikhonov regularization, Inverse Problems, 31 (2015), 055005, 34pp. doi: 10.1088/0266-5611/31/5/055005.
[5]	H. Cheng and C. L. Fu, An iteration regularization for a time-fractional inverse diffusion problem, Applied Mathematical Modelling, 36 (2012), 5642-5649. doi: 10.1016/j.apm.2012.01.016.
[6]	E. Cuesta, M. Kirane and S. A. Malik, Image structure preserving denoising using generalized fractional time integrals, Signal Processing, 92 (2012), 553-563. doi: 10.1016/j.sigpro.2011.09.001.
[7]	Y. Deng and Z. Liu, Iteration methods on sideways parabolic equations, Inverse Problems, 25 (2009), 095004, 14pp. doi: 10.1088/0266-5611/25/9/095004.
[8]	Y. Deng and Z. Liu, New fast iteration for determining surface temperature and heat flux of general sideways parabolic equation, Nonlinear Analysis: Real World Applications, 12 (2011), 156-166. doi: 10.1016/j.nonrwa.2010.06.005.
[9]	K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer Science & Business Media, 2010. doi: 10.1007/978-3-642-14574-2.
[10]	R. Du, W. R. Cao and Z. Z. Sun, A compact difference scheme for the fractional diffusion-wave equation, Applied Mathematical Modelling, 34 (2010), 2998-3007. doi: 10.1016/j.apm.2010.01.008.
[11]	H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publisher, Dordrecht, Boston, London, 1996.
[12]	G. H. Gao and Z. Z. Sun, The finite difference approximation for a class of fractional sub-diffusion equations on a space unbounded domain, Journal of Computational Physics, 236 (2013), 443-460. doi: 10.1016/j.jcp.2012.11.011.
[13]	D. Gerth, E. Klann, R. Ramlau and L. Reichel, On fractional tikhonov regularization, Journal of Inverse and Ill-posed Problems, 23 (2015), 611-625. doi: 10.1515/jiip-2014-0050.
[14]	C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations, 104p, Boston Pitman Publication, 1984.
[15]	Y. Han, X. Xiong and X. Xue, A fractional landweber method for solving backward time-fractional diffusion problem, Computers & Mathematics with Applications, 78 (2019), 81-91. doi: 10.1016/j.camwa.2019.02.017.
[16]	M. E. Hochstenbach and L. Reichel, Fractional tikhonov regularization for linear discrete ill-posed problems, BIT Numerical Mathematics, 51 (2011), 197-215. doi: 10.1007/s10543-011-0313-9.
[17]	A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer Science & Business Media, 2011. doi: 10.1007/978-1-4419-8474-6.
[18]	E. Klann, P. Maass and R. Ramlau, Two-step regularization methods for linear inverse problems, Journal of Inverse and Ill-posed Problems, 14 (2006), 583-607. doi: 10.1515/156939406778474523.
[19]	X. J. Li and C. J. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM Journal on Numerical Analysis, 47 (2009), 2108-2131. doi: 10.1137/080718942.
[20]	Y. M. Lin and C. J. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of computational physics, 225 (2007), 1533-1552. doi: 10.1016/j.jcp.2007.02.001.
[21]	J. J. Liu and M. Yamamoto, A backward problem for the time-fractional diffusion equation, Applicable Analysis, 89 (2010), 1769-1788. doi: 10.1080/00036810903479731.
[22]	R. L. Magin, Fractional Calculus in Bioengineering, volume 2(6)., Begell House Redding, 2006.
[23]	F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, 2010. doi: 10.1142/9781848163300.
[24]	R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics Reports, 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.
[25]	I. Podlubny, Fractional Differential Equations, Acad. Press, New York, 1999.
[26]	I. Podlubny and M. Kacenak, Mittag-leffler Function, the matlab routine, 2006.
[27]	M. Richter, Inverse Problems: Basics, Theory and Applications in Geophysics, Birkhäuser, 2016.
[28]	K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, Journal of Mathematical Analysis and Applications, 382 (2011), 426-447. doi: 10.1016/j.jmaa.2011.04.058.
[29]	F. Y. Song and C. J. Xu, Spectral direction splitting methods for two-dimensional space fractional diffusion equations, Journal of Computational Physics, 299 (2015), 196-214. doi: 10.1016/j.jcp.2015.07.011.
[30]	Z. Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Applied Numerical Mathematics, 56 (2006), 193-209. doi: 10.1016/j.apnum.2005.03.003.
[31]	J. G. Wang and T. Wei, An iterative method for backward time-fractional diffusion problem, Numerical Methods for Partial Differential Equations, 30 (2014), 2029-2041. doi: 10.1002/num.21887.
[32]	J. G. Wang, T. Wei and Y. B. Zhou, Tikhonov regularization method for a backward problem for the time-fractional diffusion equation, Applied Mathematical Modelling, 37 (2013), 8518-8532. doi: 10.1016/j.apm.2013.03.071.
[33]	L. Wang and J. J. Liu, Data regularization for a backward time-fractional diffusion problem, Computers & Mathematics with Applications, 64 (2012), 3613-3626. doi: 10.1016/j.camwa.2012.10.001.
[34]	T. Wei and J. G. Wang, A modified quasi-boundary value method for the backward time-fractional diffusion problem, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), 603-621. doi: 10.1051/m2an/2013107.
[35]	T. Wei and Y. Zhang, The backward problem for a time-fractional diffusion-wave equation in a bounded domain, Computers & Mathematics with Applications, 75 (2018), 3632-3648. doi: 10.1016/j.camwa.2018.02.022.
[36]	X. Xiong, X. Xue and Z. Qian, A modified iterative regularization method for ill-posed problems, Applied Numerical Mathematics, 122 (2017), 108-128. doi: 10.1016/j.apnum.2017.08.004.
[37]	F. Yang, Y. Zhang and X. X. Li, Landweber iterative method for identifying the initial value problem of the time-space fractional diffusion-wave equation, Numerical Algorithms, 83 (2020), 1509-1530. doi: 10.1007/s11075-019-00734-6.

show all references

References:

[1]	O. P. Agrawal, Formulation of euler-lagrange equations for fractional variational problems, Journal of Mathematical Analysis and Applications, 272 (2002), 368-379. doi: 10.1016/S0022-247X(02)00180-4.
[2]	O. P. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynamics, 29 (2002), 145-155. doi: 10.1023/A:1016539022492.
[3]	D. A. Benson, S. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resources Research, 36 (2000), 1403-1412. doi: 10.1029/2000WR900031.
[4]	D. Bianchi, A. Buccini, M. Donatelli and S. Serra-Capizzano, Iterated fractional tikhonov regularization, Inverse Problems, 31 (2015), 055005, 34pp. doi: 10.1088/0266-5611/31/5/055005.
[5]	H. Cheng and C. L. Fu, An iteration regularization for a time-fractional inverse diffusion problem, Applied Mathematical Modelling, 36 (2012), 5642-5649. doi: 10.1016/j.apm.2012.01.016.
[6]	E. Cuesta, M. Kirane and S. A. Malik, Image structure preserving denoising using generalized fractional time integrals, Signal Processing, 92 (2012), 553-563. doi: 10.1016/j.sigpro.2011.09.001.
[7]	Y. Deng and Z. Liu, Iteration methods on sideways parabolic equations, Inverse Problems, 25 (2009), 095004, 14pp. doi: 10.1088/0266-5611/25/9/095004.
[8]	Y. Deng and Z. Liu, New fast iteration for determining surface temperature and heat flux of general sideways parabolic equation, Nonlinear Analysis: Real World Applications, 12 (2011), 156-166. doi: 10.1016/j.nonrwa.2010.06.005.
[9]	K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer Science & Business Media, 2010. doi: 10.1007/978-3-642-14574-2.
[10]	R. Du, W. R. Cao and Z. Z. Sun, A compact difference scheme for the fractional diffusion-wave equation, Applied Mathematical Modelling, 34 (2010), 2998-3007. doi: 10.1016/j.apm.2010.01.008.
[11]	H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publisher, Dordrecht, Boston, London, 1996.
[12]	G. H. Gao and Z. Z. Sun, The finite difference approximation for a class of fractional sub-diffusion equations on a space unbounded domain, Journal of Computational Physics, 236 (2013), 443-460. doi: 10.1016/j.jcp.2012.11.011.
[13]	D. Gerth, E. Klann, R. Ramlau and L. Reichel, On fractional tikhonov regularization, Journal of Inverse and Ill-posed Problems, 23 (2015), 611-625. doi: 10.1515/jiip-2014-0050.
[14]	C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations, 104p, Boston Pitman Publication, 1984.
[15]	Y. Han, X. Xiong and X. Xue, A fractional landweber method for solving backward time-fractional diffusion problem, Computers & Mathematics with Applications, 78 (2019), 81-91. doi: 10.1016/j.camwa.2019.02.017.
[16]	M. E. Hochstenbach and L. Reichel, Fractional tikhonov regularization for linear discrete ill-posed problems, BIT Numerical Mathematics, 51 (2011), 197-215. doi: 10.1007/s10543-011-0313-9.
[17]	A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer Science & Business Media, 2011. doi: 10.1007/978-1-4419-8474-6.
[18]	E. Klann, P. Maass and R. Ramlau, Two-step regularization methods for linear inverse problems, Journal of Inverse and Ill-posed Problems, 14 (2006), 583-607. doi: 10.1515/156939406778474523.
[19]	X. J. Li and C. J. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM Journal on Numerical Analysis, 47 (2009), 2108-2131. doi: 10.1137/080718942.
[20]	Y. M. Lin and C. J. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of computational physics, 225 (2007), 1533-1552. doi: 10.1016/j.jcp.2007.02.001.
[21]	J. J. Liu and M. Yamamoto, A backward problem for the time-fractional diffusion equation, Applicable Analysis, 89 (2010), 1769-1788. doi: 10.1080/00036810903479731.
[22]	R. L. Magin, Fractional Calculus in Bioengineering, volume 2(6)., Begell House Redding, 2006.
[23]	F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, 2010. doi: 10.1142/9781848163300.
[24]	R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics Reports, 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.
[25]	I. Podlubny, Fractional Differential Equations, Acad. Press, New York, 1999.
[26]	I. Podlubny and M. Kacenak, Mittag-leffler Function, the matlab routine, 2006.
[27]	M. Richter, Inverse Problems: Basics, Theory and Applications in Geophysics, Birkhäuser, 2016.
[28]	K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, Journal of Mathematical Analysis and Applications, 382 (2011), 426-447. doi: 10.1016/j.jmaa.2011.04.058.
[29]	F. Y. Song and C. J. Xu, Spectral direction splitting methods for two-dimensional space fractional diffusion equations, Journal of Computational Physics, 299 (2015), 196-214. doi: 10.1016/j.jcp.2015.07.011.
[30]	Z. Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Applied Numerical Mathematics, 56 (2006), 193-209. doi: 10.1016/j.apnum.2005.03.003.
[31]	J. G. Wang and T. Wei, An iterative method for backward time-fractional diffusion problem, Numerical Methods for Partial Differential Equations, 30 (2014), 2029-2041. doi: 10.1002/num.21887.
[32]	J. G. Wang, T. Wei and Y. B. Zhou, Tikhonov regularization method for a backward problem for the time-fractional diffusion equation, Applied Mathematical Modelling, 37 (2013), 8518-8532. doi: 10.1016/j.apm.2013.03.071.
[33]	L. Wang and J. J. Liu, Data regularization for a backward time-fractional diffusion problem, Computers & Mathematics with Applications, 64 (2012), 3613-3626. doi: 10.1016/j.camwa.2012.10.001.
[34]	T. Wei and J. G. Wang, A modified quasi-boundary value method for the backward time-fractional diffusion problem, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), 603-621. doi: 10.1051/m2an/2013107.
[35]	T. Wei and Y. Zhang, The backward problem for a time-fractional diffusion-wave equation in a bounded domain, Computers & Mathematics with Applications, 75 (2018), 3632-3648. doi: 10.1016/j.camwa.2018.02.022.
[36]	X. Xiong, X. Xue and Z. Qian, A modified iterative regularization method for ill-posed problems, Applied Numerical Mathematics, 122 (2017), 108-128. doi: 10.1016/j.apnum.2017.08.004.
[37]	F. Yang, Y. Zhang and X. X. Li, Landweber iterative method for identifying the initial value problem of the time-space fractional diffusion-wave equation, Numerical Algorithms, 83 (2020), 1509-1530. doi: 10.1007/s11075-019-00734-6.

Figure 1. The regularized solutions with $ \alpha = 1.1$ for Examples 5.1– 5.3, corresponding to the three figures from left to right. (a)–(c) for ExFLR; (d)–(f) for ImFLR

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Figure 2. Regularized solutions with $\alpha = 1.6 $ for Examples 5.1–5.3, corresponding to the three figures from left to right. (a)–(c) for ExFLR; (d)–(f) for ImFLR

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Figure 3. (ExFLR) Regularization parameter $ m $ and relative error $ e_r $ as functions of $ \nu $, vary $ \nu $ in the range $ \{0, 0.1, \cdots, 1\} $. (a) and (d) for Example 5.1 with $ \varepsilon = 10\% $; (b) and (e) for Example 5.2 with $ \varepsilon = 0.1\% $; (c) and (f) for Example 5.3 with $ \varepsilon = 0.1\% $

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Figure 4. The computed (regularized) solutions for Example 5.1-3 (from left to right) with $ \alpha = 1.6 $: smoothness comparison of the computed solutions for three different values of $ \nu $. (a)-(c) for ExFLR; (d)-(f) for ImFLR

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Figure 5. The computed initial conditions by ImFLR and absolute errors for Example 5.4. (a)—(c) for the regularized solutions $f_4^{m, \delta}$; (d)—(f) for the absolute errors $|f_4^{m, \delta}-f_4|$

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Table 1. Examples 5.1–5.3. Relative errors and regularization parameter versus relative noise levels

		$\alpha = 1.1 $				$\alpha = 1.6 $
		ExFLR (21)		ImFLR (22)		ExFLR (21)		ImFLR (22)
$ f $	$ \varepsilon $	$ m $	$ e_r(f^m_\delta, \varepsilon) $	$ m $	$ e_r(f^m_\delta, \varepsilon) $	$ m $	$ e_r(f^m_\delta, \varepsilon) $	$ m $	$ e_r(f^m_\delta, \varepsilon) $
$ f_1 $	$ 1\% $	7324	0.0103	7328	0.0103	147	0.0188	152	0.0187
	$ 5\% $	4775	0.0506	4778	0.0506	96	0.0634	99	0.0634
	$ 10\% $	3677	0.1003	3680	0.1002	73	0.1209	76	0.1188
$ f_2 $	$ 0.1\% $	14913	0.0323	14915	0.0323	510	0.0647	511	0.0641
	$ 0.5\% $	3846	0.0674	3846	0.0674	8	0.1183	13	0.1187
	$ 1\% $	701	0.1068	706	0.1067	6	0.1211	11	0.1200
$ f_3 $	$ 0.1\% $	232223	0.2947	232224	0.2947	69253	0.2728	69254	0.2728
	$ 0.5\% $	32815	0.3827	32817	0.3827	7456	0.3753	7458	0.3752
	$ 1\% $	20594	0.4186	20595	0.4186	4695	0.4032	4697	0.4032

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		$\alpha = 1.1 $				$\alpha = 1.6 $
		ExFLR (21)		ImFLR (22)		ExFLR (21)		ImFLR (22)
$ f $	$ \varepsilon $	$ m $	$ e_r(f^m_\delta, \varepsilon) $	$ m $	$ e_r(f^m_\delta, \varepsilon) $	$ m $	$ e_r(f^m_\delta, \varepsilon) $	$ m $	$ e_r(f^m_\delta, \varepsilon) $
$ f_1 $	$ 1\% $	7324	0.0103	7328	0.0103	147	0.0188	152	0.0187
	$ 5\% $	4775	0.0506	4778	0.0506	96	0.0634	99	0.0634
	$ 10\% $	3677	0.1003	3680	0.1002	73	0.1209	76	0.1188
$ f_2 $	$ 0.1\% $	14913	0.0323	14915	0.0323	510	0.0647	511	0.0641
	$ 0.5\% $	3846	0.0674	3846	0.0674	8	0.1183	13	0.1187
	$ 1\% $	701	0.1068	706	0.1067	6	0.1211	11	0.1200
$ f_3 $	$ 0.1\% $	232223	0.2947	232224	0.2947	69253	0.2728	69254	0.2728
	$ 0.5\% $	32815	0.3827	32817	0.3827	7456	0.3753	7458	0.3752
	$ 1\% $	20594	0.4186	20595	0.4186	4695	0.4032	4697	0.4032

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An extension of the landweber regularization for a backward time fractional wave problem

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American Institute of Mathematical Sciences

An extension of the landweber regularization for a backward time fractional wave problem

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