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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 & 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[22]

R. L. Magin, Fractional Calculus in Bioengineering, volume 2(6)., Begell House Redding, 2006. Google Scholar

[23]

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, 2010. doi: 10.1142/9781848163300.  Google Scholar

[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.  Google Scholar

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I. Podlubny and M. Kacenak, Mittag-leffler Function, the matlab routine, 2006. Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publisher, Dordrecht, Boston, London, 1996.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations, 104p, Boston Pitman Publication, 1984.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

R. L. Magin, Fractional Calculus in Bioengineering, volume 2(6)., Begell House Redding, 2006. Google Scholar

[23]

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, 2010. doi: 10.1142/9781848163300.  Google Scholar

[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.  Google Scholar

[25] I. Podlubny, Fractional Differential Equations, Acad. Press, New York, 1999.   Google Scholar
[26]

I. Podlubny and M. Kacenak, Mittag-leffler Function, the matlab routine, 2006. Google Scholar

[27]

M. Richter, Inverse Problems: Basics, Theory and Applications in Geophysics, Birkhäuser, 2016.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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