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June 2018, 12(3): 773-799. doi: 10.3934/ipi.2018033

Backward problem for a time-space fractional diffusion equation

1. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

2. 

Department of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

3. 

School of Electronic and Information Engineering, Xi'an Jiaotong University, Xi'an 710049, China

4. 

Department of Bioengineering, Xi'an Jiaotong University, Xi'an 710049, China

* Corresponding author: Jigen Peng

Received  March 2016 Revised  January 2018 Published  March 2018

In this paper, a backward problem for a time-space fractional diffusion process has been considered. For this problem, we propose to construct the initial function by minimizing data residual error in Fourier space domain with variable total variation (TV) regularizing term which can protect the edges as TV regularizing term and reduce staircasing effect. The well-posedness of this optimization problem is obtained under a very general setting. Actually, we rewrite the time-space fractional diffusion equation as an abstract fractional differential equation and deduce our results by using fractional operator semigroup theory, hence, our theoretical results can be applied to other backward problems for the differential equations with more general fractional operator. Then a modified Bregman iterative algorithm has been proposed to approximate the minimizer. The new features of this algorithm is that the regularizing term altered in each step and we need not to solve the complex Euler-Lagrange equation of variable TV regularizing term (just need to solve a simple Euler-Lagrange equation). The convergence of this algorithm and the strategy of choosing parameters are also obtained. Numerical implementations are provided to support our theoretical analysis to show the flexibility of our minimization model.

Citation: Junxiong Jia, Jigen Peng, Jinghuai Gao, Yujiao Li. Backward problem for a time-space fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (3) : 773-799. doi: 10.3934/ipi.2018033
References:
[1]

R. Acar and C. R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Probl., 10 (1994), 1217-1229. doi: 10.1088/0266-5611/10/6/003.

[2]

W. Arendt, C. J. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Birkhäuser Verlag, Basel, 2001.

[3]

B. BaeumerM. Kovács and H. Sankaranarayanan, Higher order Grünwald approximations of fractional derivatives and fractional powers of operators, T. Am. Math. Soc., 367 (2015), 813-834. doi: 10.1090/S0002-9947-2014-05887-X.

[4]

B. BaeumerM. KovácsM. M. MeerschaertR. Schilling and P. Straka, Reflected spectrally negative stable processes and their governing equations, T. Am. Math. Soc., 368 (2016), 227-248. doi: 10.1090/tran/6360.

[5]

B. BaeumerM. M. Meerschaert and E. Nane, Space-time duality for fractional diffusion, J. Appl. Probab., 46 (2009), 1100-1115. doi: 10.1239/jap/1261670691.

[6]

B. BaeumerS. Kurita and M. M. Meerschaert, Inhomogeneous fractional diffusion equations, Fract. Calc. Appl. Ana., 8 (2005), 371-386.

[7]

E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, Ph. D thesis, Eindhoven University of Technology in Eindhoven, 2001.

[8]

P. Blomgren, T. F. Chan, P. Mulet, L. Vese and W. L. Wan, Variational PDE Models and Methods for Image Processing, Chapman & Hall/CRC, Boca Raton, 2000.

[9]

E. M. BolltR. ChartrandS. Esedo$\bar{\text{g}}$luP. Schultz and K. R. Vixie, Graduated adaptive image denoising: Local compromise between total variation and isotropic diffusion, Adv. Comput. Math., 31 (2009), 61-85. doi: 10.1007/s10444-008-9082-7.

[10]

T. Bui-Thanh and O. Ghattas, A scalable algorithm for MAP estimators in Bayesian inverse problems with Besov priors, Inverse Probl. Imag., 9 (2015), 27-53. doi: 10.3934/ipi.2015.9.27.

[11]

C. M. Carracedo and M. S. Alix, The Theory of Fractional Powers of Operators, North-Holland Publishing Co., Amsterdam, 2001.

[12]

J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Probl. , 25 (2009), 115002, 16pp. doi: 10.1088/0266-5611/25/11/115002.

[13]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998.

[14]

T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems, SIAM J. Imaging Sci., 2 (2009), 323-343. doi: 10.1137/080725891.

[15]

M. Haase, The Functional Calculus for Sectorial Operators, Birkhäuser Verlag, Basel, 2006.

[16]

P. HarjulehtoP. HästöV. Latvala and O. Toivanen, Critical variable exponent functionals in image restoration, Appl. Math. Lett., 26 (2013), 56-60. doi: 10.1016/j.aml.2012.03.032.

[17]

P. HarjulehtoP. Hästö and V. Latvala, Minimizers of the variable exponent, non-uniformly convex Dirichlet energy, J. Math. Pure. Appl., 89 (2008), 174-197. doi: 10.1016/j.matpur.2007.10.006.

[18]

N. Jacob, Pseudo Differential Operators and Markov Processes: Fourier Analysis and Semigroups, Imperial College Press, London, 2001.

[19]

J. JiaJ. Peng and K. Li, Well-posedness of abstract distributed-order fractional-order fractional diffusion equations, Commun. Pur. Appl. Anal., 13 (2014), 605-621. doi: 10.3934/cpaa.2014.13.605.

[20]

B. JinR. LazarovD. Sheen and Z. Zhou, Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data, Fract. Calc. Appl. Ana., 19 (2016), 69-93. doi: 10.1515/fca-2016-0005.

[21]

B. Jin and W. Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse Probl., 31 (2015), 035003, 40pp. doi: 10.1088/0266-5611/31/3/035003.

[22]

R. Klages, G. Radons and I. M. Sokolov, Anomalous Transport: Foundations and Applications, Wiley-VCH Verlag GmbH & Co. KGaA, Darmstadt, 2008. doi: 10.1002/9783527622979.

[23]

F. LiZ. Li and L. Pi, Variable exponent functionals in image restoration, Appl. Math. Comput., 216 (2010), 870-882. doi: 10.1016/j.amc.2010.01.094.

[24]

M. LiC. Chen and F. B. Li, On fractional powers of generators of fractional resolvent families, J. Funct. Anal., 259 (2010), 2702-2726. doi: 10.1016/j.jfa.2010.07.007.

[25]

J. Liu and M. Yamamoto, A backward problem for the time-fractional diffusion equation, Appl. Anal., 89 (2010), 1769-1788. doi: 10.1080/00036810903479731.

[26]

F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Soltion Frac., 7 (1996), 1461-1477. doi: 10.1016/0960-0779(95)00125-5.

[27]

F. Mainardi and M. Tomirotti, On a special function arising in the time fractional diffusion-wave equation, Transform Methods and Special Functions, Sofia'94, Proceedings of International Workshop, (1994), 171-183.

[28]

M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, Walter de Gruyter & Co, Berlin/Boston, 2012. doi: 10.1515/9783110258165.

[29]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.

[30]

S. OsherM. BurgerD. GoldfarbJ. Xu and W. Yin, An iterative regularization method for total variation-based image restoration, Multiscale Model. Sim., 4 (2005), 460-489. doi: 10.1137/040605412.

[31]

J. Peng and K. Li, A novel characteristic of solution operator for the fractional abstract Cauchy problem, J. Math. Anal. Appl., 385 (2012), 786-796. doi: 10.1016/j.jmaa.2011.07.009.

[32]

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, Inc., San Diego, CA, 1999.

[33]

I. Podlubny, Matlab program for computing Mittag-Leffler fuction $E_{α, β}(·)$, http://www.mathworks.com/matlabcentral/fileexchange/8738-mittag-leffler-function.

[34]

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.

[35]

J. Tiirola, Image decompositions using spaces of variable smoothness and integrability, SIAM J. Imaging Sci., 7 (2014), 1558-1587. doi: 10.1137/130923324.

[36]

L. Wang and J. Liu, Total variation regularization for a backward time-fractional diffusion problem, Inverse Probl. 29 (2013), 115013, 22pp. doi: 10.1088/0266-5611/29/11/115013.

[37]

Z. WangJ. GaoQ. ZhouK. Li and J. Peng, A new extension of seismic instantaneous frequency using a fractional time derivative, J. Appl. Geophys., 98 (2013), 176-181. doi: 10.1016/j.jappgeo.2013.08.016.

[38]

M. X. Wang, Operator Semigroup and Evolutionary Equations (in Chinese), Science Press, China, 2006.

[39]

E. M. Wright, The generalized Bessel function of order greater than one, Q. J. Math., 11 (1940), 36-48. doi: 10.1093/qmath/os-11.1.36.

[40]

G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371 (2002), 461-580. doi: 10.1016/S0370-1573(02)00331-9.

[41]

Y. Zhang and X. Xu, Inverse source problem for a fractional diffusion equation, Inverse Probl. 27 (2011), 035010, 12pp. doi: 10.1088/0266-5611/27/3/035010.

[42]

G. H. Zheng and T. Wei, Two regularization methods for solving a Riesz-Feller space-fractional backward diffusion problem, Inverse Probl. 26 (2010), 115017, 22pp. doi: 10.1088/0266-5611/26/11/115017.

show all references

References:
[1]

R. Acar and C. R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Probl., 10 (1994), 1217-1229. doi: 10.1088/0266-5611/10/6/003.

[2]

W. Arendt, C. J. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Birkhäuser Verlag, Basel, 2001.

[3]

B. BaeumerM. Kovács and H. Sankaranarayanan, Higher order Grünwald approximations of fractional derivatives and fractional powers of operators, T. Am. Math. Soc., 367 (2015), 813-834. doi: 10.1090/S0002-9947-2014-05887-X.

[4]

B. BaeumerM. KovácsM. M. MeerschaertR. Schilling and P. Straka, Reflected spectrally negative stable processes and their governing equations, T. Am. Math. Soc., 368 (2016), 227-248. doi: 10.1090/tran/6360.

[5]

B. BaeumerM. M. Meerschaert and E. Nane, Space-time duality for fractional diffusion, J. Appl. Probab., 46 (2009), 1100-1115. doi: 10.1239/jap/1261670691.

[6]

B. BaeumerS. Kurita and M. M. Meerschaert, Inhomogeneous fractional diffusion equations, Fract. Calc. Appl. Ana., 8 (2005), 371-386.

[7]

E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, Ph. D thesis, Eindhoven University of Technology in Eindhoven, 2001.

[8]

P. Blomgren, T. F. Chan, P. Mulet, L. Vese and W. L. Wan, Variational PDE Models and Methods for Image Processing, Chapman & Hall/CRC, Boca Raton, 2000.

[9]

E. M. BolltR. ChartrandS. Esedo$\bar{\text{g}}$luP. Schultz and K. R. Vixie, Graduated adaptive image denoising: Local compromise between total variation and isotropic diffusion, Adv. Comput. Math., 31 (2009), 61-85. doi: 10.1007/s10444-008-9082-7.

[10]

T. Bui-Thanh and O. Ghattas, A scalable algorithm for MAP estimators in Bayesian inverse problems with Besov priors, Inverse Probl. Imag., 9 (2015), 27-53. doi: 10.3934/ipi.2015.9.27.

[11]

C. M. Carracedo and M. S. Alix, The Theory of Fractional Powers of Operators, North-Holland Publishing Co., Amsterdam, 2001.

[12]

J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Probl. , 25 (2009), 115002, 16pp. doi: 10.1088/0266-5611/25/11/115002.

[13]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998.

[14]

T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems, SIAM J. Imaging Sci., 2 (2009), 323-343. doi: 10.1137/080725891.

[15]

M. Haase, The Functional Calculus for Sectorial Operators, Birkhäuser Verlag, Basel, 2006.

[16]

P. HarjulehtoP. HästöV. Latvala and O. Toivanen, Critical variable exponent functionals in image restoration, Appl. Math. Lett., 26 (2013), 56-60. doi: 10.1016/j.aml.2012.03.032.

[17]

P. HarjulehtoP. Hästö and V. Latvala, Minimizers of the variable exponent, non-uniformly convex Dirichlet energy, J. Math. Pure. Appl., 89 (2008), 174-197. doi: 10.1016/j.matpur.2007.10.006.

[18]

N. Jacob, Pseudo Differential Operators and Markov Processes: Fourier Analysis and Semigroups, Imperial College Press, London, 2001.

[19]

J. JiaJ. Peng and K. Li, Well-posedness of abstract distributed-order fractional-order fractional diffusion equations, Commun. Pur. Appl. Anal., 13 (2014), 605-621. doi: 10.3934/cpaa.2014.13.605.

[20]

B. JinR. LazarovD. Sheen and Z. Zhou, Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data, Fract. Calc. Appl. Ana., 19 (2016), 69-93. doi: 10.1515/fca-2016-0005.

[21]

B. Jin and W. Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse Probl., 31 (2015), 035003, 40pp. doi: 10.1088/0266-5611/31/3/035003.

[22]

R. Klages, G. Radons and I. M. Sokolov, Anomalous Transport: Foundations and Applications, Wiley-VCH Verlag GmbH & Co. KGaA, Darmstadt, 2008. doi: 10.1002/9783527622979.

[23]

F. LiZ. Li and L. Pi, Variable exponent functionals in image restoration, Appl. Math. Comput., 216 (2010), 870-882. doi: 10.1016/j.amc.2010.01.094.

[24]

M. LiC. Chen and F. B. Li, On fractional powers of generators of fractional resolvent families, J. Funct. Anal., 259 (2010), 2702-2726. doi: 10.1016/j.jfa.2010.07.007.

[25]

J. Liu and M. Yamamoto, A backward problem for the time-fractional diffusion equation, Appl. Anal., 89 (2010), 1769-1788. doi: 10.1080/00036810903479731.

[26]

F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Soltion Frac., 7 (1996), 1461-1477. doi: 10.1016/0960-0779(95)00125-5.

[27]

F. Mainardi and M. Tomirotti, On a special function arising in the time fractional diffusion-wave equation, Transform Methods and Special Functions, Sofia'94, Proceedings of International Workshop, (1994), 171-183.

[28]

M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, Walter de Gruyter & Co, Berlin/Boston, 2012. doi: 10.1515/9783110258165.

[29]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.

[30]

S. OsherM. BurgerD. GoldfarbJ. Xu and W. Yin, An iterative regularization method for total variation-based image restoration, Multiscale Model. Sim., 4 (2005), 460-489. doi: 10.1137/040605412.

[31]

J. Peng and K. Li, A novel characteristic of solution operator for the fractional abstract Cauchy problem, J. Math. Anal. Appl., 385 (2012), 786-796. doi: 10.1016/j.jmaa.2011.07.009.

[32]

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, Inc., San Diego, CA, 1999.

[33]

I. Podlubny, Matlab program for computing Mittag-Leffler fuction $E_{α, β}(·)$, http://www.mathworks.com/matlabcentral/fileexchange/8738-mittag-leffler-function.

[34]

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.

[35]

J. Tiirola, Image decompositions using spaces of variable smoothness and integrability, SIAM J. Imaging Sci., 7 (2014), 1558-1587. doi: 10.1137/130923324.

[36]

L. Wang and J. Liu, Total variation regularization for a backward time-fractional diffusion problem, Inverse Probl. 29 (2013), 115013, 22pp. doi: 10.1088/0266-5611/29/11/115013.

[37]

Z. WangJ. GaoQ. ZhouK. Li and J. Peng, A new extension of seismic instantaneous frequency using a fractional time derivative, J. Appl. Geophys., 98 (2013), 176-181. doi: 10.1016/j.jappgeo.2013.08.016.

[38]

M. X. Wang, Operator Semigroup and Evolutionary Equations (in Chinese), Science Press, China, 2006.

[39]

E. M. Wright, The generalized Bessel function of order greater than one, Q. J. Math., 11 (1940), 36-48. doi: 10.1093/qmath/os-11.1.36.

[40]

G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371 (2002), 461-580. doi: 10.1016/S0370-1573(02)00331-9.

[41]

Y. Zhang and X. Xu, Inverse source problem for a fractional diffusion equation, Inverse Probl. 27 (2011), 035010, 12pp. doi: 10.1088/0266-5611/27/3/035010.

[42]

G. H. Zheng and T. Wei, Two regularization methods for solving a Riesz-Feller space-fractional backward diffusion problem, Inverse Probl. 26 (2010), 115017, 22pp. doi: 10.1088/0266-5611/26/11/115017.

Figure 1.  Left: Initial data; Right: The solution of the fractional diffusion equation (2) at time $T = 1$ with $\alpha = 0.6$, $\beta = 0.9$.
Figure 2.  Left: Boundaries of the initial data; Right: Boundaries of the solution of the fractional diffusion equation (2) at time $T = 1$ with $\alpha = 0.6$, $\beta = 0.9$.
Figure 3.  Left: Original function; Middle: Recovered function by variable TV model with $\delta = 0.0005$; Right: Recovered function by variable TV model with $\delta = 0.005$ for Example 1.
Figure 4.  Initial function for Example 2.
Figure 5.  Left: Recovered function by variable TV model with $\delta = 0.0005$ for Example 2; Right: Recovered function by the variable TV model with $\delta = 0.005$ for Example 2.
Figure 6.  The curve of the relative error of the recovered data for different values of parameter $\alpha$
Table 1.  The values of RelErr of three methods for Example 1
RelErr TV model Tikhonov model Variable TV model
$\sigma = 0.0005$ $3.8283\%$ $0.3857\%$ $0.3696\%$
$\sigma = 0.005$ $8.8646\%$ $0.6559\%$ $0.6597\%$
RelErr TV model Tikhonov model Variable TV model
$\sigma = 0.0005$ $3.8283\%$ $0.3857\%$ $0.3696\%$
$\sigma = 0.005$ $8.8646\%$ $0.6559\%$ $0.6597\%$
Table 2.  The values of RelErr of three methods for Example 2
RelErr TV model Tikhonov model Variable TV model
$\sigma = 0.0005$ $13.0053\%$ $13.7772\%$ $13.0666\%$
$\sigma = 0.005$ $22.7222\%$ $25.2101\%$ $22.7810\%$
RelErr TV model Tikhonov model Variable TV model
$\sigma = 0.0005$ $13.0053\%$ $13.7772\%$ $13.0666\%$
$\sigma = 0.005$ $22.7222\%$ $25.2101\%$ $22.7810\%$
Table 3.  The values of RelErr with different parameters λ of Variable TV model for Example 2
$\lambda = 10^{11}$ $\lambda = \frac{1}{4}\times 10^{11}$ $\lambda = \frac{1}{16}\times 10^{11}$
$\sigma = 0.0005$ $\text{M} = 9$ $\text{M} = 34$ $\text{M} = 150$
$\text{RelErr} = 13.0792\%$ $\text{RelErr} = 13.0342\%$ $\text{RelErr} = 13.0275\%$
$\lambda = 10^{11}$ $\lambda = \frac{1}{4}\times 10^{11}$ $\lambda = \frac{1}{16}\times 10^{11}$
$\sigma = 0.0005$ $\text{M} = 9$ $\text{M} = 34$ $\text{M} = 150$
$\text{RelErr} = 13.0792\%$ $\text{RelErr} = 13.0342\%$ $\text{RelErr} = 13.0275\%$
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