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Spectral Optimization Methods for the Time Fractional Diffusion Inverse Problem

Spectral Optimization Methods for the Time Fractional Diffusion Inverse Problem
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Year:    2013

Numerical Mathematics: Theory, Methods and Applications, Vol. 6 (2013), Iss. 3 : pp. 499–519

Abstract

An inverse problem of reconstructing the initial condition for a time fractional diffusion equation is investigated. On the basis of the optimal control framework, the uniqueness and first order necessary optimality condition of the minimizer for the objective functional are established, and a time-space spectral method is proposed to numerically solve the resulting minimization problem. The contribution of the paper is threefold: 1) a priori error estimate for the spectral approximation is derived; 2) a conjugate gradient optimization algorithm is designed to efficiently solve the inverse problem; 3) some numerical experiments are carried out to show that the proposed method is capable to find out the optimal initial condition, and that the convergence rate of the method is exponential if the optimal initial condition is smooth.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/nmtma.2013.1207nm

Numerical Mathematics: Theory, Methods and Applications, Vol. 6 (2013), Iss. 3 : pp. 499–519

Published online:    2013-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    21

Keywords:    Time fractional diffusion equation inverse problem spectral method error estimate conjugate gradient method.

  1. Recovering the Initial Distribution for a Time-Fractional Diffusion Equation

    Al-Jamal, Mohammad F.

    Acta Applicandae Mathematicae, Vol. 149 (2017), Iss. 1 P.87

    https://doi.org/10.1007/s10440-016-0088-8 [Citations: 7]

  2. Uniformly Stable Explicitly Solvable Finite Difference Method for Fractional Diffusion Equations

    Rui, Hongxing | Huang, Jian

    East Asian Journal on Applied Mathematics, Vol. 5 (2015), Iss. 1 P.29

    https://doi.org/10.4208/eajam.030614.051114a [Citations: 4]

  3. An efficient computational framework for data assimilation of fractional-order dynamical system with sparse observations

    Xu, Qinwu

    Computers & Mathematics with Applications, Vol. 176 (2024), Iss. P.12

    https://doi.org/10.1016/j.camwa.2024.09.004 [Citations: 0]

  4. Finite element approximation of optimal control problems governed by time fractional diffusion equation

    Zhou, Zhaojie | Gong, Wei

    Computers & Mathematics with Applications, Vol. 71 (2016), Iss. 1 P.301

    https://doi.org/10.1016/j.camwa.2015.11.014 [Citations: 69]

  5. On the formulation and numerical simulation of distributed-order fractional optimal control problems

    Zaky, M.A. | Machado, J.A. Tenreiro

    Communications in Nonlinear Science and Numerical Simulation, Vol. 52 (2017), Iss. P.177

    https://doi.org/10.1016/j.cnsns.2017.04.026 [Citations: 143]

  6. Inversion of the Initial Value for a Time-Fractional Diffusion-Wave Equation by Boundary Data

    Jiang, Suzhen | Liao, Kaifang | Wei, Ting

    Computational Methods in Applied Mathematics, Vol. 20 (2020), Iss. 1 P.109

    https://doi.org/10.1515/cmam-2018-0194 [Citations: 6]

  7. A posteriori error estimates for the fractional optimal control problems

    Ye, Xingyang | Xu, Chuanju

    Journal of Inequalities and Applications, Vol. 2015 (2015), Iss. 1

    https://doi.org/10.1186/s13660-015-0662-z [Citations: 5]

  8. Numerical solution of a parabolic optimal control problem with time-fractional derivative

    Romanenko, A

    Journal of Physics: Conference Series, Vol. 1158 (2019), Iss. P.042004

    https://doi.org/10.1088/1742-6596/1158/4/042004 [Citations: 1]

  9. Time-stepping discontinuous Galerkin approximation of optimal control problem governed by time fractional diffusion equation

    Zhou, Zhaojie | Zhang, Chenyang

    Numerical Algorithms, Vol. 79 (2018), Iss. 2 P.437

    https://doi.org/10.1007/s11075-017-0445-3 [Citations: 15]

  10. An Inverse Source Problem with Sparsity Constraint for the Time-Fractional Diffusion Equation

    Ruan, Zhousheng | Yang, Zhijian | Lu, Xiliang

    Advances in Applied Mathematics and Mechanics, Vol. 8 (2016), Iss. 1 P.1

    https://doi.org/10.4208/aamm.2014.m722 [Citations: 10]

  11. On spectral Petrov-Galerkin method for solving optimal control problem governed by a two-sided fractional diffusion equation

    Li, Shengyue | Cao, Wanrong | Wang, Yibo

    Computers & Mathematics with Applications, Vol. 107 (2022), Iss. P.104

    https://doi.org/10.1016/j.camwa.2021.12.020 [Citations: 7]

  12. Error analysis of spectral approximation for space–time fractional optimal control problems with control and state constraints

    Chen, Yanping | Lin, Xiuxiu | Huang, Yunqing

    Journal of Computational and Applied Mathematics, Vol. 413 (2022), Iss. P.114293

    https://doi.org/10.1016/j.cam.2022.114293 [Citations: 3]

  13. A spectral Galerkin approximation of optimal control problem governed by fractional advection–diffusion–reaction equations

    Wang, Fangyuan | Zhang, Zhongqiang | Zhou, Zhaojie

    Journal of Computational and Applied Mathematics, Vol. 386 (2021), Iss. P.113233

    https://doi.org/10.1016/j.cam.2020.113233 [Citations: 24]

  14. An efficient and accurate numerical method for the fractional optimal control problems with fractional Laplacian and state constraint

    Zhang, Jiaqi | Yang, Yin

    Numerical Methods for Partial Differential Equations, Vol. 39 (2023), Iss. 6 P.4403

    https://doi.org/10.1002/num.23056 [Citations: 0]

  15. Error Analysis of Fully Discrete Finite Element Approximations to an Optimal Control Problem Governed by a Time-Fractional PDE

    Gunzburger, Max | Wang, Jilu

    SIAM Journal on Control and Optimization, Vol. 57 (2019), Iss. 1 P.241

    https://doi.org/10.1137/17M1155636 [Citations: 22]

  16. Spectral Galerkin approximation of optimal control problem governed by Riesz fractional differential equation

    Zhang, Lu | Zhou, Zhaojie

    Applied Numerical Mathematics, Vol. 143 (2019), Iss. P.247

    https://doi.org/10.1016/j.apnum.2019.04.003 [Citations: 21]

  17. Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint

    Jin, Bangti | Li, Buyang | Zhou, Zhi

    IMA Journal of Numerical Analysis, Vol. 40 (2020), Iss. 1 P.377

    https://doi.org/10.1093/imanum/dry064 [Citations: 20]

  18. Finite element approximation of time fractional optimal control problem with integral state constraint

    Liu, Jie | Zhou, Zhaojie

    AIMS Mathematics, Vol. 6 (2021), Iss. 1 P.979

    https://doi.org/10.3934/math.2021059 [Citations: 11]

  19. A space-time spectral method for the time fractional diffusion optimal control problems

    Ye, Xingyang | Xu, Chuanju

    Advances in Difference Equations, Vol. 2015 (2015), Iss. 1

    https://doi.org/10.1186/s13662-015-0489-4 [Citations: 12]

  20. Fractional spectral collocation method for optimal control problem governed by space fractional diffusion equation

    Li, Shengyue | Zhou, Zhaojie

    Applied Mathematics and Computation, Vol. 350 (2019), Iss. P.331

    https://doi.org/10.1016/j.amc.2019.01.018 [Citations: 6]

  21. A stable explicitly solvable numerical method for the Riesz fractional advection–dispersion equations

    Zhang, Jingyuan

    Applied Mathematics and Computation, Vol. 332 (2018), Iss. P.209

    https://doi.org/10.1016/j.amc.2018.03.060 [Citations: 3]

  22. Alternating direction implicit Galerkin finite element method for the two-dimensional fractional diffusion-wave equation

    Li, Limei | Xu, Da | Luo, Man

    Journal of Computational Physics, Vol. 255 (2013), Iss. P.471

    https://doi.org/10.1016/j.jcp.2013.08.031 [Citations: 68]

  23. Spectral Galerkin method for state constrained optimal control of fractional advection‐diffusion‐reaction equations

    Wang, Fangyuan | Zhou, Zhaojie

    Numerical Methods for Partial Differential Equations, Vol. 38 (2022), Iss. 5 P.1526

    https://doi.org/10.1002/num.22853 [Citations: 1]

  24. Finite Element Approximation of Optimal Control Problem Governed by Space Fractional Equation

    Zhou, Zhaojie | Tan, Zhiyu

    Journal of Scientific Computing, Vol. 78 (2019), Iss. 3 P.1840

    https://doi.org/10.1007/s10915-018-0829-0 [Citations: 26]

  25. An approximate method for solving fractional partial differential equation by using an embedding process

    Ziaei, E | Farahi, M H | Ahmadian, A | Senu, N | Salahshour, S

    Journal of Physics: Conference Series, Vol. 1132 (2018), Iss. P.012060

    https://doi.org/10.1088/1742-6596/1132/1/012060 [Citations: 0]

  26. A Preconditioner for Galerkin–Legendre Spectral All-at-Once System from Time-Space Fractional Diffusion Equation

    Wang, Meijuan | Zhang, Shugong

    Symmetry, Vol. 15 (2023), Iss. 12 P.2144

    https://doi.org/10.3390/sym15122144 [Citations: 0]

  27. A posteriori error estimates of spectral method for the fractional optimal control problems with non-homogeneous initial conditions

    Ye, Xingyang | Xu, Chuanju

    AIMS Mathematics, Vol. 6 (2021), Iss. 11 P.12028

    https://doi.org/10.3934/math.2021697 [Citations: 5]

  28. Spectral Galerkin Approximation of Space Fractional Optimal Control Problem with Integral State Constraint

    Wang, Fangyuan | Li, Xiaodi | Zhou, Zhaojie

    Fractal and Fractional, Vol. 5 (2021), Iss. 3 P.102

    https://doi.org/10.3390/fractalfract5030102 [Citations: 2]

  29. A Priori Error Analysis for Time-Stepping Discontinuous Galerkin Finite Element Approximation of Time Fractional Optimal Control Problem

    Zhang, Chenyang | Liu, Huipo | Zhou, Zhaojie

    Journal of Scientific Computing, Vol. 80 (2019), Iss. 2 P.993

    https://doi.org/10.1007/s10915-019-00964-9 [Citations: 12]

  30. Identifying source term in the subdiffusion equation with L 2-TV regularization *

    Fan, Bin | Xu, Chuanju

    Inverse Problems, Vol. 37 (2021), Iss. 10 P.105008

    https://doi.org/10.1088/1361-6420/ac1e7f [Citations: 4]

  31. A novel high-order ADI method for 3D fractionalconvection–diffusion equations

    Zhai, Shuying | Gui, Dongwei | Huang, Pengzhan | Feng, Xinlong

    International Communications in Heat and Mass Transfer, Vol. 66 (2015), Iss. P.212

    https://doi.org/10.1016/j.icheatmasstransfer.2015.05.028 [Citations: 14]

  32. A Space-Time Fractional Optimal Control Problem: Analysis and Discretization

    Antil, Harbir | Otárola, Enrique | Salgado, Abner J.

    SIAM Journal on Control and Optimization, Vol. 54 (2016), Iss. 3 P.1295

    https://doi.org/10.1137/15M1014991 [Citations: 57]

  33. An unconditionally stable compact ADI method for three-dimensional time-fractional convection–diffusion equation

    Zhai, Shuying | Feng, Xinlong | He, Yinnian

    Journal of Computational Physics, Vol. 269 (2014), Iss. P.138

    https://doi.org/10.1016/j.jcp.2014.03.020 [Citations: 67]

  34. Integral constraint regularization method for fractional optimal control problem with pointwise state constraint

    Wang, Fangyuan | Chen, Chuanjun | Zhou, Zhaojie

    Chaos, Solitons & Fractals, Vol. 180 (2024), Iss. P.114559

    https://doi.org/10.1016/j.chaos.2024.114559 [Citations: 0]
  35. Frontiers in PDE-Constrained Optimization

    Optimization of a Fractional Differential Equation

    Otárola, Enrique | Salgado, Abner J.

    2018

    https://doi.org/10.1007/978-1-4939-8636-1_8 [Citations: 0]

  36. Error estimate for indirect spectral approximation of optimal control problem governed by fractional diffusion equation with variable diffusivity coefficient

    Wang, Fangyuan | Zheng, Xiangcheng | Zhou, Zhaojie

    Applied Numerical Mathematics, Vol. 170 (2021), Iss. P.146

    https://doi.org/10.1016/j.apnum.2021.07.021 [Citations: 0]