Identification of Matrix Diffusion Coefficient in a Parabolic PDE
-
Subhankar Mondal
and
M. Thamban Nair
Abstract
An inverse problem of identifying the diffusion coefficient in matrix form in a parabolic PDE is considered. Following the idea of natural linearization, considered by Cao and Pereverzev (2006), the nonlinear inverse problem is transformed into a problem of solving an operator equation where the operator involved is linear. Solving the linear operator equation turns out to be an ill-posed problem. The method of Tikhonov regularization is employed for obtaining stable approximations and its finite-dimensional analysis is done based on the Galerkin method, for which an orthogonal projection on the space of matrices with entries from is defined. Since the error estimates in Tikhonov regularization method rely heavily on the adjoint operator, an explicit representation of adjoint of the linear operator involved is obtained. For choosing the regularizing parameter, the adaptive technique is employed in order to obtain order optimal rate of convergence. For the relaxed noisy data, we describe a procedure for obtaining a smoothed version so as to obtain the error estimates. Numerical experiments are carried out for a few illustrative examples.
Acknowledgements
The authors thank the anonymous referees for many useful suggestions on the first draft of this paper.
References
[1] R. A. Adams, Sobolev Spaces, Pure Appl. Math. 65, Academic Press, New York, 1975. Search in Google Scholar
[2] J.-P. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris 256 (1963), 5042–5044. Search in Google Scholar
[3] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Texts Appl. Math. 15, Springer, New York, 2008. 10.1007/978-0-387-75934-0Search in Google Scholar
[4] H. Cao and S. V. Pereverzev, Natural linearization for the identification of a diffusion coefficient in a quasi-linear parabolic system from short-time observations, Inverse Problems 22 (2006), no. 6, 2311–2330. 10.1088/0266-5611/22/6/024Search in Google Scholar
[5] Q. Chen, A. Engström and J. Agren, On negative diagonal elements in the diffusion coefficient matrix of multicomponent systems, J. Phase Equilib. Diffus. 39 (2018), 592–596. 10.1007/s11669-018-0648-xSearch in Google Scholar
[6] P. Clément, Approximation by finite element functions using local regularization, Rev. Franç. Automat. Informat. Rech. Opérat. 9 (1975), no. R2, 77–84. 10.1051/m2an/197509R200771Search in Google Scholar
[7] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Math. Appl. 375, Kluwer Academic, Dordrecht, 1996. 10.1007/978-94-009-1740-8Search in Google Scholar
[8] H. W. Engl and J. Zou, A new approach to convergence rate analysis of Tikhonov regularization for parameter identification in heat conduction, Inverse Problems 16 (2000), no. 6, 1907–1923. 10.1088/0266-5611/16/6/319Search in Google Scholar
[9] L. C. Evans, Partial Differential Equations, 2nd ed., Grad. Stud. Math. 19, American Mathematical Society, Providence, 2010. Search in Google Scholar
[10] P. Favaro, S. Osher, S. Soatto and L. Vese, 3D shape from anisotropic diffusion, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, IEEE Press, Piscataway (2003), 10.1109/cvpr.2003.1211352. 10.1109/cvpr.2003.1211352Search in Google Scholar
[11] S. George and M. T. Nair, A modified Newton–Lavrentiev regularization for nonlinear ill-posed Hammerstein-type operator equations, J. Complexity 24 (2008), no. 2, 228–240. 10.1016/j.jco.2007.08.001Search in Google Scholar
[12] A. Ghafiri, J. Chaoufi, C. Vallee, E. H. Arjdal, J. C. Dupre, A. Germaneau, K. Atchonouglo and H. Fatmaoui, Identification of thermal parameters by treating the inverse problem, Internat. J. Comput. Appl. 87 (2014), no. 11, 1–5. 10.5120/15249-3719Search in Google Scholar
[13] M. Hanke and O. Scherzer, Error analysis of an equation error method for the identification of the diffusion coefficient in a quasi-linear parabolic differential equation, SIAM J. Appl. Math. 59 (1999), no. 3, 1012–1027. 10.1137/S0036139997331628Search in Google Scholar
[14] M. Heidernätsch, M. Bauer and G. Radonsa, Characterizing N-dimensional anisotropic Brownian motion by the distribution of diffusivities, J. Chem. Phys. 139 (2013), Article ID 184105. 10.1063/1.4828860Search in Google Scholar PubMed
[15] M. Hinze and T. N. T. Quyen, Matrix coefficient identification in an elliptic equation with the convex energy functional method, Inverse Problems 32 (2016), no. 8, Article ID 085007. 10.1088/0266-5611/32/8/085007Search in Google Scholar
[16] S. Kesavan, Topics in Functional Analysis and Applications, John Wiley & Sons, New York, 1989. Search in Google Scholar
[17] M. T. Nair, Functional Analysis: A First Course, PHI-Learning, New Delhi, 2002. Search in Google Scholar
[18] M. T. Nair, Linear Operator Equations: Approximation and Regularization, World Scientific, Hackensack, 2009. 10.1142/7055Search in Google Scholar
[19] M. T. Nair and S. Das Roy, A linear regularization method for a nonlinear parameter identification problem, J. Inverse Ill-Posed Probl. 25 (2017), no. 6, 687–701. 10.1515/jiip-2015-0091Search in Google Scholar
[20] S. Pereverzev and E. Schock, On the adaptive selection of the parameter in regularization of ill-posed problems, SIAM J. Numer. Anal. 43 (2005), no. 5, 2060–2076. 10.1137/S0036142903433819Search in Google Scholar
[21] R. Serrano, An alternative proof of the Aubin-Lions lemma, Arch. Math. (Basel) 101 (2013), no. 3, 253–257. 10.1007/s00013-013-0552-xSearch in Google Scholar
[22] J. Simon, Compact sets in the space , Ann. Mat. Pura Appl. (4) 146 (1986), 65–96. 10.1007/BF01762360Search in Google Scholar
[23] X.-C. Tai and T. Kärkkäinen, Identification of a nonlinear parameter in a parabolic equation from a linear equation, Comput. Appl. Math. 14 (1995), no. 2, 157–184. Search in Google Scholar
[24] G. Yuan and M. Yamamoto, Lipschitz stability in the determination of the principal part of a parabolic equation, ESAIM Control Optim. Calc. Var. 15 (2009), no. 3, 525–554. 10.1051/cocv:2008043Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Abstract
An inverse problem of identifying the diffusion coefficient in matrix form in a parabolic PDE is considered. Following the idea of natural linearization, considered by Cao and Pereverzev (2006), the nonlinear inverse problem is transformed into a problem of solving an operator equation where the operator involved is linear. Solving the linear operator equation turns out to be an ill-posed problem. The method of Tikhonov regularization is employed for obtaining stable approximations and its finite-dimensional analysis is done based on the Galerkin method, for which an orthogonal projection on the space of matrices with entries from is defined. Since the error estimates in Tikhonov regularization method rely heavily on the adjoint operator, an explicit representation of adjoint of the linear operator involved is obtained. For choosing the regularizing parameter, the adaptive technique is employed in order to obtain order optimal rate of convergence. For the relaxed noisy data, we describe a procedure for obtaining a smoothed version so as to obtain the error estimates. Numerical experiments are carried out for a few illustrative examples.
Acknowledgements
The authors thank the anonymous referees for many useful suggestions on the first draft of this paper.
References
[1] R. A. Adams, Sobolev Spaces, Pure Appl. Math. 65, Academic Press, New York, 1975. Search in Google Scholar
[2] J.-P. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris 256 (1963), 5042–5044. Search in Google Scholar
[3] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Texts Appl. Math. 15, Springer, New York, 2008. 10.1007/978-0-387-75934-0Search in Google Scholar
[4] H. Cao and S. V. Pereverzev, Natural linearization for the identification of a diffusion coefficient in a quasi-linear parabolic system from short-time observations, Inverse Problems 22 (2006), no. 6, 2311–2330. 10.1088/0266-5611/22/6/024Search in Google Scholar
[5] Q. Chen, A. Engström and J. Agren, On negative diagonal elements in the diffusion coefficient matrix of multicomponent systems, J. Phase Equilib. Diffus. 39 (2018), 592–596. 10.1007/s11669-018-0648-xSearch in Google Scholar
[6] P. Clément, Approximation by finite element functions using local regularization, Rev. Franç. Automat. Informat. Rech. Opérat. 9 (1975), no. R2, 77–84. 10.1051/m2an/197509R200771Search in Google Scholar
[7] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Math. Appl. 375, Kluwer Academic, Dordrecht, 1996. 10.1007/978-94-009-1740-8Search in Google Scholar
[8] H. W. Engl and J. Zou, A new approach to convergence rate analysis of Tikhonov regularization for parameter identification in heat conduction, Inverse Problems 16 (2000), no. 6, 1907–1923. 10.1088/0266-5611/16/6/319Search in Google Scholar
[9] L. C. Evans, Partial Differential Equations, 2nd ed., Grad. Stud. Math. 19, American Mathematical Society, Providence, 2010. Search in Google Scholar
[10] P. Favaro, S. Osher, S. Soatto and L. Vese, 3D shape from anisotropic diffusion, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, IEEE Press, Piscataway (2003), 10.1109/cvpr.2003.1211352. 10.1109/cvpr.2003.1211352Search in Google Scholar
[11] S. George and M. T. Nair, A modified Newton–Lavrentiev regularization for nonlinear ill-posed Hammerstein-type operator equations, J. Complexity 24 (2008), no. 2, 228–240. 10.1016/j.jco.2007.08.001Search in Google Scholar
[12] A. Ghafiri, J. Chaoufi, C. Vallee, E. H. Arjdal, J. C. Dupre, A. Germaneau, K. Atchonouglo and H. Fatmaoui, Identification of thermal parameters by treating the inverse problem, Internat. J. Comput. Appl. 87 (2014), no. 11, 1–5. 10.5120/15249-3719Search in Google Scholar
[13] M. Hanke and O. Scherzer, Error analysis of an equation error method for the identification of the diffusion coefficient in a quasi-linear parabolic differential equation, SIAM J. Appl. Math. 59 (1999), no. 3, 1012–1027. 10.1137/S0036139997331628Search in Google Scholar
[14] M. Heidernätsch, M. Bauer and G. Radonsa, Characterizing N-dimensional anisotropic Brownian motion by the distribution of diffusivities, J. Chem. Phys. 139 (2013), Article ID 184105. 10.1063/1.4828860Search in Google Scholar PubMed
[15] M. Hinze and T. N. T. Quyen, Matrix coefficient identification in an elliptic equation with the convex energy functional method, Inverse Problems 32 (2016), no. 8, Article ID 085007. 10.1088/0266-5611/32/8/085007Search in Google Scholar
[16] S. Kesavan, Topics in Functional Analysis and Applications, John Wiley & Sons, New York, 1989. Search in Google Scholar
[17] M. T. Nair, Functional Analysis: A First Course, PHI-Learning, New Delhi, 2002. Search in Google Scholar
[18] M. T. Nair, Linear Operator Equations: Approximation and Regularization, World Scientific, Hackensack, 2009. 10.1142/7055Search in Google Scholar
[19] M. T. Nair and S. Das Roy, A linear regularization method for a nonlinear parameter identification problem, J. Inverse Ill-Posed Probl. 25 (2017), no. 6, 687–701. 10.1515/jiip-2015-0091Search in Google Scholar
[20] S. Pereverzev and E. Schock, On the adaptive selection of the parameter in regularization of ill-posed problems, SIAM J. Numer. Anal. 43 (2005), no. 5, 2060–2076. 10.1137/S0036142903433819Search in Google Scholar
[21] R. Serrano, An alternative proof of the Aubin-Lions lemma, Arch. Math. (Basel) 101 (2013), no. 3, 253–257. 10.1007/s00013-013-0552-xSearch in Google Scholar
[22] J. Simon, Compact sets in the space , Ann. Mat. Pura Appl. (4) 146 (1986), 65–96. 10.1007/BF01762360Search in Google Scholar
[23] X.-C. Tai and T. Kärkkäinen, Identification of a nonlinear parameter in a parabolic equation from a linear equation, Comput. Appl. Math. 14 (1995), no. 2, 157–184. Search in Google Scholar
[24] G. Yuan and M. Yamamoto, Lipschitz stability in the determination of the principal part of a parabolic equation, ESAIM Control Optim. Calc. Var. 15 (2009), no. 3, 525–554. 10.1051/cocv:2008043Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- A General Error Estimate For Parabolic Variational Inequalities
- Functional A Posteriori Error Estimates for the Parabolic Obstacle Problem
- Multi-Scale Paraxial Models to Approximate Vlasov–Maxwell Equations
- Finite Element Penalty Method for the Oldroyd Model of Order One with Non-smooth Initial Data
- Quasi-Newton Iterative Solution of Non-Orthotropic Elliptic Problems in 3D with Boundary Nonlinearity
- Improving Regularization Techniques for Incompressible Fluid Flows via Defect Correction
- Fully Discrete Finite Element Approximation of the MHD Flow
- Sparse Data-Driven Quadrature Rules via ℓ p -Quasi-Norm Minimization
- Identification of Matrix Diffusion Coefficient in a Parabolic PDE
- A Finite Element Method for Two-Phase Flow with Material Viscous Interface
- On the Finite Element Approximation of Fourth-Order Singularly Perturbed Eigenvalue Problems
- Application of Fourier Truncation Method to Numerical Differentiation for Bivariate Functions
- Factorized Schemes for First and Second Order Evolution Equations with Fractional Powers of Operators
Articles in the same Issue
- Frontmatter
- A General Error Estimate For Parabolic Variational Inequalities
- Functional A Posteriori Error Estimates for the Parabolic Obstacle Problem
- Multi-Scale Paraxial Models to Approximate Vlasov–Maxwell Equations
- Finite Element Penalty Method for the Oldroyd Model of Order One with Non-smooth Initial Data
- Quasi-Newton Iterative Solution of Non-Orthotropic Elliptic Problems in 3D with Boundary Nonlinearity
- Improving Regularization Techniques for Incompressible Fluid Flows via Defect Correction
- Fully Discrete Finite Element Approximation of the MHD Flow
- Sparse Data-Driven Quadrature Rules via ℓ p -Quasi-Norm Minimization
- Identification of Matrix Diffusion Coefficient in a Parabolic PDE
- A Finite Element Method for Two-Phase Flow with Material Viscous Interface
- On the Finite Element Approximation of Fourth-Order Singularly Perturbed Eigenvalue Problems
- Application of Fourier Truncation Method to Numerical Differentiation for Bivariate Functions
- Factorized Schemes for First and Second Order Evolution Equations with Fractional Powers of Operators
