Abstract
In this research we study a problem of identifying of the right-hand side in a parabolic equation dependent on spatial variables in multidimensional domain. For numerical solution of the set inverse initial-boundary problem we use a conjugate gradient method with purely implicit time approximation. The results of the computational experiment performed on model problems with quasi-real solutions, including those with noise in input data are being discussed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Cannon, J.R.: Determination of an unknown heat source from overspecified boundary data. SIAM J. Numer. Anal. 5, 275–286 (1968)
D’haeyer, S., Johansson, B.T., Slodicka, M.: Reconstruction of a spacewise-dependent heat source in a time-dependent heat diffusion process. IMA J. Appl. Math. 79(1), 33–53 (2014)
Isakov, V.: Inverse Source Problems: Mathematical Surveys and Monographs, vol. 34. American Mathematical Society (AMS), Providence (1990)
Isakov, V.: Inverse parabolic problems with the final overdetermination. Commun. Pure Appl. Math. 44(2), 185–209 (1991)
Johansson, T., Lesnic, D.: Determination of a spacewise dependent heat source. J. Comput. Appl. Math. 209(1), 66–80 (2007)
Johansson, B.T., Lesnic, D.: A variational method for identifying a spacewise-dependent heat source. IMA J. Appl. Math. 72, 748–760 (2007)
Erdem, A., Lesnic, D., Hasanov, A.: Identification of a spacewise dependent heat source. Appl. Math. Model. 37, 10231–10244 (2013)
Kabanikhin, S.I.: Inverse and Ill-Posed Problems: Theory and Applications. De Gruyter, Berlin (2011)
Lions, J.-L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. I. Springer, Berlin (1972). https://doi.org/10.1007/978-3-642-65217-2
Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)
Samarskii, A.A.: The Theory of Difference Schemes. Marcel Dekker, New York (2001)
Samarskii, A.A., Vabishchevich, P.N.: Numerical Methods for Solving Inverse Problems of Mathematical Physics. De Gruyter, Berlin (2007)
Samarskii, A.A., Vabishchevich, P.N., Vasil’ev, V.I.: Iterative solution of a retrospective inverse problem of heat conduction. Mat. Model. 9(5), 119–127 (1997)
Vabishchevich, P.N.: Additive Operator-Difference Schemes. Walter de Gruyter GmbH, Berlin (2014)
Vabishchevich, P.N.: Iterative computational identification of a space-wise dependent source in parabolic equation. Inverse Probl. Sci. Eng. 25(8), 1168–1190 (2017)
Vasil’ev, V.I., Kardashevsky, A.M.: Iterative solution of the retrospective inverse problem for a parabolic equation using the conjugate gradient method. In: Dimov, I., Faragó, I., Vulkov, L. (eds.) Numerical Analysis and Its Applications. LNCS, vol. 10187, pp. 698–705. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-319-57099-0_80
Acknowledgments
The authors express their sincere gratitude to Professor P.N. Vabishchevich for constructive comments and fruitful discussions. The research was supported by the Government of the Russian Federation (project â„–Â 14.Y26.31.0013).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this paper
Cite this paper
Vasil’ev, V.I., Popov, V.V., Kardashevsky, A.M. (2018). Conjugate Gradient Method for Identification of a Spacewise Heat Source. In: Lirkov, I., Margenov, S. (eds) Large-Scale Scientific Computing. LSSC 2017. Lecture Notes in Computer Science(), vol 10665. Springer, Cham. https://doi.org/10.1007/978-3-319-73441-5_66
Download citation
DOI: https://doi.org/10.1007/978-3-319-73441-5_66
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-73440-8
Online ISBN: 978-3-319-73441-5
eBook Packages: Computer ScienceComputer Science (R0)