Numerical identification of a Robin coefficient in parabolic problems

Jin, Bangti; Lu, Xiliang

doi:10.1090/S0025-5718-2012-02559-2

Numerical identification of a Robin coefficient in parabolic problems
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by Bangti Jin and Xiliang Lu PDF

Math. Comp. 81 (2012), 1369-1398 Request permission

Abstract:

This paper studies a regularization approach for an inverse problem of estimating a spatially-and-temporally dependent Robin coefficient arising in the analysis of convective heat transfer. The parameter-to-state map is analyzed, especially a differentiability result is established. A regularization approach is proposed, and the properties, e.g., existence and optimality system, of the functional are investigated. A finite element method is adopted for discretizing the continuous optimization problem, and the convergence of the finite element approximations as the mesh size and temporal step size tend to zero is established. Numerical results by the conjugate gradient method for one- and two-dimensional problems are presented.

References

Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
Oleg M. Alifanov, Inverse heat transfer problems, Springer-Verlag, Berlin, 1994.
James V. Beck, Ben Blackwell, and Charles R. St. Clair, Inverse Heat Conduction: Ill-Posed Problems, Wiley-Interscience, New York, 1985.
Mourad Bellassoued, Jin Cheng, and Mourad Choulli, Stability estimate for an inverse boundary coefficient problem in thermal imaging, J. Math. Anal. Appl. 343 (2008), no. 1, 328–336. MR 2412131 (2009e:35299)
James H. Bramble, Joseph E. Pasciak, and Olaf Steinbach, On the stability of the $L^2$ projection in $H^1(\Omega )$, Math. Comp. 71 (2002), no. 237, 147–156. MR 1862992 (2002h:65175)
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954, DOI 10.1007/978-0-387-75934-0
James H. Bramble and Jinchao Xu, Some estimates for a weighted $L^2$ projection, Math. Comp. 56 (1991), no. 194, 463–476. MR 1066830 (91k:65140)
S. Chaabane, I. Fellah, M. Jaoua, and J. Leblond, Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems, Inverse Problems 20 (2004), no. 1, 47–59. MR 2044605 (2005a:35276)
K. Chrysafinos, M. D. Gunzburger, and L. S. Hou, Semidiscrete approximations of optimal Robin boundary control problems constrained by semilinear parabolic PDE, J. Math. Anal. Appl. 323 (2006), no. 2, 891–912. MR 2260151 (2007h:49041)
S. Chantasiriwan, Inverse heat conduction problem of determining time-dependent heat transfer coefficient, Int. J. Heat Mass Transfer 42 (1999), no. 23, 4275–4285.
Xu-Yan Chen, A strong unique continuation theorem for parabolic equations, Math. Ann. 311 (1998), no. 4, 603–630. MR 1637972, DOI 10.1007/s002080050202
Mourad Choulli, On the determination of an unknown boundary function in a parabolic equation, Inverse Problems 15 (1999), no. 3, 659–667. MR 1696926, DOI 10.1088/0266-5611/15/3/302
Slim Chaabane and Mohamed Jaoua, Identification of Robin coefficients by the means of boundary measurements, Inverse Problems 15 (1999), no. 6, 1425–1438. MR 1733209, DOI 10.1088/0266-5611/15/6/303
Han-Taw Chen and Xin-Yi Wu, Investigation of heat transfer coefficient in two-dimensional transient inverse heat conduction problems using the hybrid inverse scheme, Int. J. Numer. Methods Engrg. 73 (2008), no. 1, 107–122. MR 2378451 (2009b:65242)
Lawrence C. Evans and Ronald F. Gariepy, Measure Theory and Fine Properties of Functions, CRC, Boca Raton, 1992. MR 1158660 (93f:28001)
Lawrence C. Evans, Partial Differential Equations, AMS, Providence, R.I., 1998.
Konrad Gröger, $W^{1,p}$-estimates of solutions to evolution equations corresponding to nonsmooth second order elliptic differential operators, Nonlinear Anal. 18 (1992), no. 6, 569–577. MR 1154481, DOI 10.1016/0362-546X(92)90211-V
D. Hömberg, C. Meyer, J. Rehberg, and W. Ring, Optimal control for the thermistor problem, SIAM J. Control Optim. 48 (2009/10), no. 5, 3449–3481. MR 2599927, DOI 10.1137/080736259
Bangti Jin, Conjugate gradient method for the Robin inverse problem associated with the Laplace equation, Internat. J. Numer. Methods Engrg. 71 (2007), no. 4, 433–453. MR 2332815, DOI 10.1002/nme.1949
Bangti Jin and Jun Zou, Inversion of Robin coefficient by a spectral stochastic finite element approach, J. Comput. Phys. 227 (2008), no. 6, 3282–3306. MR 2392734, DOI 10.1016/j.jcp.2007.11.042
—, Numerical estimation of piecewise constant Robin coefficient, SIAM J. Control Optim. 48 (2009), no. 3, 1977–2002. MR 2516196
Bangti Jin and Jun Zou, Numerical estimation of the Robin coefficient in a stationary diffusion equation, IMA J. Numer. Anal. 30 (2010), no. 3, 677–701. MR 2670110, DOI 10.1093/imanum/drn066
Refahi Abou Khachfe and Yvon Jarny, Determination of heat sources and heat transfer coefficient for two-dimensional heat flow – numerical and experimental study, Int. J. Heat Mass Transfer 44 (2001), no. 7, 1309–1322.
Ian Knowles, A variational algorithm for electrical impedance tomography, Inverse Problems 14 (1998), no. 6, 1513–1525. MR 1662464, DOI 10.1088/0266-5611/14/6/010
F Kreith (ed.), The CRC Handbook of Thermal Engineering, CRC, Bota Raton, 2000.
David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 567696
Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
S. Lenhart and D. G. Wilson, Optimal control of a heat transfer problem with convective boundary condition, J. Optim. Theory Appl. 79 (1993), no. 3, 581–597. MR 1255288 (94j:49005)
Sigeru Mizohata, Unicité du prolongement des solutions pour quelques opérateurs différentiels paraboliques, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 31 (1958), 219–239 (French). MR 106347, DOI 10.1215/kjm/1250776858
Dinh Nho Hào, A noncharacteristic Cauchy problem for linear parabolic equations and related inverse problems. I. Solvability, Inverse Problems 10 (1994), no. 2, 295–315. MR 1269009
Dinh Nho Hào and H.-J. Reinhardt, On a sideways parabolic equation, Inverse Problems 13 (1997), no. 2, 297–309. MR 1445920, DOI 10.1088/0266-5611/13/2/007
A. M. Osman and J. V. Beck, Nonlinear inverse problem for the estimation of time-and-space dependent heat transfer coefficients, J. Thermophys. 3 (1988), no. 2, 146–152.
T. T. M. Onyango, D. B. Ingham, and D. Lesnic, Reconstruction of heat transfer coefficients using the boundary element method, Comput. Math. Appl. 56 (2008), no. 1, 114–126. MR 2427690 (2009f:80017)
Jian Su and Geoffrey Hewitt, Inverse heat conduction problem of estimating time-varying heat transfer coefficient, Numer. Heat Transfer Part A 45 (2004), no. 8, 777–789.
M. Slodička, D. Lesnic, and T. T. M. Onyango, Determination of a time-dependent heat transfer coefficient in a nonlinear inverse heat conduction problem, Inverse Probl. Sci. Eng. 18 (2010), no. 1, 65–81. MR 2598682, DOI 10.1080/17415970903234273
Marián Slodička and Roger Van Keer, Determination of a Robin coefficient in semilinear parabolic problems by means of boundary measurements, Inverse Problems 18 (2002), no. 1, 139–152. MR 1893587, DOI 10.1088/0266-5611/18/1/310
Andrey N. Tikhonov and Vasiliy Y. Arsenin, Solutions of ill-posed problems, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C.; John Wiley & Sons, New York-Toronto, Ont.-London, 1977. Translated from the Russian; Preface by translation editor Fritz John. MR 0455365
F. M. White, Heat and mass transfer, Addison-Wesley, Reading, 1988.
Jianli Xie and Jun Zou, Numerical reconstruction of heat fluxes, SIAM J. Numer. Anal. 43 (2005), no. 4, 1504–1535. MR 2182138 (2006e:35344)
Fenglian Yang, Liang Yan, and Ting Wei, The identification of a Robin coefficient by a conjugate gradient method, Int. J. Numer. Methods Engrg. 78 (2009), no. 7, 800–816. MR 2514342
J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces, Appl. Math. Optim. 5 (1979), no. 1, 49–62. MR 526427, DOI 10.1007/BF01442543