On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach

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February 2012, 6(1): 25-38. doi: 10.3934/ipi.2012.6.25

On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach

Roman Chapko ^1, and B. Tomas Johansson ^2,

1.	Faculty of Applied Mathematics and Computer Science, Ivan Franko National University of Lviv, 79000 Lviv, Ukraine
2.	School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT

Received January 2011 Revised December 2011 Published February 2012

Abstract
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We investigate the problem of determining the stationary temperature field on an inclusion from given Cauchy data on an accessible exterior boundary. On this accessible part the temperature (or the heat flux) is known, and, additionally, on a portion of this exterior boundary the heat flux (or temperature) is also given. We propose a direct boundary integral approach in combination with Tikhonov regularization for the stable determination of the temperature and flux on the inclusion. To determine these quantities on the inclusion, boundary integral equations are derived using Green's functions, and properties of these equations are shown in an $L^2$-setting. An effective way of discretizing these boundary integral equations based on the Nyström method and trigonometric approximations, is outlined. Numerical examples are included, both with exact and noisy data, showing that accurate approximations can be obtained with small computational effort, and the accuracy is increasing with the length of the portion of the boundary where the additionally data is given.

Keywords: Cauchy problem, Laplace equation, Green's function, single- and double layer potentials, Logarithmic- and hypersingularities, Tikhonov regularization., Integral equation of the first kind.

Mathematics Subject Classification: Primary: 35R25; Secondary: 35J05, 65R2.

Citation: Roman Chapko, B. Tomas Johansson. On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach. Inverse Problems and Imaging, 2012, 6 (1) : 25-38. doi: 10.3934/ipi.2012.6.25

References:

[1]	A.-P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math., 80 (1958), 16-36.
[2]	F. Cakoni and R. Kress, Integral equations for inverse problems in corrosion detection from partial Cauchy data, Inverse Probl. Imaging, 1 (2007), 229-245. doi: 10.3934/ipi.2007.1.229.
[3]	F. Cakoni, R. Kress and C. Schuft, Integral equations for shape and impedance reconstruction in corrosion detection, Inverse Problems, 26 (2010), 095012, 24 pp.
[4]	H. Cao, M. V. Klibanov and S. V. Pereverzev, A Carleman estimate and the balancing principle in the quasi-reversibility method for solving the Cauchy problem for the Laplace equation, Inverse Problems, 25 (2009), 035005, 21 pp.
[5]	T. Carleman, Sur un problème d'unicité pur les systèmes d'équations aux dérivées partielles à deux variables indépendantes, (French) Ark. Mat., Astr. Fys., 26 (1939), 9 pp.
[6]	R. Chapko and B. T. Johansson, An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite domains, Inverse Probl. Imaging, 2 (2008), 317-333.
[7]	R. Chapko and R. Kress, A hybrid method for inverse boundary value problems in potential theory, J. Inverse Ill-Posed Probl., 13 (2005), 27-40. doi: 10.1515/1569394053583711.
[8]	J. Cheng, Y. C. Hon, T. Wei and M. Yamamoto, Numerical computation of a Cauchy problem for Laplace's equation, ZAMM Z. Angew. Math. Mech., 81 (2001), 665-674. doi: 10.1002/1521-4001(200110)81:10<665::AID-ZAMM665>3.0.CO;2-V.
[9]	X. Escriva, T. N. Baranger and N. H. Tlatli, Leak identification in porous media by solving the Cauchy problem, C. R. Mecanique, 335 (2007), 401-406. doi: 10.1016/j.crme.2007.04.001.
[10]	Dinh Nho Hào, "Methods for Inverse Heat Conduction Problems," Habilitationsschrift, University of Siegen, Siegen, 1996, Methoden und Verfahren der Mathematischen Physik, 43, Peter Lang, Frankfurt am Main, 1998.
[11]	Dinh Nho Hào, B. T. Johansson, D. Lesnic and Pham Minh Hien, A variational method and approximations of a Cauchy problem for elliptic equations, J. Algorithms Comput. Technol., 4 (2010), 89-119. doi: 10.1260/1748-3018.4.1.89.
[12]	M. Hanke and M. Bruhl, Recent progress in electrical impedance tomography. Special section on imaging, Inverse Problems, 19 (2003), S65-S90. doi: 10.1088/0266-5611/19/6/055.
[13]	P. C. Hansen, The L-curve and its use in the numerical treatment of inverse problems, in "Computational Inverse Problems in Electrocardiology" (ed. P. Johnston), WIT Press, Southampton, (2001), 119-142.
[14]	B. He, Y. Wang and D. Wu, Estimating cortical potentials from scalp EEG's in a realistically shaped inhomogeneous head model by means of the boundary element method, IEEE Trans. Biomedical Engn., 46 (1999), 1264-1268. doi: 10.1109/10.790505.
[15]	J. Helsing and B. T. Johansson, Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques, Inverse Probl. Sci. Engn., 18 (2010), 381-399. doi: 10.1080/17415971003624322.
[16]	D. Holder, "Electrical Impedance Tomography: Methods, History and Applications," Institute of Physics, Bristol, 2005.
[17]	Y. C. Hon and T. Wei, A meshless scheme for solving inverse problems of Laplace equation, in "Recent Development in Theories & Numerics," World Sci. Publ., River Edge, NJ, (2003), 291-300.
[18]	V. A. Kozlov and V. G. Maz'ya, Iterative procedures for solving ill-posed boundary value problems that preserve differential equations, Algebra i Analiz, 1 (1989), 144-170; Translation in Leningrad Math. J., 1 (1990), 1207-1228.
[19]	R. Kress, On the numerical solution of a hypersingular integral equation in scattering theory, J. Comp. Appl. Math., 61 (1995), 345-360. doi: 10.1016/0377-0427(94)00073-7.
[20]	R. Kress and W. Rundell, Nonlinear integral equations and the iterative solution for an inverse boundary value problem, Inverse Problems, 21 (2005), 1207-1223. doi: 10.1088/0266-5611/21/4/002.
[21]	P. K. Kythe, "Green's Functions and Linear Differential Equations: Theory, Applications, and Computation," Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, CRC Press, Boca Raton, FL, 2011.
[22]	M. M. Lavrentiev, "Some Improperly Posed Problems of Mathematical Physics," Springer Verlag, Berlin, 1967.
[23]	J. Y. Lee and J. R. Yoon, A numerical method for Cauchy problem using singular value decomposition, Comm. Korean Math. Soc. 16 (2001), 487-508.
[24]	L. Marin, Relaxation procedures for an iterative MFS algorithm for two-dimensional steady-state isotropic heat conduction Cauchy problems, Engng. Anal. Bound. Elem., 35 (2011), 415-429. doi: 10.1016/j.enganabound.2010.07.011.
[25]	L. Marin and D. Lesnic, The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations, Comput. & Structures, 83 (2005), 267-278. doi: 10.1016/j.compstruc.2004.10.005.
[26]	L. E. Payne, "Improperly Posed Problems in Partial Differential Equations," Regional Conference Series in Applied Mathematics, No. 22, SIAM, Philadelphia, Pa., 1975.
[27]	T. Regińska, Regularization of discrete ill-posed problems, BIT, 44 (2004), 119-133. doi: 10.1023/B:BITN.0000025090.68586.5e.
[28]	N. Tarchanov, "The Cauchy Problem for Solutions of Elliptic Equations," Mathematical Topics, 7, Akademie Verlag, Berlin, 1995.
[29]	A. Zeb, L. Elliott, D. B. Ingham and D. Lesnic, Solution of the Cauchy problem for Laplace equation, in "First UK Conference on Boundary Integral Methods" (eds. L. Elliott, D. B. Ingham and D. Lesnic), Leeds University Press, (1997), 297-307.

show all references

References:

[1]	A.-P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math., 80 (1958), 16-36.
[2]	F. Cakoni and R. Kress, Integral equations for inverse problems in corrosion detection from partial Cauchy data, Inverse Probl. Imaging, 1 (2007), 229-245. doi: 10.3934/ipi.2007.1.229.
[3]	F. Cakoni, R. Kress and C. Schuft, Integral equations for shape and impedance reconstruction in corrosion detection, Inverse Problems, 26 (2010), 095012, 24 pp.
[4]	H. Cao, M. V. Klibanov and S. V. Pereverzev, A Carleman estimate and the balancing principle in the quasi-reversibility method for solving the Cauchy problem for the Laplace equation, Inverse Problems, 25 (2009), 035005, 21 pp.
[5]	T. Carleman, Sur un problème d'unicité pur les systèmes d'équations aux dérivées partielles à deux variables indépendantes, (French) Ark. Mat., Astr. Fys., 26 (1939), 9 pp.
[6]	R. Chapko and B. T. Johansson, An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite domains, Inverse Probl. Imaging, 2 (2008), 317-333.
[7]	R. Chapko and R. Kress, A hybrid method for inverse boundary value problems in potential theory, J. Inverse Ill-Posed Probl., 13 (2005), 27-40. doi: 10.1515/1569394053583711.
[8]	J. Cheng, Y. C. Hon, T. Wei and M. Yamamoto, Numerical computation of a Cauchy problem for Laplace's equation, ZAMM Z. Angew. Math. Mech., 81 (2001), 665-674. doi: 10.1002/1521-4001(200110)81:10<665::AID-ZAMM665>3.0.CO;2-V.
[9]	X. Escriva, T. N. Baranger and N. H. Tlatli, Leak identification in porous media by solving the Cauchy problem, C. R. Mecanique, 335 (2007), 401-406. doi: 10.1016/j.crme.2007.04.001.
[10]	Dinh Nho Hào, "Methods for Inverse Heat Conduction Problems," Habilitationsschrift, University of Siegen, Siegen, 1996, Methoden und Verfahren der Mathematischen Physik, 43, Peter Lang, Frankfurt am Main, 1998.
[11]	Dinh Nho Hào, B. T. Johansson, D. Lesnic and Pham Minh Hien, A variational method and approximations of a Cauchy problem for elliptic equations, J. Algorithms Comput. Technol., 4 (2010), 89-119. doi: 10.1260/1748-3018.4.1.89.
[12]	M. Hanke and M. Bruhl, Recent progress in electrical impedance tomography. Special section on imaging, Inverse Problems, 19 (2003), S65-S90. doi: 10.1088/0266-5611/19/6/055.
[13]	P. C. Hansen, The L-curve and its use in the numerical treatment of inverse problems, in "Computational Inverse Problems in Electrocardiology" (ed. P. Johnston), WIT Press, Southampton, (2001), 119-142.
[14]	B. He, Y. Wang and D. Wu, Estimating cortical potentials from scalp EEG's in a realistically shaped inhomogeneous head model by means of the boundary element method, IEEE Trans. Biomedical Engn., 46 (1999), 1264-1268. doi: 10.1109/10.790505.
[15]	J. Helsing and B. T. Johansson, Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques, Inverse Probl. Sci. Engn., 18 (2010), 381-399. doi: 10.1080/17415971003624322.
[16]	D. Holder, "Electrical Impedance Tomography: Methods, History and Applications," Institute of Physics, Bristol, 2005.
[17]	Y. C. Hon and T. Wei, A meshless scheme for solving inverse problems of Laplace equation, in "Recent Development in Theories & Numerics," World Sci. Publ., River Edge, NJ, (2003), 291-300.
[18]	V. A. Kozlov and V. G. Maz'ya, Iterative procedures for solving ill-posed boundary value problems that preserve differential equations, Algebra i Analiz, 1 (1989), 144-170; Translation in Leningrad Math. J., 1 (1990), 1207-1228.
[19]	R. Kress, On the numerical solution of a hypersingular integral equation in scattering theory, J. Comp. Appl. Math., 61 (1995), 345-360. doi: 10.1016/0377-0427(94)00073-7.
[20]	R. Kress and W. Rundell, Nonlinear integral equations and the iterative solution for an inverse boundary value problem, Inverse Problems, 21 (2005), 1207-1223. doi: 10.1088/0266-5611/21/4/002.
[21]	P. K. Kythe, "Green's Functions and Linear Differential Equations: Theory, Applications, and Computation," Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, CRC Press, Boca Raton, FL, 2011.
[22]	M. M. Lavrentiev, "Some Improperly Posed Problems of Mathematical Physics," Springer Verlag, Berlin, 1967.
[23]	J. Y. Lee and J. R. Yoon, A numerical method for Cauchy problem using singular value decomposition, Comm. Korean Math. Soc. 16 (2001), 487-508.
[24]	L. Marin, Relaxation procedures for an iterative MFS algorithm for two-dimensional steady-state isotropic heat conduction Cauchy problems, Engng. Anal. Bound. Elem., 35 (2011), 415-429. doi: 10.1016/j.enganabound.2010.07.011.
[25]	L. Marin and D. Lesnic, The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations, Comput. & Structures, 83 (2005), 267-278. doi: 10.1016/j.compstruc.2004.10.005.
[26]	L. E. Payne, "Improperly Posed Problems in Partial Differential Equations," Regional Conference Series in Applied Mathematics, No. 22, SIAM, Philadelphia, Pa., 1975.
[27]	T. Regińska, Regularization of discrete ill-posed problems, BIT, 44 (2004), 119-133. doi: 10.1023/B:BITN.0000025090.68586.5e.
[28]	N. Tarchanov, "The Cauchy Problem for Solutions of Elliptic Equations," Mathematical Topics, 7, Akademie Verlag, Berlin, 1995.
[29]	A. Zeb, L. Elliott, D. B. Ingham and D. Lesnic, Solution of the Cauchy problem for Laplace equation, in "First UK Conference on Boundary Integral Methods" (eds. L. Elliott, D. B. Ingham and D. Lesnic), Leeds University Press, (1997), 297-307.

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