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On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach
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 |
References:
[1] |
A.-P. Calderón, Uniqueness in the Cauchy problem for partial differential equations,, Amer. J. Math., 80 (1958), 16.
|
[2] |
F. Cakoni and R. Kress, Integral equations for inverse problems in corrosion detection from partial Cauchy data,, Inverse Probl. Imaging, 1 (2007), 229.
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).
|
[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).
|
[5] |
T. Carleman, Sur un problème d'unicité pur les systèmes d'équations aux dérivées partielles à deux variables indépendantes, (French), Ark. Mat., 26 (1939).
|
[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.
|
[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.
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.
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.
doi: 10.1016/j.crme.2007.04.001. |
[10] |
Dinh Nho Hào, "Methods for Inverse Heat Conduction Problems,", Habilitationsschrift, 43 (1996).
|
[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.
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).
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, (2001), 119. Google Scholar |
[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.
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.
doi: 10.1080/17415971003624322. |
[16] |
D. Holder, "Electrical Impedance Tomography: Methods, History and Applications,", Institute of Physics, (2005). Google Scholar |
[17] |
Y. C. Hon and T. Wei, A meshless scheme for solving inverse problems of Laplace equation,, in, (2003), 291.
|
[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.
|
[19] |
R. Kress, On the numerical solution of a hypersingular integral equation in scattering theory,, J. Comp. Appl. Math., 61 (1995), 345.
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.
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, (2011).
|
[22] |
M. M. Lavrentiev, "Some Improperly Posed Problems of Mathematical Physics,", Springer Verlag, (1967). Google Scholar |
[23] |
J. Y. Lee and J. R. Yoon, A numerical method for Cauchy problem using singular value decomposition,, Comm. Korean Math. Soc. {\bf 16} (2001), 16 (2001), 487. Google Scholar |
[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.
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.
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, (1975).
|
[27] |
T. Regińska, Regularization of discrete ill-posed problems,, BIT, 44 (2004), 119.
doi: 10.1023/B:BITN.0000025090.68586.5e. |
[28] |
N. Tarchanov, "The Cauchy Problem for Solutions of Elliptic Equations,", Mathematical Topics, 7 (1995).
|
[29] |
A. Zeb, L. Elliott, D. B. Ingham and D. Lesnic, Solution of the Cauchy problem for Laplace equation,, in, (1997), 297. Google Scholar |
show all references
References:
[1] |
A.-P. Calderón, Uniqueness in the Cauchy problem for partial differential equations,, Amer. J. Math., 80 (1958), 16.
|
[2] |
F. Cakoni and R. Kress, Integral equations for inverse problems in corrosion detection from partial Cauchy data,, Inverse Probl. Imaging, 1 (2007), 229.
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).
|
[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).
|
[5] |
T. Carleman, Sur un problème d'unicité pur les systèmes d'équations aux dérivées partielles à deux variables indépendantes, (French), Ark. Mat., 26 (1939).
|
[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.
|
[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.
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.
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.
doi: 10.1016/j.crme.2007.04.001. |
[10] |
Dinh Nho Hào, "Methods for Inverse Heat Conduction Problems,", Habilitationsschrift, 43 (1996).
|
[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.
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).
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, (2001), 119. Google Scholar |
[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.
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.
doi: 10.1080/17415971003624322. |
[16] |
D. Holder, "Electrical Impedance Tomography: Methods, History and Applications,", Institute of Physics, (2005). Google Scholar |
[17] |
Y. C. Hon and T. Wei, A meshless scheme for solving inverse problems of Laplace equation,, in, (2003), 291.
|
[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.
|
[19] |
R. Kress, On the numerical solution of a hypersingular integral equation in scattering theory,, J. Comp. Appl. Math., 61 (1995), 345.
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.
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, (2011).
|
[22] |
M. M. Lavrentiev, "Some Improperly Posed Problems of Mathematical Physics,", Springer Verlag, (1967). Google Scholar |
[23] |
J. Y. Lee and J. R. Yoon, A numerical method for Cauchy problem using singular value decomposition,, Comm. Korean Math. Soc. {\bf 16} (2001), 16 (2001), 487. Google Scholar |
[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.
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.
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, (1975).
|
[27] |
T. Regińska, Regularization of discrete ill-posed problems,, BIT, 44 (2004), 119.
doi: 10.1023/B:BITN.0000025090.68586.5e. |
[28] |
N. Tarchanov, "The Cauchy Problem for Solutions of Elliptic Equations,", Mathematical Topics, 7 (1995).
|
[29] |
A. Zeb, L. Elliott, D. B. Ingham and D. Lesnic, Solution of the Cauchy problem for Laplace equation,, in, (1997), 297. Google Scholar |
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