Identification of the source for full parabolic equations
Abstract
In this work, we consider the problem of identifying the time independent source for full parabolic equations in Rn from noisy data. This is an ill-posed problem in the sense of Hadamard. To compensate the factor that causes the instability, a family of parametric regularization operators is introduced, where the rule to select the value of the regularization parameter is included. This rule, known as regularization parameter choice rule, depends on the data noise level and the degree of smoothness that it is assumed for the source. The proof for the stability and convergence of the regularization criteria is presented and a Hölder type bound is obtained for the estimation error. Numerical examples are included to illustrate the effectiveness of this regularization approach.
Keyword : inverse and ill-posed problem, regularization operator, transport equation, Fourier transform
How to Cite
Umbricht, G. F. (2021). Identification of the source for full parabolic equations. Mathematical Modelling and Analysis, 26(3), 339-357. https://doi.org/10.3846/mma.2021.12700
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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W. Cheng, C.L. Fu and Z. Qian. Two regularization methods for a spherically symmetric inverse heat conduction problem. Applied Mathematical Modelling, 32(4):432–442, 2008. https://doi.org/10.1016/j.apm.2006.12.012
F.F. Dou and C.L. Fu. Determining an unknown source in the heat equation by a wavelet dual least squares method. Applied Mathematics Letters, 22(5):661–667, 2009. https://doi.org/10.1016/j.aml.2008.08.003
F.F. Dou, C.L. Fu and F.L. Yang. Optimal error bound and Fourier regularization for identifying an unknown source in the heat equation. Journal of Computational and Applied Mathematics, 230(2):728–737, 2009. https://doi.org/10.1016/j.cam.2009.01.008
L. Eldén, F. Berntsson and T. Reginska. Wavelet and Fourier methods for solving the sideways heat equation. SIAM Journal on Scientific Computing, 21(6):2187– 2205, 2000. https://doi.org/10.1137/s1064827597331394
H. Engel, M. Hanke and A. Neubauer. Regularization of inverse problems. Kluwer Academic Publisher, The Netherlands, 1996.
A. Farcas, L. Elliott, D.B. Ingham, D. Lesnic and S. Mera. A dual reciprocity boundary element method for the regularized numerical solution of the inverse source problem associated to the Poisson equation. Inverse Problems in Engineering, 11(2):123–139, 2003. https://doi.org/10.1080/1068276031000074267
A. Farcas and D. Lesnic. The boundary-element method for the determination of a heat source dependent on one variable. Journal of Engineering Mathematics, 54(4):375–388, 2006. https://doi.org/10.1007/s10665-005-9023-0
C.L. Fu. Simplified Tikhonov and Fourier regularization methods on a general sideways parabolic equation. Journal of Computational and Applied Mathematics, 167(2):449–463, 2004. https://doi.org/10.1016/s0377-0427(03)00932-4
J. Hadamard. Lectures on Cauchy problem in linear differential equations. Yale University Press, New Haven, 1923.
P.C. Hansen and D.P. O’Leary. The use of the L-curve in the regularization of discrete ill-posed problems. SIAM Journal on Scientific Computing, 14(6):1487– 1503, 1993. https://doi.org/10.1137/0914086
Y.C. Hon, M. Li and Y.A. Melnikov. Inverse source identification by Green’ s function. Engineering Analysis with Boundary Elements, 34(4):352–358, 2010. https://doi.org/10.1016/j.enganabound.2009.09.009
B. Jin and L. Marin. The method of fundamental solutions for inverse source problems associated with the steady-state heat conduction. International Journal for Numerical Methods in Engineering, 69(8):1570–1589, 2007. https://doi.org/10.1002/nme.1826
B.T. Johansson and D. Lesnic. Determination of a spacewise dependent heat source. Journal of Computational and Applied Mathematics, 209(1):66–80, 2007. https://doi.org/10.1016/j.cam.2006.10.026
B.T. Johansson and D. Lesnic. A procedure for determining a spacewise dependent heat source and the initial temperature. Applicable Analysis, 87(3):265–276, 2008. https://doi.org/10.1080/00036810701858193
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T.T. Le and L.H. Nguyen. A convergent numerical method to recover the initial condition of nonlinear parabolic equations from lateral Cauchy data. Journal of Inverse and Ill-posed Problems, 14(3):287–300, 2020. https://doi.org/10.1515/jiip-2020-0028
Q. Li and L.H. Nguyen. Recovering the initial condition of parabolic equations from lateral Cauchy data via the quasi-reversibility method. Inverse Problems in Science and Engineering, 28(4):580–598, 2019. https://doi.org/10.1080/17415977.2019.1643850
X.X. Li, H.Z. Guo, S.M. Wan and F. Yang. Inverse source identification by the modified regularization method on Poisson equation. Journal of Applied Mathematics, 2012(1):1–13, 2012. https://doi.org/10.1155/2012/971952
C.S. Liu. An two-stage LGSM to identify time dependent heat source through an internal measurement of temperature. International Journal of Heat and Mass Transfer, 52(7):1635–1642, 2009. https://doi.org/10.1016/j.ijheatmasstransfer.2008.09.021
R.A.F. Macleod, W.G. Dirks, Y. Matsuo, M. Kaufmann, H. Milch and H.G. Drexler. Widespread intraspecies cross-contamination of human tumor cell lines arising at source. International Journal of Cancer, 83(4):555– 563, 1999. https://doi.org/10.1002/(SICI)1097-0215(19991112)83:4<555::AID-IJC19>3.0.CO;2-2
G.L. Mazzieri, D. Spies and K.G. Temperini. Mixed spatially varying L2-BV regularization of inverse ill-posed problems. Journal of Inverse and Ill-Posed Problems, 23(6):571–585, 2015. https://doi.org/10.1515/jiip-2014-0034
T. Nara and S. Ando. A projective method for an inverse source problem of the Poisson equation. Inverse Problems, 19(2):355–369, 2003. https://doi.org/10.1088/0266-5611/19/2/307
T. Ohe and K. Ohnaka. A precise estimation method for locations in an inverse logarithmic potential problem for point mass models. Applied Mathematical Modelling,18(8):446–452,1994. https://doi.org/10.1016/0307-904x(94)90306-9
C. Pao. Parabolic systems in unbounded domains I. Existence and dynamics. Journal of Mathematical Analysis and Applications, 217(1):129–160, 1998. https://doi.org/10.1006/jmaa.1997.5706
Z. Qian, C.L. Fu and X.L. Feng. A modified method for high order numerical derivatives. Applied Mathematics and Computation, 182(2):1191–1200, 2006. https://doi.org/10.1016/j.amc.2006.04.059
Z. Qian, C.L. Fu and R. Shi. A modified method for a backward heat conduction problem. Applied Mathematics and Computation, 185(1):564–573, 2007. https://doi.org/10.1016/j.amc.2006.07.055
K. Rashedi and S.A. Yousef. Ritz-Galerkin method for solving a class of inverse problems in the parabolic equation. International Journal of Nonlinear Science, 12(1):498–502, 2011. https://doi.org/10.1080/17415977.2012.701627
E.G. Savateev. On problems of determining the source function in a parabolic equation. Journal of Inverse and Ill-Posed Problems, 3(1):83–102, 1995. https://doi.org/10.1515/jiip.1995.3.1.83
I.F. Sivergina, M.P. Polis and I. Kolmanovsky. Source identification for parabolic equations. Mathematics of Control, Signals, and Systems, 16(2):141–157, 2003. https://doi.org/10.1007/s00498-003-0136-6
Y. Sun and Y. Kagawa. Identification of electric charge distribution using dual reciprocity boundary element models. IEEE Transactions on Magnetics, 33(2):1970–1973, 1997. https://doi.org/10.1109/20.582682
G.S. Tan, Y.J. Cheng and X.Q. Wang. Determining magnitude of groundwater pollution sources by data compatibility analysis. Inverse Problems in Science and Engineering, 14(3):287–300, 2006. https://doi.org/10.1080/17415970500485153
D.D. Trong, N.T. Long and P.N.D Alain. Nonhomogeneous heat equation: identification and regularization for the inhomogeneous term. Journal of Mathematical Analysis and Applications, 312(1):93–104, 2005. https://doi.org/10.1016/j.jmaa.2005.03.037
D.D. Trong, P.H. Quan and P.N.D Alain. Determination of a twodimensional heat source: uniqueness, regularization and error estimate. Journal of Computational and Applied Mathematics, 191(1):50–67, 2006. https://doi.org/10.1016/j.cam.2005.04.022
M. Yamamoto. Conditional stability in determination of force terms of heat equations in a rectangle. Mathematical and Computer Modelling, 18(1):79–88, 1993. https://doi.org/10.1016/0895-7177(93)90081-9
L. Yan, C.L. Fu and F.F. Dou. A computational method for identifying a spacewise-dependent heat source. International Journal for Numerical Methods in Biomedical Engineering, 26(1):597–608, 2010. https://doi.org/10.1002/cnm.1155
L. Yan, C.L. Fu and F.L. Yang. The method of fundamental solutions for the inverse heat source problem. Engineering Analysis with Boundary Elements, 32(3):216–222, 2008. https://doi.org/10.1016/j.enganabound.2007.08.002
L. Yan, F.L. Yang and C.L. Fu. A methless method for solving an inverse spacewise-dependent heat source problem. Journal of Computational Physics, 228(1):123–136, 2009. https://doi.org/10.1016/j.jcp.2008.09.001
F. Yang and C.L. Fu. The method of simplified Tikhonov regularization for dealing with the inverse time-dependent heat source problem. Computers & Mathematics with Applications, 60(5):1228–1236, 2010. https://doi.org/10.1016/j.camwa.2010.06.004
F. Yang and C.L. Fu. A simplified Tikhonov regularization method for the heat source. Applied Mathematical Modelling, 34(11):3286–3299, 2010. https://doi.org/10.1016/j.apm.2010.02.020
F. Yang and C.L. Fu. Two regularization methods to identify timedependent heat source through an internal measurement of temperature. Mathematical and Computer Modelling, 53(5):793–804, 2011. https://doi.org/10.1016/j.mcm.2010.10.016
F. Yang and C.L. Fu. The modified regularization method for identifying the unknown source on Poisson equation. Applied Mathematical Modelling, 36(2):756– 763, 2012. https://doi.org/10.1016/j.apm.2011.07.008
F. Yang and C.L. Fu. A mollification regularization method for the inverse spatial-dependent heat source problem. Journal of Computational and Applied Mathematics, 255(1):555–567, 2014. https://doi.org/10.1016/j.cam.2013.06.012
Y. Zeng and J.G. Anderson. A composite source model of the 1994 Northridge earthquake using genetic algorithms. Bulletin of the Seismological Society of America, 86(1):71–83, 1996. https://doi.org/10.1155/2012/971952
Z. Zhao and Z. Meng. A modified Tikhonov regularization method for a backward heat equation. Inverse Problems in Science and Engineering, 19(8):1175–1182, 2011. https://doi.org/10.1080/17415977.2011.605885
Z. Zhao, O. Xie and Z. Meng. Determination of an unknown source in the heat equation by the method of Tikhonov regularization in Hilbert scales. Journal of Applied Mathematics and Physics, 2(1):10–17, 2014. https://doi.org/10.4236/jamp.2014.22002
G.C. Beroza and P. Spudich. Linearized inversion for fault rupture behavior: application to the 1984 Morgan Hill, California, earthquake. Journal of Geophysical Research: Solid Earth, 93(6):6275–6296, 1988. https://doi.org/10.1029/jb093ib06p06275
J.R. Cannon and P. Duchateau. Structural identification of an unknown source term in a heat equation. Inverse Problems, 14(3):535–551, 1998. https://doi.org/10.1088/0266-5611/14/3/010
W. Cheng, C.L. Fu and Z. Qian. A modified Tikhonov regularization method for a spherically symmetric three-dimensional inverse heat conduction problem. Mathematics and Computers in Simulation, 75(3):97–112, 2007. https://doi.org/10.1016/j.matcom.2006.09.005
W. Cheng, C.L. Fu and Z. Qian. Two regularization methods for a spherically symmetric inverse heat conduction problem. Applied Mathematical Modelling, 32(4):432–442, 2008. https://doi.org/10.1016/j.apm.2006.12.012
F.F. Dou and C.L. Fu. Determining an unknown source in the heat equation by a wavelet dual least squares method. Applied Mathematics Letters, 22(5):661–667, 2009. https://doi.org/10.1016/j.aml.2008.08.003
F.F. Dou, C.L. Fu and F.L. Yang. Optimal error bound and Fourier regularization for identifying an unknown source in the heat equation. Journal of Computational and Applied Mathematics, 230(2):728–737, 2009. https://doi.org/10.1016/j.cam.2009.01.008
L. Eldén, F. Berntsson and T. Reginska. Wavelet and Fourier methods for solving the sideways heat equation. SIAM Journal on Scientific Computing, 21(6):2187– 2205, 2000. https://doi.org/10.1137/s1064827597331394
H. Engel, M. Hanke and A. Neubauer. Regularization of inverse problems. Kluwer Academic Publisher, The Netherlands, 1996.
A. Farcas, L. Elliott, D.B. Ingham, D. Lesnic and S. Mera. A dual reciprocity boundary element method for the regularized numerical solution of the inverse source problem associated to the Poisson equation. Inverse Problems in Engineering, 11(2):123–139, 2003. https://doi.org/10.1080/1068276031000074267
A. Farcas and D. Lesnic. The boundary-element method for the determination of a heat source dependent on one variable. Journal of Engineering Mathematics, 54(4):375–388, 2006. https://doi.org/10.1007/s10665-005-9023-0
C.L. Fu. Simplified Tikhonov and Fourier regularization methods on a general sideways parabolic equation. Journal of Computational and Applied Mathematics, 167(2):449–463, 2004. https://doi.org/10.1016/s0377-0427(03)00932-4
J. Hadamard. Lectures on Cauchy problem in linear differential equations. Yale University Press, New Haven, 1923.
P.C. Hansen and D.P. O’Leary. The use of the L-curve in the regularization of discrete ill-posed problems. SIAM Journal on Scientific Computing, 14(6):1487– 1503, 1993. https://doi.org/10.1137/0914086
Y.C. Hon, M. Li and Y.A. Melnikov. Inverse source identification by Green’ s function. Engineering Analysis with Boundary Elements, 34(4):352–358, 2010. https://doi.org/10.1016/j.enganabound.2009.09.009
B. Jin and L. Marin. The method of fundamental solutions for inverse source problems associated with the steady-state heat conduction. International Journal for Numerical Methods in Engineering, 69(8):1570–1589, 2007. https://doi.org/10.1002/nme.1826
B.T. Johansson and D. Lesnic. Determination of a spacewise dependent heat source. Journal of Computational and Applied Mathematics, 209(1):66–80, 2007. https://doi.org/10.1016/j.cam.2006.10.026
B.T. Johansson and D. Lesnic. A procedure for determining a spacewise dependent heat source and the initial temperature. Applicable Analysis, 87(3):265–276, 2008. https://doi.org/10.1080/00036810701858193
A. Kirsch. An introduction to the mathematical theory of inverse problems. Springer, New York, 2011.
R. Lattès and J.L. Lions. The method of quasi-reversibility: applications to partial differential equations. Elservier, New York, 1969.
T.T. Le and L.H. Nguyen. A convergent numerical method to recover the initial condition of nonlinear parabolic equations from lateral Cauchy data. Journal of Inverse and Ill-posed Problems, 14(3):287–300, 2020. https://doi.org/10.1515/jiip-2020-0028
Q. Li and L.H. Nguyen. Recovering the initial condition of parabolic equations from lateral Cauchy data via the quasi-reversibility method. Inverse Problems in Science and Engineering, 28(4):580–598, 2019. https://doi.org/10.1080/17415977.2019.1643850
X.X. Li, H.Z. Guo, S.M. Wan and F. Yang. Inverse source identification by the modified regularization method on Poisson equation. Journal of Applied Mathematics, 2012(1):1–13, 2012. https://doi.org/10.1155/2012/971952
C.S. Liu. An two-stage LGSM to identify time dependent heat source through an internal measurement of temperature. International Journal of Heat and Mass Transfer, 52(7):1635–1642, 2009. https://doi.org/10.1016/j.ijheatmasstransfer.2008.09.021
R.A.F. Macleod, W.G. Dirks, Y. Matsuo, M. Kaufmann, H. Milch and H.G. Drexler. Widespread intraspecies cross-contamination of human tumor cell lines arising at source. International Journal of Cancer, 83(4):555– 563, 1999. https://doi.org/10.1002/(SICI)1097-0215(19991112)83:4<555::AID-IJC19>3.0.CO;2-2
G.L. Mazzieri, D. Spies and K.G. Temperini. Mixed spatially varying L2-BV regularization of inverse ill-posed problems. Journal of Inverse and Ill-Posed Problems, 23(6):571–585, 2015. https://doi.org/10.1515/jiip-2014-0034
T. Nara and S. Ando. A projective method for an inverse source problem of the Poisson equation. Inverse Problems, 19(2):355–369, 2003. https://doi.org/10.1088/0266-5611/19/2/307
T. Ohe and K. Ohnaka. A precise estimation method for locations in an inverse logarithmic potential problem for point mass models. Applied Mathematical Modelling,18(8):446–452,1994. https://doi.org/10.1016/0307-904x(94)90306-9
C. Pao. Parabolic systems in unbounded domains I. Existence and dynamics. Journal of Mathematical Analysis and Applications, 217(1):129–160, 1998. https://doi.org/10.1006/jmaa.1997.5706
Z. Qian, C.L. Fu and X.L. Feng. A modified method for high order numerical derivatives. Applied Mathematics and Computation, 182(2):1191–1200, 2006. https://doi.org/10.1016/j.amc.2006.04.059
Z. Qian, C.L. Fu and R. Shi. A modified method for a backward heat conduction problem. Applied Mathematics and Computation, 185(1):564–573, 2007. https://doi.org/10.1016/j.amc.2006.07.055
K. Rashedi and S.A. Yousef. Ritz-Galerkin method for solving a class of inverse problems in the parabolic equation. International Journal of Nonlinear Science, 12(1):498–502, 2011. https://doi.org/10.1080/17415977.2012.701627
E.G. Savateev. On problems of determining the source function in a parabolic equation. Journal of Inverse and Ill-Posed Problems, 3(1):83–102, 1995. https://doi.org/10.1515/jiip.1995.3.1.83
I.F. Sivergina, M.P. Polis and I. Kolmanovsky. Source identification for parabolic equations. Mathematics of Control, Signals, and Systems, 16(2):141–157, 2003. https://doi.org/10.1007/s00498-003-0136-6
Y. Sun and Y. Kagawa. Identification of electric charge distribution using dual reciprocity boundary element models. IEEE Transactions on Magnetics, 33(2):1970–1973, 1997. https://doi.org/10.1109/20.582682
G.S. Tan, Y.J. Cheng and X.Q. Wang. Determining magnitude of groundwater pollution sources by data compatibility analysis. Inverse Problems in Science and Engineering, 14(3):287–300, 2006. https://doi.org/10.1080/17415970500485153
D.D. Trong, N.T. Long and P.N.D Alain. Nonhomogeneous heat equation: identification and regularization for the inhomogeneous term. Journal of Mathematical Analysis and Applications, 312(1):93–104, 2005. https://doi.org/10.1016/j.jmaa.2005.03.037
D.D. Trong, P.H. Quan and P.N.D Alain. Determination of a twodimensional heat source: uniqueness, regularization and error estimate. Journal of Computational and Applied Mathematics, 191(1):50–67, 2006. https://doi.org/10.1016/j.cam.2005.04.022
M. Yamamoto. Conditional stability in determination of force terms of heat equations in a rectangle. Mathematical and Computer Modelling, 18(1):79–88, 1993. https://doi.org/10.1016/0895-7177(93)90081-9
L. Yan, C.L. Fu and F.F. Dou. A computational method for identifying a spacewise-dependent heat source. International Journal for Numerical Methods in Biomedical Engineering, 26(1):597–608, 2010. https://doi.org/10.1002/cnm.1155
L. Yan, C.L. Fu and F.L. Yang. The method of fundamental solutions for the inverse heat source problem. Engineering Analysis with Boundary Elements, 32(3):216–222, 2008. https://doi.org/10.1016/j.enganabound.2007.08.002
L. Yan, F.L. Yang and C.L. Fu. A methless method for solving an inverse spacewise-dependent heat source problem. Journal of Computational Physics, 228(1):123–136, 2009. https://doi.org/10.1016/j.jcp.2008.09.001
F. Yang and C.L. Fu. The method of simplified Tikhonov regularization for dealing with the inverse time-dependent heat source problem. Computers & Mathematics with Applications, 60(5):1228–1236, 2010. https://doi.org/10.1016/j.camwa.2010.06.004
F. Yang and C.L. Fu. A simplified Tikhonov regularization method for the heat source. Applied Mathematical Modelling, 34(11):3286–3299, 2010. https://doi.org/10.1016/j.apm.2010.02.020
F. Yang and C.L. Fu. Two regularization methods to identify timedependent heat source through an internal measurement of temperature. Mathematical and Computer Modelling, 53(5):793–804, 2011. https://doi.org/10.1016/j.mcm.2010.10.016
F. Yang and C.L. Fu. The modified regularization method for identifying the unknown source on Poisson equation. Applied Mathematical Modelling, 36(2):756– 763, 2012. https://doi.org/10.1016/j.apm.2011.07.008
F. Yang and C.L. Fu. A mollification regularization method for the inverse spatial-dependent heat source problem. Journal of Computational and Applied Mathematics, 255(1):555–567, 2014. https://doi.org/10.1016/j.cam.2013.06.012
Y. Zeng and J.G. Anderson. A composite source model of the 1994 Northridge earthquake using genetic algorithms. Bulletin of the Seismological Society of America, 86(1):71–83, 1996. https://doi.org/10.1155/2012/971952
Z. Zhao and Z. Meng. A modified Tikhonov regularization method for a backward heat equation. Inverse Problems in Science and Engineering, 19(8):1175–1182, 2011. https://doi.org/10.1080/17415977.2011.605885
Z. Zhao, O. Xie and Z. Meng. Determination of an unknown source in the heat equation by the method of Tikhonov regularization in Hilbert scales. Journal of Applied Mathematics and Physics, 2(1):10–17, 2014. https://doi.org/10.4236/jamp.2014.22002