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October  2019, 13(5): 1007-1021. doi: 10.3934/ipi.2019045

Identifiability of diffusion coefficients for source terms of non-uniform sign

1. 

Institut für Mathematik, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, 55128 Mainz, Germany

2. 

Institute for Numerical Simulation, University of Bonn, Endenicher Allee 19B, 53115 Bonn, Germany

3. 

Department of Mathematics, University of Transport and Communications, No.3 Cau Giay Street, Lang Thuong Ward, Dong Da District, Hanoi, Vietnam

* Corresponding author

Received  November 2018 Revised  April 2019 Published  July 2019

Fund Project: The authors acknowledge support by the Hausdorff Center of Mathematics, University of Bonn.

The problem of recovering a diffusion coefficient $ a $ in a second-order elliptic partial differential equation from a corresponding solution $ u $ for a given right-hand side $ f $ is considered, with particular focus on the case where $ f $ is allowed to take both positive and negative values. Identifiability of $ a $ from $ u $ is shown under mild smoothness requirements on $ a $, $ f $, and on the spatial domain $ D $, assuming that either the gradient of $ u $ is nonzero almost everywhere, or that $ f $ as a distribution does not vanish on any open subset of $ D $. Further results of this type under essentially minimal regularity conditions are obtained for the case of $ D $ being an interval, including detailed information on the continuity properties of the mapping from $ u $ to $ a $.

Citation: Markus Bachmayr, Van Kien Nguyen. Identifiability of diffusion coefficients for source terms of non-uniform sign. Inverse Problems and Imaging, 2019, 13 (5) : 1007-1021. doi: 10.3934/ipi.2019045
References:
[1]

G. Alessandrini, On the identification of the leading coefficient of an elliptic equation, Boll. Un. Mat. Ital. C (6), 4 (1985), 87-111. 

[2]

G. Alessandrini, An identification problem for an elliptic equation in two variables, Ann. Mat. Pura Appl. (4), 145 (1986), 265-295.  doi: 10.1007/BF01790543.

[3]

A. BonitoA. CohenR. DeVoreG. Petrova and G. Welper, Diffusion coefficients estimation for elliptic partial differential equations, SIAM J. Math. Anal., 49 (2017), 1570-1592.  doi: 10.1137/16M1094476.

[4]

G. Chavent and K. Kunisch, The output least squares identifiability of the diffusion coefficient from an $H^1$-observation in a 2-D elliptic equation, A tribute to J. L. Lions, ESAIM Control Optim. Calc. Var., 8 (2002), 423-440.  doi: 10.1051/cocv:2002028.

[5]

G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Rational Mech. Anal., 147 (1999), 89-118.  doi: 10.1007/s002050050146.

[6]

G.-Q. ChenM. Torres and W. P. Ziemer, Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws, Communications on Pure and Applied Mathematics, 62 (2009), 242-304.  doi: 10.1002/cpa.20262.

[7]

C. Chicone and J. Gerlach, A note on the identifiability of distributed parameters in elliptic equations, SIAM J. Math. Anal., 18 (1987), 1378-1384.  doi: 10.1137/0518099.

[8]

R. S. Falk, Error estimates for the numerical identification of a variable coefficient, Math. Comp., 40 (1983), 537-546.  doi: 10.1090/S0025-5718-1983-0689469-3.

[9]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977.

[10]

N. Honda, J. McLaughlin and G. Nakamura, Conditional stability for a single interior measurement, Inverse Problems, 30 (2014), 055001l, 19pp. doi: 10.1088/0266-5611/30/5/055001.

[11]

K. Ito and K. Kunisch, On the injectivity and linearization of the coefficient-to-solution mapping for elliptic boundary value problems, J. Math. Anal. Appl., 188 (1994), 1040-1066.  doi: 10.1006/jmaa.1994.1479.

[12]

R. V. Kohn and B. D. Lowe, A variational method for parameter identification, RAIRO Modél. Math. Anal. Numér., 22 (1988), 119-158.  doi: 10.1051/m2an/1988220101191.

[13]

K. Kunisch, Inherent identifiability of parameters in elliptic differential equations, J. Math. Anal. Appl., 132 (1988), 453-472.  doi: 10.1016/0022-247X(88)90074-1.

[14]

K. Kunisch and L. W. White, Identifiability under approximation for an elliptic boundary value problem, SIAM J. Control Optim., 25 (1987), 279-297.  doi: 10.1137/0325017.

[15] F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems. An Introduction to Geometric Measure Theory, Cambridge University Press, 2012.  doi: 10.1017/CBO9781139108133.
[16]

P. Marcellini, Identificazione di un coefficiente in una equazione differenziale ordinaria del secondo ordine, Ricerche Mat, 31 (1982), 223-243. 

[17]

G. R. Richter, An inverse problem for the steady state diffusion equation, SIAM J. Appl. Math., 41 (1981), 210-221.  doi: 10.1137/0141016.

[18]

G. R. Richter, Numerical identification of a spatially varying diffusion coefficient, Math. Comp., 36 (1981), 375-386.  doi: 10.1090/S0025-5718-1981-0606502-3.

[19]

V. Volterra, Sui Principii del Calcolo Integrale, Giornale di Matematiche, 1881.

show all references

References:
[1]

G. Alessandrini, On the identification of the leading coefficient of an elliptic equation, Boll. Un. Mat. Ital. C (6), 4 (1985), 87-111. 

[2]

G. Alessandrini, An identification problem for an elliptic equation in two variables, Ann. Mat. Pura Appl. (4), 145 (1986), 265-295.  doi: 10.1007/BF01790543.

[3]

A. BonitoA. CohenR. DeVoreG. Petrova and G. Welper, Diffusion coefficients estimation for elliptic partial differential equations, SIAM J. Math. Anal., 49 (2017), 1570-1592.  doi: 10.1137/16M1094476.

[4]

G. Chavent and K. Kunisch, The output least squares identifiability of the diffusion coefficient from an $H^1$-observation in a 2-D elliptic equation, A tribute to J. L. Lions, ESAIM Control Optim. Calc. Var., 8 (2002), 423-440.  doi: 10.1051/cocv:2002028.

[5]

G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Rational Mech. Anal., 147 (1999), 89-118.  doi: 10.1007/s002050050146.

[6]

G.-Q. ChenM. Torres and W. P. Ziemer, Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws, Communications on Pure and Applied Mathematics, 62 (2009), 242-304.  doi: 10.1002/cpa.20262.

[7]

C. Chicone and J. Gerlach, A note on the identifiability of distributed parameters in elliptic equations, SIAM J. Math. Anal., 18 (1987), 1378-1384.  doi: 10.1137/0518099.

[8]

R. S. Falk, Error estimates for the numerical identification of a variable coefficient, Math. Comp., 40 (1983), 537-546.  doi: 10.1090/S0025-5718-1983-0689469-3.

[9]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977.

[10]

N. Honda, J. McLaughlin and G. Nakamura, Conditional stability for a single interior measurement, Inverse Problems, 30 (2014), 055001l, 19pp. doi: 10.1088/0266-5611/30/5/055001.

[11]

K. Ito and K. Kunisch, On the injectivity and linearization of the coefficient-to-solution mapping for elliptic boundary value problems, J. Math. Anal. Appl., 188 (1994), 1040-1066.  doi: 10.1006/jmaa.1994.1479.

[12]

R. V. Kohn and B. D. Lowe, A variational method for parameter identification, RAIRO Modél. Math. Anal. Numér., 22 (1988), 119-158.  doi: 10.1051/m2an/1988220101191.

[13]

K. Kunisch, Inherent identifiability of parameters in elliptic differential equations, J. Math. Anal. Appl., 132 (1988), 453-472.  doi: 10.1016/0022-247X(88)90074-1.

[14]

K. Kunisch and L. W. White, Identifiability under approximation for an elliptic boundary value problem, SIAM J. Control Optim., 25 (1987), 279-297.  doi: 10.1137/0325017.

[15] F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems. An Introduction to Geometric Measure Theory, Cambridge University Press, 2012.  doi: 10.1017/CBO9781139108133.
[16]

P. Marcellini, Identificazione di un coefficiente in una equazione differenziale ordinaria del secondo ordine, Ricerche Mat, 31 (1982), 223-243. 

[17]

G. R. Richter, An inverse problem for the steady state diffusion equation, SIAM J. Appl. Math., 41 (1981), 210-221.  doi: 10.1137/0141016.

[18]

G. R. Richter, Numerical identification of a spatially varying diffusion coefficient, Math. Comp., 36 (1981), 375-386.  doi: 10.1090/S0025-5718-1981-0606502-3.

[19]

V. Volterra, Sui Principii del Calcolo Integrale, Giornale di Matematiche, 1881.

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