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Recovering a large number of diffusion constants in a parabolic equation from energy measurements
Politecnico di Milano, Piazza Leonardo da Vinci 32, 2013 Milano, Italy |
$\left(H, \left\langle { \cdot , \cdot } \right\rangle \right)$ |
$A_{i}:D(A_i) \to H$ |
$i = 1,···,n$ |
$u:[0,T] \to H$ |
$n$ |
$α_{1},···,α_{n} > 0$ |
$ u'(t) + α_{1} A_{1}u(t) + ··· + α_{n} A_{n}u(t) = 0, ~~~t ∈ (0,T), ~~~u(0) = x,$ |
$\left\langle A_{1} u(T),u(T)\right\rangle = \varphi_{1}, ~~~··· ~~~,\left\langle A_{n} u(T),u(T)\right\rangle = \varphi_{n},$ |
$\varphi_{i}$ |
$A_{i}$ |
$x ∈ H$ |
References:
[1] |
M. Akamatsu, G. Nakamura and S. Steinberg,
Identification of Lam6 coefficients from boundary observations, Inverse Problems, 7 (1991), 335-354.
doi: 10.1088/0266-5611/7/3/003. |
[2] |
E. A. Artyukhin and A. S. Okhapkin,
Determination of the parameters in the generalized heat-conduction equation from transient experimental data, J. Eng. Phys. Thermophys., 42 (1982), 693-698.
doi: 10.1007/BF00835106. |
[3] |
H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983; (English Translation) Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. |
[4] |
J. R. Cannon,
Determination of certain parameters in heat conduction problems, J. Math. Anal. Appl., 8 (1964), 188-201.
doi: 10.1016/0022-247X(64)90061-7. |
[5] |
K. -C. Chang, Methods in Nonlinear Analysis, Monographs in Mathematics, Springer-Verlag, Berlin, 2005. |
[6] |
D. Gale and H. Nikaido,
The jacobian matrix and global univalence of mappings, Math. Annalen, 159 (1965), 81-93.
doi: 10.1007/BF01360282. |
[7] |
T. Kato, Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften, 132, Springer-Verlag, Berlin and New York, 1966; 2nd ed., 1976. |
[8] |
A. Lorenzi, Recovering two constants in a parabolic linear equation, Journal of Physics: Conference Series, 73 (2007).
doi: 10.1088/1742-6596/73/1/012014. |
[9] |
A. Lorenzi and G. Mola,
Identification of a real constant in linear evolution equations in Hilbert spaces, Inverse Problems and Imaging, 5 (2011), 695-714.
doi: 10.3934/ipi.2011.5.695. |
[10] |
A. Lorenzi and G. Mola, Recovering the reaction and the diffusion coefficients in a linear parabolic equation,
Inverse Problems, 28 (2012), 075006, 23 pp. |
[11] |
L. Lorenzi,
An identification problem for the Ornstein-Uhlenbeck operator, Journal of Inverse and Ill-posed Problems, 19 (2011), 293-326.
|
[12] |
A. Sh. Lyubanova,
Identification of a constant coefficient in an elliptic equation, Appl. Anal., 87 (2008), 1121-1128.
doi: 10.1080/00036810802189654. |
[13] |
G. Mola,
Identification of the diffusion coefficient in linear evolution equations in Hilbert spaces, J. Abstract Differential Equations and Applications, 2 (2011), 14-28.
|
[14] |
G. Mola, N. Okazawa and T. Yokota, Reconstruction of two constant coefficients in linear anisotropic diffusion model, Inverse Problems, 32 (2016), 115016, 22 pp. |
[15] |
G. Mola, N. Okazawa, J. Prüss and T. Yokota,
Semigroup-theoretic approach to identification of linear diffusion coefficients, Discrete Continuous Dynamical Systems, Series S, 9 (2016), 777-790.
doi: 10.3934/dcdss.2016028. |
[16] |
G. Nakamura and G. Uhlmann,
Identification of Lame parameters by boundary measurements, American Journal of Mathematics, 115 (1993), 1161-1187.
doi: 10.2307/2375069. |
[17] |
N. Okazawa,
On the perturbation of linear operators in Banach and Hilbert spaces, J. Math. Soc. Japan, 34 (1982), 677-701.
doi: 10.2969/jmsj/03440677. |
[18] |
S. J. L. van Eijndhoven and J. de Graaf,
A Fundamental Approach to the Generalized Eigenvalue Problem for Self-Adjoint Operators, J. Functional Analysis, 63 (1985), 74-85.
doi: 10.1016/0022-1236(85)90098-9. |
[19] |
M. Yamamoto, Determination of constant parameters in some semilinear parabolic equations, Ill-posed Problems in Natural Sciences (Moscow, 1991), VSP, Utrecht, 1992,439-445. |
show all references
References:
[1] |
M. Akamatsu, G. Nakamura and S. Steinberg,
Identification of Lam6 coefficients from boundary observations, Inverse Problems, 7 (1991), 335-354.
doi: 10.1088/0266-5611/7/3/003. |
[2] |
E. A. Artyukhin and A. S. Okhapkin,
Determination of the parameters in the generalized heat-conduction equation from transient experimental data, J. Eng. Phys. Thermophys., 42 (1982), 693-698.
doi: 10.1007/BF00835106. |
[3] |
H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983; (English Translation) Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. |
[4] |
J. R. Cannon,
Determination of certain parameters in heat conduction problems, J. Math. Anal. Appl., 8 (1964), 188-201.
doi: 10.1016/0022-247X(64)90061-7. |
[5] |
K. -C. Chang, Methods in Nonlinear Analysis, Monographs in Mathematics, Springer-Verlag, Berlin, 2005. |
[6] |
D. Gale and H. Nikaido,
The jacobian matrix and global univalence of mappings, Math. Annalen, 159 (1965), 81-93.
doi: 10.1007/BF01360282. |
[7] |
T. Kato, Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften, 132, Springer-Verlag, Berlin and New York, 1966; 2nd ed., 1976. |
[8] |
A. Lorenzi, Recovering two constants in a parabolic linear equation, Journal of Physics: Conference Series, 73 (2007).
doi: 10.1088/1742-6596/73/1/012014. |
[9] |
A. Lorenzi and G. Mola,
Identification of a real constant in linear evolution equations in Hilbert spaces, Inverse Problems and Imaging, 5 (2011), 695-714.
doi: 10.3934/ipi.2011.5.695. |
[10] |
A. Lorenzi and G. Mola, Recovering the reaction and the diffusion coefficients in a linear parabolic equation,
Inverse Problems, 28 (2012), 075006, 23 pp. |
[11] |
L. Lorenzi,
An identification problem for the Ornstein-Uhlenbeck operator, Journal of Inverse and Ill-posed Problems, 19 (2011), 293-326.
|
[12] |
A. Sh. Lyubanova,
Identification of a constant coefficient in an elliptic equation, Appl. Anal., 87 (2008), 1121-1128.
doi: 10.1080/00036810802189654. |
[13] |
G. Mola,
Identification of the diffusion coefficient in linear evolution equations in Hilbert spaces, J. Abstract Differential Equations and Applications, 2 (2011), 14-28.
|
[14] |
G. Mola, N. Okazawa and T. Yokota, Reconstruction of two constant coefficients in linear anisotropic diffusion model, Inverse Problems, 32 (2016), 115016, 22 pp. |
[15] |
G. Mola, N. Okazawa, J. Prüss and T. Yokota,
Semigroup-theoretic approach to identification of linear diffusion coefficients, Discrete Continuous Dynamical Systems, Series S, 9 (2016), 777-790.
doi: 10.3934/dcdss.2016028. |
[16] |
G. Nakamura and G. Uhlmann,
Identification of Lame parameters by boundary measurements, American Journal of Mathematics, 115 (1993), 1161-1187.
doi: 10.2307/2375069. |
[17] |
N. Okazawa,
On the perturbation of linear operators in Banach and Hilbert spaces, J. Math. Soc. Japan, 34 (1982), 677-701.
doi: 10.2969/jmsj/03440677. |
[18] |
S. J. L. van Eijndhoven and J. de Graaf,
A Fundamental Approach to the Generalized Eigenvalue Problem for Self-Adjoint Operators, J. Functional Analysis, 63 (1985), 74-85.
doi: 10.1016/0022-1236(85)90098-9. |
[19] |
M. Yamamoto, Determination of constant parameters in some semilinear parabolic equations, Ill-posed Problems in Natural Sciences (Moscow, 1991), VSP, Utrecht, 1992,439-445. |
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