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On eigenelements sensitivity for compact self-adjoint operators and applications
1. | Université de La Rochelle, Avenue M. Crpeau, 17042 La Rochelle, France, France, France |
References:
[1] |
N. Akkari, A. Hamdouni, E. Liberge and M. Jazar, A mathematical and numerical study of the sensitivity of a reduced order model by POD (ROM-POD), for a 2D incompressible fluid flow, Journal of Computational and Applied Mathematics, 270 (2014), 522-530.
doi: 10.1016/j.cam.2013.11.025. |
[2] |
N. Akkari, A. Hamdouni and M. Jazar, Mathematical and numerical results on the sensitivity of the POD approximation relative to the Burgers equation, Applied Mathematics and Computation, 247 (2014), 951-961.
doi: 10.1016/j.amc.2014.09.005. |
[3] |
N. Akkari, A. Hamdouni, E. Liberge and M. Jazzar, On the sensitivity of the POD technique for a parameterized quasi-nonlinear parabolic equation, Advanced Modeling and Simulation in Engineering Sciences, 1 (2014), p14.
doi: 10.1186/s40323-014-0014-4. |
[4] |
C. Allery, C. Béghein and A. Hamdouni, On investigation of particle dispersion by a POD approach, Int. Applied Mechanics, 44 (2008), 110-119.
doi: 10.1007/s10778-008-0025-2. |
[5] |
R. Bhatia and L. Elsner, The Hoffman-Wielandt inequality in infinite dimensions, Proc. Indian Acad. Sci. (Math. Sci.), 104 (1994), 483-494.
doi: 10.1007/BF02867116. |
[6] |
C. Davis and W. M. Kahan, The rotation of eigenvectors by a perturbation. III, SIAM J. Numer. Anal., 7 (1970), 1-46.
doi: 10.1137/0707001. |
[7] |
B. Denis de Senneville, A. El Hamidi and C. Moonen, A direct PCA-based approach for real-time description of physiological organ deformations, IEEE Transactions on Medical Imaging, 34 (2014), 974-982.
doi: 10.1109/TMI.2014.2371995. |
[8] |
R. Everson and L. Sirovich, Karhunen-Loeve procedure for gappy data, Journal of the Optical Society of America A: Optics, Image Science and Vision, 12 (1995), 1657-1664.
doi: 10.1364/JOSAA.12.001657. |
[9] |
E. Liberge and A. Hamdouni, Reduced order modelling method via proper orthogonal decomposition (POD) for flow around an oscillating cylinder, Journal of Fluids and Structures, 26 (2010), 292-311.
doi: 10.1016/j.jfluidstructs.2009.10.006. |
[10] |
A. Hay, J. Borggaard and D. Pelletier, Improved low-order modeling from sensitivity analysis of the proper orthogonal decomposition, J. Fluid Mech., 629 (2009), 41-72.
doi: 10.1017/S0022112009006363. |
[11] |
J. Hoffman and H. W. Wielandt, The variation of the spectrum of a normal matrix, Duke Math. J., 20 (1953), 37-39.
doi: 10.1215/S0012-7094-53-02004-3. |
[12] |
D. Hömberg and S. Volkwein, Control of laser surface hardening by a reduced-order approach utilizing proper orthogonal decomposition, Math. Comput. Model., 38 (2003), 1003-1028.
doi: 10.1016/S0895-7177(03)90102-6. |
[13] |
K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics, SIAM J. Numer. Anal., 40 (2002), 492-515.
doi: 10.1137/S0036142900382612. |
[14] |
K. Kunisch and S. Volkwein, Control of Burgers equation by a reduced order approach using proper orthogonal decomposition, J. Optim. Theory Appl., 102 (1999), 345-371.
doi: 10.1023/A:1021732508059. |
[15] |
T. Lassila and G. Rozza, Parametric free-form shape design with PDE models and reduced basis models, Comput. Methods Appl. Mech. Engrg., 199 (2010), 1583-1592.
doi: 10.1016/j.cma.2010.01.007. |
[16] |
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1980. |
[17] |
M. Pomarède, Investigation et Application des Méthodes D'ordre Réduit pour les Calculs D'éoulements dans les Faisceaux Tubulaires D'Échangeurs de Chaleur, PhD thesis, University of La Rochelle, 2012. |
[18] |
S. Roujol, M. Ries, B. Quesson, C. Moonen and B. Denis de Senneville, Real-time MR-thermometry and dosimetry for interventional guidance on abdominal organs, Magnetic Resonance in Medicine, 63 (2010), 1080-1087.
doi: 10.1002/mrm.22309. |
[19] |
B. Rousselet and D. Chenais, Continuité et différentiabilité d'éléments propres: Application à l'optimisation de structures, Appl. Math. Optim., 22 (1990), 27-59.
doi: 10.1007/BF01447319. |
[20] |
S. Volkwein, Optimal control of a phase-field model using the proper orthogonal decomposition, Z. Angew. Math. Mech., 81 (2001), 83-97.
doi: 10.1002/1521-4001(200102)81:2<83::AID-ZAMM83>3.0.CO;2-R. |
show all references
References:
[1] |
N. Akkari, A. Hamdouni, E. Liberge and M. Jazar, A mathematical and numerical study of the sensitivity of a reduced order model by POD (ROM-POD), for a 2D incompressible fluid flow, Journal of Computational and Applied Mathematics, 270 (2014), 522-530.
doi: 10.1016/j.cam.2013.11.025. |
[2] |
N. Akkari, A. Hamdouni and M. Jazar, Mathematical and numerical results on the sensitivity of the POD approximation relative to the Burgers equation, Applied Mathematics and Computation, 247 (2014), 951-961.
doi: 10.1016/j.amc.2014.09.005. |
[3] |
N. Akkari, A. Hamdouni, E. Liberge and M. Jazzar, On the sensitivity of the POD technique for a parameterized quasi-nonlinear parabolic equation, Advanced Modeling and Simulation in Engineering Sciences, 1 (2014), p14.
doi: 10.1186/s40323-014-0014-4. |
[4] |
C. Allery, C. Béghein and A. Hamdouni, On investigation of particle dispersion by a POD approach, Int. Applied Mechanics, 44 (2008), 110-119.
doi: 10.1007/s10778-008-0025-2. |
[5] |
R. Bhatia and L. Elsner, The Hoffman-Wielandt inequality in infinite dimensions, Proc. Indian Acad. Sci. (Math. Sci.), 104 (1994), 483-494.
doi: 10.1007/BF02867116. |
[6] |
C. Davis and W. M. Kahan, The rotation of eigenvectors by a perturbation. III, SIAM J. Numer. Anal., 7 (1970), 1-46.
doi: 10.1137/0707001. |
[7] |
B. Denis de Senneville, A. El Hamidi and C. Moonen, A direct PCA-based approach for real-time description of physiological organ deformations, IEEE Transactions on Medical Imaging, 34 (2014), 974-982.
doi: 10.1109/TMI.2014.2371995. |
[8] |
R. Everson and L. Sirovich, Karhunen-Loeve procedure for gappy data, Journal of the Optical Society of America A: Optics, Image Science and Vision, 12 (1995), 1657-1664.
doi: 10.1364/JOSAA.12.001657. |
[9] |
E. Liberge and A. Hamdouni, Reduced order modelling method via proper orthogonal decomposition (POD) for flow around an oscillating cylinder, Journal of Fluids and Structures, 26 (2010), 292-311.
doi: 10.1016/j.jfluidstructs.2009.10.006. |
[10] |
A. Hay, J. Borggaard and D. Pelletier, Improved low-order modeling from sensitivity analysis of the proper orthogonal decomposition, J. Fluid Mech., 629 (2009), 41-72.
doi: 10.1017/S0022112009006363. |
[11] |
J. Hoffman and H. W. Wielandt, The variation of the spectrum of a normal matrix, Duke Math. J., 20 (1953), 37-39.
doi: 10.1215/S0012-7094-53-02004-3. |
[12] |
D. Hömberg and S. Volkwein, Control of laser surface hardening by a reduced-order approach utilizing proper orthogonal decomposition, Math. Comput. Model., 38 (2003), 1003-1028.
doi: 10.1016/S0895-7177(03)90102-6. |
[13] |
K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics, SIAM J. Numer. Anal., 40 (2002), 492-515.
doi: 10.1137/S0036142900382612. |
[14] |
K. Kunisch and S. Volkwein, Control of Burgers equation by a reduced order approach using proper orthogonal decomposition, J. Optim. Theory Appl., 102 (1999), 345-371.
doi: 10.1023/A:1021732508059. |
[15] |
T. Lassila and G. Rozza, Parametric free-form shape design with PDE models and reduced basis models, Comput. Methods Appl. Mech. Engrg., 199 (2010), 1583-1592.
doi: 10.1016/j.cma.2010.01.007. |
[16] |
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1980. |
[17] |
M. Pomarède, Investigation et Application des Méthodes D'ordre Réduit pour les Calculs D'éoulements dans les Faisceaux Tubulaires D'Échangeurs de Chaleur, PhD thesis, University of La Rochelle, 2012. |
[18] |
S. Roujol, M. Ries, B. Quesson, C. Moonen and B. Denis de Senneville, Real-time MR-thermometry and dosimetry for interventional guidance on abdominal organs, Magnetic Resonance in Medicine, 63 (2010), 1080-1087.
doi: 10.1002/mrm.22309. |
[19] |
B. Rousselet and D. Chenais, Continuité et différentiabilité d'éléments propres: Application à l'optimisation de structures, Appl. Math. Optim., 22 (1990), 27-59.
doi: 10.1007/BF01447319. |
[20] |
S. Volkwein, Optimal control of a phase-field model using the proper orthogonal decomposition, Z. Angew. Math. Mech., 81 (2001), 83-97.
doi: 10.1002/1521-4001(200102)81:2<83::AID-ZAMM83>3.0.CO;2-R. |
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