On eigenelements sensitivity for compact self-adjoint operators and applications

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April 2016, 9(2): 445-455. doi: 10.3934/dcdss.2016006

On eigenelements sensitivity for compact self-adjoint operators and applications

Abdallah El Hamidi ^1, , Aziz Hamdouni ^1, and Marwan Saleh ^1,

1.	Université de La Rochelle, Avenue M. Crpeau, 17042 La Rochelle, France, France, France

Received May 2015 Revised November 2015 Published March 2016

Abstract
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In this manuscript, we present optimal sensitivity results of eigenvalues and eigenspaces with respect to self-adjoint compact operators. We show that while eigenvalues depend in a Lipschitzian way in compact operators, the eigenspaces are only locally Lipschitz. Our results generalize to arbitrary dimension eigenspaces the results obtained in [19] for one-dimensional eigenspaces sensitivity and thus simplify the celebrate results by Davis and Kahan [6] developed for general Hermitian operator perturbations. Moreover, Proper Orthogonal Decomposition bases sensitivity is carried out in the case of time-interval perturbations, spatial perturbations (Gappy-POD) or parameter perturbations.

Keywords: sensitivity, compact operators., Proper orthogonal decomposition.

Mathematics Subject Classification: Primary: 15A18, 15A42; Secondary: 65F1.

Citation: Abdallah El Hamidi, Aziz Hamdouni, Marwan Saleh. On eigenelements sensitivity for compact self-adjoint operators and applications. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 445-455. doi: 10.3934/dcdss.2016006

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. 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. 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). 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. 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. 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. 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. 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, 12 (1995), 1657. 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. 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. 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. 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. 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. 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. 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. 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, (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. 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. 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. doi: 10.1002/1521-4001(200102)81:2<83::AID-ZAMM83>3.0.CO;2-R.

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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. 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. 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). 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. 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. 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. 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. 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, 12 (1995), 1657. 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. 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. 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. 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. 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. 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. 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. 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, (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. 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. 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. doi: 10.1002/1521-4001(200102)81:2<83::AID-ZAMM83>3.0.CO;2-R.

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