American Institute of Mathematical Sciences

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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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

T. Kato, Perturbation Theory for Linear Operators,, Springer-Verlag, (1980). Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. doi: 10.1016/j.cam.2013.11.025. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

T. Kato, Perturbation Theory for Linear Operators,, Springer-Verlag, (1980). Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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