American Institute of Mathematical Sciences

Advanced
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 and 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), 522-530. 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.

[1]

Lars Grüne, Luca Mechelli, Simon Pirkelmann, Stefan Volkwein. Performance estimates for economic model predictive control and their application in proper orthogonal decomposition-based implementations. Mathematical Control and Related Fields, 2021, 11 (3) : 579-599. doi: 10.3934/mcrf.2021013
[2]

Meixin Xiong, Liuhong Chen, Ju Ming, Jaemin Shin. Accelerating the Bayesian inference of inverse problems by using data-driven compressive sensing method based on proper orthogonal decomposition. Electronic Research Archive, 2021, 29 (5) : 3383-3403. doi: 10.3934/era.2021044
[3]

Yaiza Canzani, A. Rod Gover, Dmitry Jakobson, Raphaël Ponge. Nullspaces of conformally invariant operators. Applications to $\boldsymbol{Q_k}$-curvature. Electronic Research Announcements, 2013, 20: 43-50. doi: 10.3934/era.2013.20.43
[4]

Saikat Mazumdar. Struwe's decomposition for a polyharmonic operator on a compact Riemannian manifold with or without boundary. Communications on Pure and Applied Analysis, 2017, 16 (1) : 311-330. doi: 10.3934/cpaa.2017015
[5]

P. Cerejeiras, U. Kähler, M. M. Rodrigues, N. Vieira. Hodge type decomposition in variable exponent spaces for the time-dependent operators: the Schrödinger case. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2253-2272. doi: 10.3934/cpaa.2014.13.2253
[6]

Sören Bartels, Nico Weber. Parameter learning and fractional differential operators: Applications in regularized image denoising and decomposition problems. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021048
[7]

Mirela Kohr, Cornel Pintea, Wolfgang L. Wendland. Neumann-transmission problems for pseudodifferential Brinkman operators on Lipschitz domains in compact Riemannian manifolds. Communications on Pure and Applied Analysis, 2014, 13 (1) : 175-202. doi: 10.3934/cpaa.2014.13.175
[8]

Gengsheng Wang, Yashan Xu. Advantages for controls imposed in a proper subset. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2427-2439. doi: 10.3934/dcdsb.2013.18.2427
[9]

Chinmay Kumar Giri. Index-proper nonnegative splittings of matrices. Numerical Algebra, Control and Optimization, 2016, 6 (2) : 103-113. doi: 10.3934/naco.2016002
[10]

Guangzhou Chen, Xiaotong Zhang. Constructions of irredundant orthogonal arrays. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021051
[11]

K. T. Arasu, Manil T. Mohan. Optimization problems with orthogonal matrix constraints. Numerical Algebra, Control and Optimization, 2018, 8 (4) : 413-440. doi: 10.3934/naco.2018026
[12]

Palle E. T. Jorgensen and Steen Pedersen. Orthogonal harmonic analysis of fractal measures. Electronic Research Announcements, 1998, 4: 35-42.

[13]

Golamreza Zamani Eskandani, Hamid Vaezi. Hyers--Ulam--Rassias stability of derivations in proper Jordan $CQ^{*}$-algebras. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1469-1477. doi: 10.3934/dcds.2011.31.1469
[14]

Delfim F. M. Torres. Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 491-500. doi: 10.3934/cpaa.2004.3.491
[15]

Kequan Zhao, Xinmin Yang. Characterizations of the $E$-Benson proper efficiency in vector optimization problems. Numerical Algebra, Control and Optimization, 2013, 3 (4) : 643-653. doi: 10.3934/naco.2013.3.643
[16]

Marius Durea, Elena-Andreea Florea, Radu Strugariu. Henig proper efficiency in vector optimization with variable ordering structure. Journal of Industrial and Management Optimization, 2019, 15 (2) : 791-815. doi: 10.3934/jimo.2018071
[17]

Litismita Jena, Sabyasachi Pani. Index-range monotonicity and index-proper splittings of matrices. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 379-388. doi: 10.3934/naco.2013.3.379
[18]

Siwei Yu, Jianwei Ma, Stanley Osher. Geometric mode decomposition. Inverse Problems and Imaging, 2018, 12 (4) : 831-852. doi: 10.3934/ipi.2018035
[19]

Stefano Bianchini, Daniela Tonon. A decomposition theorem for $BV$ functions. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1549-1566. doi: 10.3934/cpaa.2011.10.1549
[20]

Andrew M. Zimmer. Compact asymptotically harmonic manifolds. Journal of Modern Dynamics, 2012, 6 (3) : 377-403. doi: 10.3934/jmd.2012.6.377

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (253)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy