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

On eigenelements sensitivity for compact self-adjoint operators and applications

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.
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

[1]

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

[2]

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

[3]

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 & Applied Analysis, 2014, 13 (6) : 2253-2272. doi: 10.3934/cpaa.2014.13.2253

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

Hui Ma, Dongxu Qi, Ruixia Song, Tianjun Wang. The complete orthogonal V-system and its applications. Communications on Pure & Applied Analysis, 2007, 6 (3) : 853-871. doi: 10.3934/cpaa.2007.6.853

[17]

Cuiling Fan, Koji Momihara. Unified combinatorial constructions of optimal optical orthogonal codes. Advances in Mathematics of Communications, 2014, 8 (1) : 53-66. doi: 10.3934/amc.2014.8.53

[18]

T. L. Alderson, K. E. Mellinger. Geometric constructions of optimal optical orthogonal codes. Advances in Mathematics of Communications, 2008, 2 (4) : 451-467. doi: 10.3934/amc.2008.2.451

[19]

Xinpeng Wang, Bingo Wing-Kuen Ling, Wei-Chao Kuang, Zhijing Yang. Orthogonal intrinsic mode functions via optimization approach. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2019098

[20]

Michel Duprez, Guillaume Olive. Compact perturbations of controlled systems. Mathematical Control & Related Fields, 2018, 8 (2) : 397-410. doi: 10.3934/mcrf.2018016

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]