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Equation-free computation of coarse-grained center manifolds of microscopic simulators
On dynamic mode decomposition: Theory and applications
1. | Dept. of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, United States, United States, United States |
2. | Dept. of Applied Mathematics, University of Washington, Seattle, WA 98195, United States, United States |
References:
[1] |
S. Bagheri, Koopman-mode decomposition of the cylinder wake,, J. Fluid Mech., 726 (2013), 596.
doi: 10.1017/jfm.2013.249. |
[2] |
B. A. Belson, J. H. Tu and C. W. Rowley, A Parallelized Model Reduction Library,, ACM T. Math. Software, (2013). Google Scholar |
[3] |
M. B. Blumenthal, Predictability of a coupled ocean-atmosphere model,, J. Climate, 4 (1991), 766.
doi: 10.1175/1520-0442(1991)004<0766:POACOM>2.0.CO;2. |
[4] |
K. K. Chen, J. H. Tu and C. W. Rowley, Variants of dynamic mode decomposition: Boundary condition, Koopman, and Fourier analyses,, J. Nonlinear Sci., 22 (2012), 887.
doi: 10.1007/s00332-012-9130-9. |
[5] |
T. Colonius and K. Taira, A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions,, Comput. Method Appl. M., 197 (2008), 2131.
doi: 10.1016/j.cma.2007.08.014. |
[6] |
D. Duke, D. Honnery and J. Soria, Experimental investigation of nonlinear instabilities in annular liquid sheets,, J. Fluid Mech., 691 (2012), 594.
doi: 10.1017/jfm.2011.516. |
[7] |
D. Duke, J. Soria and D. Honnery, An error analysis of the dynamic mode decomposition,, Exp. Fluids, 52 (2012), 529.
doi: 10.1007/s00348-011-1235-7. |
[8] |
P. J. Goulart, A. Wynn and D. Pearson, Optimal mode decomposition for high dimensional systems,, In Proceedings of the 51st IEEE Conference on Decision and Control, 2012 (2012), 4965.
doi: 10.1109/CDC.2012.6426995. |
[9] |
M. Grilli, P. J. Schmid, S. Hickel and N. A. Adams, Analysis of unsteady behaviour in shockwave turbulent boundary layer interaction,, J. Fluid Mech., 700 (2012), 16.
doi: 10.1017/jfm.2012.37. |
[10] |
K. Hasselmann, PIPs and POPs: The reduction of complex dynamical-systems using Principal Interaction and Oscillation Patterns,, J. Geophys. Res.-Atmos., 93 (1988), 11015.
doi: 10.1029/JD093iD09p11015. |
[11] |
B. L. Ho and R. E. Kalman, Effective construction of linear state-variable models from input/output data,, Proceedings of the Third Annual Allerton Conference on Circuit and System Theory, (1965), 449.
|
[12] |
P. J. Holmes, J. L. Lumley, G. Berkooz and C. W. Rowley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry,, Cambridge University Press, (2012).
doi: 10.1017/CBO9780511919701. |
[13] |
H. Hotelling, Analysis of a complex of statistical variables into principal components,, J. Educ. Psychol., 24 (1933), 417.
doi: 10.1037/h0071325. |
[14] |
H. Hotelling, Analysis of a complex of statistical variables into principal components,, J. Educ. Psychol., 24 (1933), 498. Google Scholar |
[15] |
M. R. Jovanović, P. J. Schmid and J. W. Nichols, Sparsity-promoting dynamic mode decomposition,, Phys. Fluids, 26 (2014).
doi: 10.1063/1.4863670. |
[16] |
J. N. Juang and R. S. Pappa, An eigensystem realization-algorithm for modal parameter-identification and model-reduction,, J. Guid. Control Dynam., 8 (1985), 620.
doi: 10.2514/3.20031. |
[17] |
E. N. Lorenz, Empirical orthogonal functions and statistical weather prediction,, Technical report, (1956). Google Scholar |
[18] |
Z. Ma, S. Ahuja and C. W. Rowley, Reduced-order models for control of fluids using the eigensystem realization algorithm,, Theor. Comp. Fluid Dyn., 25 (2011), 233.
doi: 10.1007/s00162-010-0184-8. |
[19] |
L. Massa, R. Kumar and P. Ravindran, Dynamic mode decomposition analysis of detonation waves,, Phys. Fluids, (2012). Google Scholar |
[20] |
I. Mezić, Spectral properties of dynamical systems, model reduction and decompositions,, Nonlin. Dynam., 41 (2005), 309.
doi: 10.1007/s11071-005-2824-x. |
[21] |
I. Mezić, Analysis of fluid flows via spectral properties of the Koopman operator,, Annu. Rev. Fluid Mech., 45 (2013), 357.
doi: 10.1146/annurev-fluid-011212-140652. |
[22] |
T. W. Muld, G. Efraimsson and D. S. Henningson, Flow structures around a high-speed train extracted using proper orthogonal decomposition and dynamic mode decomposition,, Comput. Fluids, 57 (2012), 87.
doi: 10.1016/j.compfluid.2011.12.012. |
[23] |
B. R. Noack, K. Afanasiev, M. Morzyński, G. Tadmor and F. Thiele, A hierarchy of low-dimensional models for the transient and post-transient cylinder wake,, J. Fluid Mech., 497 (2003), 335.
doi: 10.1017/S0022112003006694. |
[24] |
B. R. Noack, G. Tadmor and M. Morzyński, Actuation models and dissipative control in empirical Galerkin models of fluid flows,, In Proceedings of the American Control Conference, (2004), 5722. Google Scholar |
[25] |
K. Pearson, LIII. on lines and planes of closest fit to systems of points in space,, Philos. Mag., 2 (1901), 559.
doi: 10.1080/14786440109462720. |
[26] |
C. Penland, Random forcing and forecasting using Principal Oscillation Pattern analysis,, Mon. Weather Rev., 117 (1989), 2165. Google Scholar |
[27] |
C. Penland and T. Magorian, Prediction of Niño 3 sea-surface temperatures using linear inverse modeling,, J. Climate, 6 (1993), 1067. Google Scholar |
[28] |
C. W. Rowley, Model reduction for fluids, using balanced proper orthogonal decomposition,, Int. J. Bifurcat. Chaos, 15 (2005), 997.
doi: 10.1142/S0218127405012429. |
[29] |
C. W. Rowley, I. Mezic, S. Bagheri, P. Schlatter and D. S. Henningson, Spectral analysis of nonlinear flows,, J. Fluid Mech., 641 (2009), 115.
doi: 10.1017/S0022112009992059. |
[30] |
P. J. Schmid, Dynamic mode decomposition of numerical and experimental data,, J. Fluid Mech., 656 (2010), 5.
doi: 10.1017/S0022112010001217. |
[31] |
P. J. Schmid, Application of the dynamic mode decomposition to experimental data,, Exp. Fluids, 50 (2011), 1123.
doi: 10.1007/s00348-010-0911-3. |
[32] |
P. J. Schmid, L. Li, M. P. Juniper and O. Pust, Applications of the dynamic mode decomposition,, Theor. Comp. Fluid Dyn., 25 (2011), 249.
doi: 10.1007/s00162-010-0203-9. |
[33] |
P. J. Schmid and J. Sesterhenn, Dynamic mode decomposition of numerical and experimental data,, Journal of Fluid Mechanics, 656 (2010), 5.
doi: 10.1017/S0022112010001217. |
[34] |
P. J. Schmid, D. Violato and F. Scarano, Decomposition of time-resolved tomographic PIV,, Exp. Fluids, 52 (2012), 1567.
doi: 10.1007/s00348-012-1266-8. |
[35] |
A. Seena and H. J. Sung, Dynamic mode decomposition of turbulent cavity flows for self-sustained oscillations,, Int. J. Heat Fluid Fl., 32 (2011), 1098.
doi: 10.1016/j.ijheatfluidflow.2011.09.008. |
[36] |
O. Semeraro, G. Bellani and F. Lundell, Analysis of time-resolved PIV measurements of a confined turbulent jet using POD and Koopman modes,, Exp. Fluids, 53 (2012), 1203.
doi: 10.1007/s00348-012-1354-9. |
[37] |
J. R. Singler, Optimality of balanced proper orthogonal decomposition for data reconstruction,, Numerical Functional Analysis and Optimization, 31 (2010), 852.
doi: 10.1080/01630563.2010.500022. |
[38] |
L. Sirovich, Turbulence and the dynamics of coherent structures. 2. Symmetries and transformations,, Q. Appl. Math., 45 (1987), 573.
|
[39] |
K. Taira and T. Colonius, The immersed boundary method: A projection approach,, J. Comput. Phys., 225 (2007), 2118.
doi: 10.1016/j.jcp.2007.03.005. |
[40] |
L. N. Trefethen and D. Bau III, Numerical Linear Algebra,, SIAM, (1997).
doi: 10.1137/1.9780898719574. |
[41] |
J. H. Tu, J. Griffin, A. Hart, C. W. Rowley, L. N. Cattafesta III and L. S. Ukeiley, Integration of non-time-resolved PIV and time-resolved velocity point sensors for dynamic estimation of velocity fields,, Exp. Fluids, 54 (2013).
doi: 10.2514/6.2012-33. |
[42] |
J. H. Tu and C. W. Rowley, An improved algorithm for balanced POD through an analytic treatment of impulse response tails,, J. Comput. Phys., 231 (2012), 5317.
doi: 10.1016/j.jcp.2012.04.023. |
[43] |
J. H. Tu, C. W. Rowley, E. Aram and R. Mittal, Koopman spectral analysis of separated flow over a finite-thickness flat plate with elliptical leading edge,, AIAA Paper 2011-38, (2011), 2011.
doi: 10.2514/6.2011-38. |
[44] |
H. von Storch, G. Bürger, R. Schnur and J. S. von Storch, Principal oscillation patterns: A review,, J. Climate, 8 (1995), 377. Google Scholar |
[45] |
M. O. Williams, I. G. Kevrekidis and C. W. Rowley, A data-driven approximation of the Koopman operator: extending dynamic mode decomposition,, , (2014). Google Scholar |
[46] |
A. Wynn, D. Pearson, B. Ganapathisubramani and P. J. Goulart, Optimal mode decomposition for unsteady flows,, J. Fluid Mech., 733 (2013), 473.
doi: 10.1017/jfm.2013.426. |
show all references
References:
[1] |
S. Bagheri, Koopman-mode decomposition of the cylinder wake,, J. Fluid Mech., 726 (2013), 596.
doi: 10.1017/jfm.2013.249. |
[2] |
B. A. Belson, J. H. Tu and C. W. Rowley, A Parallelized Model Reduction Library,, ACM T. Math. Software, (2013). Google Scholar |
[3] |
M. B. Blumenthal, Predictability of a coupled ocean-atmosphere model,, J. Climate, 4 (1991), 766.
doi: 10.1175/1520-0442(1991)004<0766:POACOM>2.0.CO;2. |
[4] |
K. K. Chen, J. H. Tu and C. W. Rowley, Variants of dynamic mode decomposition: Boundary condition, Koopman, and Fourier analyses,, J. Nonlinear Sci., 22 (2012), 887.
doi: 10.1007/s00332-012-9130-9. |
[5] |
T. Colonius and K. Taira, A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions,, Comput. Method Appl. M., 197 (2008), 2131.
doi: 10.1016/j.cma.2007.08.014. |
[6] |
D. Duke, D. Honnery and J. Soria, Experimental investigation of nonlinear instabilities in annular liquid sheets,, J. Fluid Mech., 691 (2012), 594.
doi: 10.1017/jfm.2011.516. |
[7] |
D. Duke, J. Soria and D. Honnery, An error analysis of the dynamic mode decomposition,, Exp. Fluids, 52 (2012), 529.
doi: 10.1007/s00348-011-1235-7. |
[8] |
P. J. Goulart, A. Wynn and D. Pearson, Optimal mode decomposition for high dimensional systems,, In Proceedings of the 51st IEEE Conference on Decision and Control, 2012 (2012), 4965.
doi: 10.1109/CDC.2012.6426995. |
[9] |
M. Grilli, P. J. Schmid, S. Hickel and N. A. Adams, Analysis of unsteady behaviour in shockwave turbulent boundary layer interaction,, J. Fluid Mech., 700 (2012), 16.
doi: 10.1017/jfm.2012.37. |
[10] |
K. Hasselmann, PIPs and POPs: The reduction of complex dynamical-systems using Principal Interaction and Oscillation Patterns,, J. Geophys. Res.-Atmos., 93 (1988), 11015.
doi: 10.1029/JD093iD09p11015. |
[11] |
B. L. Ho and R. E. Kalman, Effective construction of linear state-variable models from input/output data,, Proceedings of the Third Annual Allerton Conference on Circuit and System Theory, (1965), 449.
|
[12] |
P. J. Holmes, J. L. Lumley, G. Berkooz and C. W. Rowley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry,, Cambridge University Press, (2012).
doi: 10.1017/CBO9780511919701. |
[13] |
H. Hotelling, Analysis of a complex of statistical variables into principal components,, J. Educ. Psychol., 24 (1933), 417.
doi: 10.1037/h0071325. |
[14] |
H. Hotelling, Analysis of a complex of statistical variables into principal components,, J. Educ. Psychol., 24 (1933), 498. Google Scholar |
[15] |
M. R. Jovanović, P. J. Schmid and J. W. Nichols, Sparsity-promoting dynamic mode decomposition,, Phys. Fluids, 26 (2014).
doi: 10.1063/1.4863670. |
[16] |
J. N. Juang and R. S. Pappa, An eigensystem realization-algorithm for modal parameter-identification and model-reduction,, J. Guid. Control Dynam., 8 (1985), 620.
doi: 10.2514/3.20031. |
[17] |
E. N. Lorenz, Empirical orthogonal functions and statistical weather prediction,, Technical report, (1956). Google Scholar |
[18] |
Z. Ma, S. Ahuja and C. W. Rowley, Reduced-order models for control of fluids using the eigensystem realization algorithm,, Theor. Comp. Fluid Dyn., 25 (2011), 233.
doi: 10.1007/s00162-010-0184-8. |
[19] |
L. Massa, R. Kumar and P. Ravindran, Dynamic mode decomposition analysis of detonation waves,, Phys. Fluids, (2012). Google Scholar |
[20] |
I. Mezić, Spectral properties of dynamical systems, model reduction and decompositions,, Nonlin. Dynam., 41 (2005), 309.
doi: 10.1007/s11071-005-2824-x. |
[21] |
I. Mezić, Analysis of fluid flows via spectral properties of the Koopman operator,, Annu. Rev. Fluid Mech., 45 (2013), 357.
doi: 10.1146/annurev-fluid-011212-140652. |
[22] |
T. W. Muld, G. Efraimsson and D. S. Henningson, Flow structures around a high-speed train extracted using proper orthogonal decomposition and dynamic mode decomposition,, Comput. Fluids, 57 (2012), 87.
doi: 10.1016/j.compfluid.2011.12.012. |
[23] |
B. R. Noack, K. Afanasiev, M. Morzyński, G. Tadmor and F. Thiele, A hierarchy of low-dimensional models for the transient and post-transient cylinder wake,, J. Fluid Mech., 497 (2003), 335.
doi: 10.1017/S0022112003006694. |
[24] |
B. R. Noack, G. Tadmor and M. Morzyński, Actuation models and dissipative control in empirical Galerkin models of fluid flows,, In Proceedings of the American Control Conference, (2004), 5722. Google Scholar |
[25] |
K. Pearson, LIII. on lines and planes of closest fit to systems of points in space,, Philos. Mag., 2 (1901), 559.
doi: 10.1080/14786440109462720. |
[26] |
C. Penland, Random forcing and forecasting using Principal Oscillation Pattern analysis,, Mon. Weather Rev., 117 (1989), 2165. Google Scholar |
[27] |
C. Penland and T. Magorian, Prediction of Niño 3 sea-surface temperatures using linear inverse modeling,, J. Climate, 6 (1993), 1067. Google Scholar |
[28] |
C. W. Rowley, Model reduction for fluids, using balanced proper orthogonal decomposition,, Int. J. Bifurcat. Chaos, 15 (2005), 997.
doi: 10.1142/S0218127405012429. |
[29] |
C. W. Rowley, I. Mezic, S. Bagheri, P. Schlatter and D. S. Henningson, Spectral analysis of nonlinear flows,, J. Fluid Mech., 641 (2009), 115.
doi: 10.1017/S0022112009992059. |
[30] |
P. J. Schmid, Dynamic mode decomposition of numerical and experimental data,, J. Fluid Mech., 656 (2010), 5.
doi: 10.1017/S0022112010001217. |
[31] |
P. J. Schmid, Application of the dynamic mode decomposition to experimental data,, Exp. Fluids, 50 (2011), 1123.
doi: 10.1007/s00348-010-0911-3. |
[32] |
P. J. Schmid, L. Li, M. P. Juniper and O. Pust, Applications of the dynamic mode decomposition,, Theor. Comp. Fluid Dyn., 25 (2011), 249.
doi: 10.1007/s00162-010-0203-9. |
[33] |
P. J. Schmid and J. Sesterhenn, Dynamic mode decomposition of numerical and experimental data,, Journal of Fluid Mechanics, 656 (2010), 5.
doi: 10.1017/S0022112010001217. |
[34] |
P. J. Schmid, D. Violato and F. Scarano, Decomposition of time-resolved tomographic PIV,, Exp. Fluids, 52 (2012), 1567.
doi: 10.1007/s00348-012-1266-8. |
[35] |
A. Seena and H. J. Sung, Dynamic mode decomposition of turbulent cavity flows for self-sustained oscillations,, Int. J. Heat Fluid Fl., 32 (2011), 1098.
doi: 10.1016/j.ijheatfluidflow.2011.09.008. |
[36] |
O. Semeraro, G. Bellani and F. Lundell, Analysis of time-resolved PIV measurements of a confined turbulent jet using POD and Koopman modes,, Exp. Fluids, 53 (2012), 1203.
doi: 10.1007/s00348-012-1354-9. |
[37] |
J. R. Singler, Optimality of balanced proper orthogonal decomposition for data reconstruction,, Numerical Functional Analysis and Optimization, 31 (2010), 852.
doi: 10.1080/01630563.2010.500022. |
[38] |
L. Sirovich, Turbulence and the dynamics of coherent structures. 2. Symmetries and transformations,, Q. Appl. Math., 45 (1987), 573.
|
[39] |
K. Taira and T. Colonius, The immersed boundary method: A projection approach,, J. Comput. Phys., 225 (2007), 2118.
doi: 10.1016/j.jcp.2007.03.005. |
[40] |
L. N. Trefethen and D. Bau III, Numerical Linear Algebra,, SIAM, (1997).
doi: 10.1137/1.9780898719574. |
[41] |
J. H. Tu, J. Griffin, A. Hart, C. W. Rowley, L. N. Cattafesta III and L. S. Ukeiley, Integration of non-time-resolved PIV and time-resolved velocity point sensors for dynamic estimation of velocity fields,, Exp. Fluids, 54 (2013).
doi: 10.2514/6.2012-33. |
[42] |
J. H. Tu and C. W. Rowley, An improved algorithm for balanced POD through an analytic treatment of impulse response tails,, J. Comput. Phys., 231 (2012), 5317.
doi: 10.1016/j.jcp.2012.04.023. |
[43] |
J. H. Tu, C. W. Rowley, E. Aram and R. Mittal, Koopman spectral analysis of separated flow over a finite-thickness flat plate with elliptical leading edge,, AIAA Paper 2011-38, (2011), 2011.
doi: 10.2514/6.2011-38. |
[44] |
H. von Storch, G. Bürger, R. Schnur and J. S. von Storch, Principal oscillation patterns: A review,, J. Climate, 8 (1995), 377. Google Scholar |
[45] |
M. O. Williams, I. G. Kevrekidis and C. W. Rowley, A data-driven approximation of the Koopman operator: extending dynamic mode decomposition,, , (2014). Google Scholar |
[46] |
A. Wynn, D. Pearson, B. Ganapathisubramani and P. J. Goulart, Optimal mode decomposition for unsteady flows,, J. Fluid Mech., 733 (2013), 473.
doi: 10.1017/jfm.2013.426. |
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