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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), 596-623.
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 (accepted). |
[3] |
M. B. Blumenthal, Predictability of a coupled ocean-atmosphere model, J. Climate, 4 (1991), 766-784.
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), 887-915.
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), 2131-2146.
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), 594-604.
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), 529-542.
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), 4965-4970.
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), 16-28.
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), 11015-11021.
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), 449-459. |
[12] |
P. J. Holmes, J. L. Lumley, G. Berkooz and C. W. Rowley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, UK, 2nd edition, 2012.
doi: 10.1017/CBO9780511919701. |
[13] |
H. Hotelling, Analysis of a complex of statistical variables into principal components, J. Educ. Psychol., 24 (1933), 417-441.
doi: 10.1037/h0071325. |
[14] |
H. Hotelling, Analysis of a complex of statistical variables into principal components, J. Educ. Psychol., 24 (1933), 498-520. |
[15] |
M. R. Jovanović, P. J. Schmid and J. W. Nichols, Sparsity-promoting dynamic mode decomposition, Phys. Fluids, 26 (2014), 024103, arXiv:1309.4165v1.
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), 620-627.
doi: 10.2514/3.20031. |
[17] |
E. N. Lorenz, Empirical orthogonal functions and statistical weather prediction, Technical report, Massachusetts Institute of Technology, Dec. 1956. |
[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), 233-247.
doi: 10.1007/s00162-010-0184-8. |
[19] |
L. Massa, R. Kumar and P. Ravindran, Dynamic mode decomposition analysis of detonation waves, Phys. Fluids, 24, June 2012. |
[20] |
I. Mezić, Spectral properties of dynamical systems, model reduction and decompositions, Nonlin. Dynam., 41 (2005), 309-325.
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), 357-378.
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), 87-97.
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), 335-363.
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), 5722-5727. |
[25] |
K. Pearson, LIII. on lines and planes of closest fit to systems of points in space, Philos. Mag., 2 (1901), 559-572.
doi: 10.1080/14786440109462720. |
[26] |
C. Penland, Random forcing and forecasting using Principal Oscillation Pattern analysis, Mon. Weather Rev., 117 (1989), 2165-2185. |
[27] |
C. Penland and T. Magorian, Prediction of Niño 3 sea-surface temperatures using linear inverse modeling, J. Climate, 6 (1993), 1067-1076. |
[28] |
C. W. Rowley, Model reduction for fluids, using balanced proper orthogonal decomposition, Int. J. Bifurcat. Chaos, 15 (2005), 997-1013.
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), 115-127.
doi: 10.1017/S0022112009992059. |
[30] |
P. J. Schmid, Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656 (2010), 5-28.
doi: 10.1017/S0022112010001217. |
[31] |
P. J. Schmid, Application of the dynamic mode decomposition to experimental data, Exp. Fluids, 50 (2011), 1123-1130.
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), 249-259.
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), 5-28.
doi: 10.1017/S0022112010001217. |
[34] |
P. J. Schmid, D. Violato and F. Scarano, Decomposition of time-resolved tomographic PIV, Exp. Fluids, 52 (2012), 1567-1579.
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), 1098-1110.
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), 1203-1220.
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), 852-869.
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), 573-582. |
[39] |
K. Taira and T. Colonius, The immersed boundary method: A projection approach, J. Comput. Phys., 225 (2007), 2118-2137.
doi: 10.1016/j.jcp.2007.03.005. |
[40] |
L. N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, Philadelphia, 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), pp1429.
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), 5317-5333.
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, 49th AIAA Aerospace Sciences Meeting and Exhibit, Jan. 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), 377-400. |
[45] |
M. O. Williams, I. G. Kevrekidis and C. W. Rowley, A data-driven approximation of the Koopman operator: extending dynamic mode decomposition, arXiv:1408.4408, 2014. |
[46] |
A. Wynn, D. Pearson, B. Ganapathisubramani and P. J. Goulart, Optimal mode decomposition for unsteady flows, J. Fluid Mech., 733 (2013), 473-503.
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), 596-623.
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 (accepted). |
[3] |
M. B. Blumenthal, Predictability of a coupled ocean-atmosphere model, J. Climate, 4 (1991), 766-784.
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), 887-915.
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), 2131-2146.
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), 594-604.
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), 529-542.
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), 4965-4970.
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), 16-28.
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), 11015-11021.
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), 449-459. |
[12] |
P. J. Holmes, J. L. Lumley, G. Berkooz and C. W. Rowley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, UK, 2nd edition, 2012.
doi: 10.1017/CBO9780511919701. |
[13] |
H. Hotelling, Analysis of a complex of statistical variables into principal components, J. Educ. Psychol., 24 (1933), 417-441.
doi: 10.1037/h0071325. |
[14] |
H. Hotelling, Analysis of a complex of statistical variables into principal components, J. Educ. Psychol., 24 (1933), 498-520. |
[15] |
M. R. Jovanović, P. J. Schmid and J. W. Nichols, Sparsity-promoting dynamic mode decomposition, Phys. Fluids, 26 (2014), 024103, arXiv:1309.4165v1.
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), 620-627.
doi: 10.2514/3.20031. |
[17] |
E. N. Lorenz, Empirical orthogonal functions and statistical weather prediction, Technical report, Massachusetts Institute of Technology, Dec. 1956. |
[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), 233-247.
doi: 10.1007/s00162-010-0184-8. |
[19] |
L. Massa, R. Kumar and P. Ravindran, Dynamic mode decomposition analysis of detonation waves, Phys. Fluids, 24, June 2012. |
[20] |
I. Mezić, Spectral properties of dynamical systems, model reduction and decompositions, Nonlin. Dynam., 41 (2005), 309-325.
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), 357-378.
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), 87-97.
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), 335-363.
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), 5722-5727. |
[25] |
K. Pearson, LIII. on lines and planes of closest fit to systems of points in space, Philos. Mag., 2 (1901), 559-572.
doi: 10.1080/14786440109462720. |
[26] |
C. Penland, Random forcing and forecasting using Principal Oscillation Pattern analysis, Mon. Weather Rev., 117 (1989), 2165-2185. |
[27] |
C. Penland and T. Magorian, Prediction of Niño 3 sea-surface temperatures using linear inverse modeling, J. Climate, 6 (1993), 1067-1076. |
[28] |
C. W. Rowley, Model reduction for fluids, using balanced proper orthogonal decomposition, Int. J. Bifurcat. Chaos, 15 (2005), 997-1013.
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), 115-127.
doi: 10.1017/S0022112009992059. |
[30] |
P. J. Schmid, Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656 (2010), 5-28.
doi: 10.1017/S0022112010001217. |
[31] |
P. J. Schmid, Application of the dynamic mode decomposition to experimental data, Exp. Fluids, 50 (2011), 1123-1130.
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), 249-259.
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), 5-28.
doi: 10.1017/S0022112010001217. |
[34] |
P. J. Schmid, D. Violato and F. Scarano, Decomposition of time-resolved tomographic PIV, Exp. Fluids, 52 (2012), 1567-1579.
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), 1098-1110.
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), 1203-1220.
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), 852-869.
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), 573-582. |
[39] |
K. Taira and T. Colonius, The immersed boundary method: A projection approach, J. Comput. Phys., 225 (2007), 2118-2137.
doi: 10.1016/j.jcp.2007.03.005. |
[40] |
L. N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, Philadelphia, 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), pp1429.
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), 5317-5333.
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, 49th AIAA Aerospace Sciences Meeting and Exhibit, Jan. 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), 377-400. |
[45] |
M. O. Williams, I. G. Kevrekidis and C. W. Rowley, A data-driven approximation of the Koopman operator: extending dynamic mode decomposition, arXiv:1408.4408, 2014. |
[46] |
A. Wynn, D. Pearson, B. Ganapathisubramani and P. J. Goulart, Optimal mode decomposition for unsteady flows, J. Fluid Mech., 733 (2013), 473-503.
doi: 10.1017/jfm.2013.426. |
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