Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
126 tokens/sec
GPT-4o
28 tokens/sec
Gemini 2.5 Pro Pro
42 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

A data driven Koopman-Schur decomposition for computational analysis of nonlinear dynamics (2312.15837v2)

Published 26 Dec 2023 in math.NA and cs.NA

Abstract: This paper introduces a new theoretical and computational framework for a data driven Koopman mode analysis of nonlinear dynamics. To alleviate the potential problem of ill-conditioned eigenvectors in the existing implementations of the Dynamic Mode Decomposition (DMD) and the Extended Dynamic Mode Decomposition (EDMD), the new method introduces a Koopman-Schur decomposition that is entirely based on unitary transformations. The analysis in terms of the eigenvectors as modes of a Koopman operator compression is replaced with a modal decomposition in terms of a flag of invariant subspaces that correspond to selected eigenvalues. The main computational tool from the numerical linear algebra is the partial ordered Schur decomposition that provides convenient orthonormal bases for these subspaces. In the case of real data, a real Schur form is used and the computation is based on real orthogonal transformations. The new computational scheme is presented in the framework of the Extended DMD and the kernel trick is used.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (57)
  1. S. Ahuja and C. W. Rowley. Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators. Journal of Fluid Mechanics, 645:447–478, 2010.
  2. LAPACK Users’ Guide (Third Ed.). Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1999.
  3. Ergodic theory, dynamic mode decomposition, and computation of spectral properties of the koopman operator. SIAM Journal on Applied Dynamical Systems, 16(4):2096–2126, 2017.
  4. T. Askham and J. Kutz. Variable projection methods for an optimized dynamic mode decomposition. SIAM Journal on Applied Dynamical Systems, 17(1):380–416, 2018.
  5. On swapping diagonal blocks in real Schur form. Linear Algebra and its Applications, 186:75–95, 1993.
  6. Estimation of perturbations in robotic behavior using dynamic mode decomposition. Advanced Robotics, 29(5):331–343, 2015.
  7. Modern koopman theory for dynamical systems. SIAM Review, 64(2):229–340, 2022.
  8. Variants of dynamic mode decomposition: Boundary condition, Koopman, and Fourier analyses. Journal of Nonlinear Science, 22(6):887–915, April 2012.
  9. The geometric approach for constructing sinai–ruelle–bowen measures. Journal of Statistical Physics, 166(3-4):467–493, 2017.
  10. Residual dynamic mode decomposition: robust and verified koopmanism. Journal of Fluid Mechanics, 955:A21, 2023.
  11. Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems. arXiv e-prints, page arXiv:2111.14889, November 2021.
  12. Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition. Experiments in Fluids, 57(3):42, 2016.
  13. J. W. Demmel. A Numerical Analyst’s Jordan Canonical Form. PhD thesis, Center for Pure and Applied Mathematics, University of California, Berkeley, 1983.
  14. Z. Drmač. Dynamic Mode Decomposition-A Numerical Linear Algebra Perspective, pages 161–194. Springer International Publishing, Cham, 2020.
  15. Data driven modal decompositions: analysis and enhancements. SIAM Journal on Scientific Computing, 40(4):A2253–A2285, 2018.
  16. Z. Drmač. A LAPACK implementation of the Dynamic Mode Decomposition I. Technical report, Department of Mathematics, University of Zagreb, Croatia, and AIMdyn Inc. Santa Barbara, CA, October 2022. LAPACK Working Note 298.
  17. Z. Drmač. A LAPACK implementation of the Dynamic Mode Decomposition II. Technical report, Department of Mathematics, University of Zagreb, Croatia, and AIMdyn Inc. Santa Barbara, CA, November 2022. LAPACK Working Note 300.
  18. Zlatko Drmač. Hermitian dynamic mode decomposition – numerical analysis and software solution. ACM Trans. Math. Soft. (in revision), 00:72–83, 2023.
  19. Zlatko Drmač. A LAPACK implementation of the Dynamic Mode Decomposition. ACM Trans. Math. Soft. (in revision), 00:??–??, 2023.
  20. On least squares problems with certain Vandermonde–Khatri–Rao structure with applications to DMD. SIAM Journal on Scientific Computing, 42(5):A3250–A3284, 2020.
  21. Identification of nonlinear systems using the infinitesimal generator of the koopman semigroup – a numerical implementation of the mauroy-goncalves method. Mathematics, 9(17), 2021.
  22. Nelson Dunford et al. Spectral operators. Pacific Journal of Mathematics, 4(3):321–354, 1954.
  23. Generalized Functions, Volume 2: Spaces of Fundamental and Generalized Functions, volume 261. American Mathematical Soc., 2016.
  24. Detection and analysis of combustion instability from hi-speed flame images using dynamic mode decomposition. In ASME. Dynamic Systems and Control Conference, Volume 1, 2016.
  25. Rank degeneracy and least squares problems. Technical Report CS-TR-76-559, Stanford, CA, USA, 1976.
  26. De-biasing the dynamic mode decomposition for applied Koopman spectral analysis. ArXiv e-prints, February 2015.
  27. Dynamic mode decomposition for large and streaming datasets. Physics of Fluids, 26(11):111701, November 2014.
  28. Matrix Analysis. Cambridge University Press, 1990.
  29. Topics in Matrix Analysis. Cambridge University Press, 1991.
  30. Low–rank and sparse dynamic mode decomposition. In Annual Research Briefs 2012, pages 139–152. Center for Turbulence Research, Stanford University, 2012.
  31. Sparsity-promoting dynamic mode decomposition. Phys. Fluids, 26(2):024103 (22 pages), 2014.
  32. On convergence of Extended Dynamic Mode Decomposition to the Koopman operator. Journal of Nonlinear Science, 28:687–710, April 2018.
  33. Daniel Kressner. Block algorithms for reordering standard and generalized schur forms. ACM Trans. Math. Softw., 32(4):521–532, dec 2006.
  34. Fumi-Yuki Maeda. Function of generalized scalar operators. Journal of Science of the Hiroshima University, Series AI (Mathematics), 26(2):71–76, 1962.
  35. The Koopman Operator in Systems and Control: Concepts, Methodologies, and Applications. 01 2020.
  36. Koopman operator in systems and control. Springer, 2020.
  37. Igor Mezić. Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dynamics, 41:309–325, 2005.
  38. Igor Mezić. Spectrum of the koopman operator, spectral expansions in functional spaces, and state-space geometry. Journal of Nonlinear Science, 30(5):2091–2145, 2020.
  39. Igor Mezić. On numerical approximations of the koopman operator. Mathematics, 10(7):1180, 2022.
  40. A Koopman Operator-Based Prediction Algorithm and its Application to COVID-19 Pandemic. arXiv e-prints, page arXiv:2304.13601, April 2023.
  41. Dynamic mode decomposition with control. SIAM Journal on Applied Dynamical Systems, 15(1):142–161, 2016.
  42. Robert R Rogers. Triangular form for bounded linear operators. Journal of functional analysis, 88(1):135–152, 1990.
  43. Applications of the dynamic mode decomposition. Theoretical and Computational Fluid Dynamics, 25(1):249–259, 2011.
  44. Dynamic Mode Decomposition of numerical and experimental data. Journal of Fluid Mechanics, 656, 11 2008.
  45. Peter J Schmid. Nonmodal stability theory. Annual review of fluid mechanics, 39(1):129–162, 2007.
  46. Balachandra Suri. Quasi-two-dimensional Kolmogorov flow: Bifurcations and exact coherent structures. PhD thesis, Georgia Institute of Technology, 2017.
  47. Forecasting fluid flows using the geometry of turbulence. Phys. Rev. Lett., 118:114501, Mar 2017.
  48. Applied koopman operator theory for power systems technology. Nonlinear Theory and Its Applications, IEICE, 7(4):430–459, 2016.
  49. Bayesian dynamic mode decomposition. In Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, IJCAI-17, pages 2814–2821, 2017.
  50. Learning Koopman invariant subspaces for dynamic mode decomposition. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems 30, pages 1130–1140. Curran Associates, Inc., 2017.
  51. Sparse nonnegative dynamic mode decomposition. In 2017 IEEE International Conference on Image Processing (ICIP), pages 2682–2686, Sep. 2017.
  52. Subspace dynamic mode decomposition for stochastic Koopman analysis. Phys. Rev. E, 96:033310, Sep 2017.
  53. A numerically stable dynamic mode decomposition algorithm for nearly defective systems. IEEE Control Systems Letters, 5(1):67–72, 2021.
  54. Bifurcations in a quasi-two-dimensional kolmogorov-like flow. Journal of Fluid Mechanics, 828:837–866, 10 2017.
  55. A. van der Sluis. Condition numbers and equilibration of matrices. Numerische Mathematik, 14:14–23, 1969.
  56. A data-driven approximation of the Koopman operator: extending dynamic mode decomposition. Journal of Nonlinear Science, 25(6):1307–1346, 2015.
  57. Optimal mode decomposition for unsteady flows. Journal of Fluid Mechanics, 733:473–503, 2013.
Citations (5)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now