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Variationally consistent Hamiltonian model reduction (2404.15315v2)

Published 3 Apr 2024 in math.NA, cs.NA, and math.DS

Abstract: Though ubiquitous as first-principles models for conservative phenomena, Hamiltonian systems present numerous challenges for model reduction even in relatively simple, linear cases. Here, we present a method for the projection-based model reduction of canonical Hamiltonian systems that is variationally consistent for any choice of linear reduced basis: Hamiltonian models project to Hamiltonian models. Applicable in both intrusive and nonintrusive settings, the proposed method is energy-conserving and symplectic, with error provably decomposable into a data projection term and a term measuring deviation from canonical form. Examples from linear elasticity with realistic material parameters are used to demonstrate the advantages of a variationally consistent approach, highlighting the steady convergence exhibited by consistent models where previous methods reliant on inconsistent techniques or specially designed bases exhibit unacceptably large errors.

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References (41)
  1. Structure preserving model reduction of parametric Hamiltonian systems. SIAM Journal on Scientific Computing, 39(6):A2616–A2644, 2017.
  2. M. Afkham and J. Hesthaven. Structure-preserving model-reduction of dissipative Hamiltonian systems. J. Sci. Comput., 81:3–11, 2019.
  3. The Schwarz alternating method for the seamless coupling of nonlinear reduced order models and full order models. ArXiv pre-print, https://arxiv.org/abs/2210.12551, 2022.
  4. Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms. Computer Methods in Applied Mechanics and Engineering, 372:113433, 2020.
  5. Symplectic model reduction of Hamiltonian systems on nonlinear manifolds. ArXiv pre-print, https://arxiv.org/abs/2112.10815, 2021.
  6. Optimal bases for symplectic model order reduction of canonizable linear Hamiltonian systems. IFAC-PapersOnLine, 55(20):463–468, 2022. 10th Vienna International Conference on Mathematical Modelling MATHMOD 2022.
  7. C. E. Caligan and C. Chandre. Conservative dissipation: How important is the Jacobi identity in the dynamics? Chaos, 26(5):053101, May 2016.
  8. Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics. SIAM Journal on Scientific Computing, 37(2):B153–B184, 2015.
  9. Explicit synchronous partitioned scheme for coupled reduced order models based on composite reduced bases. Computer Methods in Applied Mechanics and Engineering, 417:116398, 2023. A Special Issue in Honor of the Lifetime Achievements of T. J. R. Hughes.
  10. A Novel Partitioned Approach for Reduced Order Model—Finite Element Model (ROM-FEM) and ROM-ROM Coupling, pages 475–489.
  11. R. Dhari. Speed of sound in solids calculator, 2024.
  12. Improving the accuracy and scalability of large-scale physics-based data-driven reduced modeling via domain decomposition. ArXiv pre-print, https://arxiv.org/abs/2311.00883, 2023.
  13. Structure-preserving, stability, and accuracy properties of the energy-conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models. International Journal for Numerical Methods in Engineering, 102(5):1077–1110, 2015.
  14. Gradient preserving operator inference: Data-driven reduced-order models for equations with gradient structure. ArXiv pre-print, https://arxiv.org/abs/2401.12138, 2024.
  15. Structure-preserving Galerkin POD reduced-order modeling of Hamiltonian systems. Computer Methods in Applied Mechanics and Engineering, 315:780–798, 2017.
  16. Energetically consistent model reduction for metriplectic systems. Computer Methods in Applied Mechanics and Engineering, 404:115709, 2023.
  17. A. Gruber and I. Tezaur. Canonical and noncanonical Hamiltonian operator inference. Computer Methods in Applied Mechanics and Engineering, 416:116334, 2023.
  18. Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data. Computer Methods in Applied Mechanics and Engineering, 196(4):1030–1047, 2007.
  19. Structure-preserving model order reduction of Hamiltonian systems. ArXiv pre-print, https://arxiv.org/abs/2109.12367, 2021.
  20. Turbulence, coherent structures, dynamical systems and symmetry. Cambridge University Press, 1996.
  21. Structure-preserving model reduction for mechanical systems. Physica D: Nonlinear Phenomena, 184(1):304–318, 2003. Complexity and Nonlinearity in Physical Systems – A Special Issue to Honor Alan Newell.
  22. J. E. Marsden and T. Ratiu. Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems. Springer, 1998.
  23. Data-driven reduced-order models via regularised operator inference for a single-injector combustion process. Journal of the Royal Society of New Zealand, 51(2):194–211, 2021.
  24. The Schwarz alternating method in solid mechanics. Computer Methods in Applied Mechanics and Engineering, 319:19–51, 2017.
  25. The Schwarz alternating method for transient solid dynamics. International Journal for Numerical Methods in Engineering, 123(21):5036–5071, 2022.
  26. E. J. Parish and K. Duraisamy. Non-Markovian closure models for large eddy simulations using the Mori-Zwanzig formalism. Phys. Rev. Fluids, 2:014604, Jan 2017.
  27. B. Peherstorfer. Sampling low-dimensional Markovian dynamics for preasymptotically recovering reduced models from data with operator inference. SIAM Journal on Scientific Computing, 42(5):A3489–A3515, 2023/01/11 2020.
  28. B. Peherstorfer and K. Willcox. Data-driven operator inference for nonintrusive projection-based model reduction. Computer Methods in Applied Mechanics and Engineering, 306:196–215, 2016.
  29. L. Peng and K. Mohseni. Symplectic model reduction of Hamiltonian systems. SIAM Journal on Scientific Computing, 38(1):A1–A27, 2022/11/05 2016.
  30. A. Pinkus. n𝑛nitalic_n-widths in Approximation Theory, volume 7. Springer, Berlin, 1985.
  31. Lift & learn: Physics-informed machine learning for large-scale nonlinear dynamical systems. Physica D: Nonlinear Phenomena, 406:132401, 2020.
  32. Albany: Using agile components to develop a flexible, generic multiphysics analysis code. Int. J. Multiscale Comput. Engng., 14:415–438, 2016.
  33. Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference. Computer Methods in Applied Mechanics and Engineering, 404:115836, 2023.
  34. K. Schacke. On the Kronecker product. Master’s thesis, University of Waterloo.
  35. H. Sharma and B. Kramer. Preserving Lagrangian structure in data-driven reduced-order modeling of large-scale dynamical systems. Physica D: Nonlinear Phenomena, page 134128, 2024.
  36. Symplectic model reduction of Hamiltonian systems using data-driven quadratic manifolds. Computer Methods in Applied Mechanics and Engineering, 417:116402, 2023.
  37. Lagrangian operator inference enhanced with structure-preserving machine learning for nonintrusive model reduction of mechanical systems. Computer Methods in Applied Mechanics and Engineering, 423:116865, 2024.
  38. Hamiltonian operator inference: Physics-preserving learning of reduced-order models for canonical Hamiltonian systems. Physica D: Nonlinear Phenomena, 431:133122, 2022.
  39. L. Sirovich. Turbulence and the dynamics of coherent structures part I: coherent structures. Quarterly of Applied Mathematics, 45(3):561–571, 1987.
  40. K. Sockwell. Mass Conserving Hamiltonian-Structure Preserving Reduced Order Modeling for the Rotating Shallow Water Equations Discretized by a Mimetic Spatial Scheme. PhD thesis, Florida State University, 2019.
  41. Data-driven identification of quadratic representations for nonlinear Hamiltonian systems using weakly symplectic liftings. ArXiv pre-print, https://arxiv.org/abs/2308.01084, 2024.
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