Open Access
Volume 56, Number 2, March-April 2022
Page(s) 617 - 650
Published online 08 March 2022
  1. P. Benner, D. Kressner and V. Mehrmann, Skew-Hamiltonian and Hamiltonian eigenvalue problems: theory, algorithms and applications. Springer, Dordrecht (2005) 3–39. [Google Scholar]
  2. N. Cagniart, Y. Maday and B. Stamm, Model order reduction for problems with large convection effects. In: Contributions to partial differential equations and applications. Vol. 47 of Comput. Methods Appl. Sci. Springer, Cham (2019) 131–150. [CrossRef] [Google Scholar]
  3. J.G. Caputo, N.K. Efremidis and C. Hang, Fourier-mode dynamics for the nonlinear schrödinger equation in one-dimensional bounded domains. Phys. Rev. E 84 (2011) 036601. [CrossRef] [PubMed] [Google Scholar]
  4. K. Carlberg, Adaptive h-refinement for reduced-order models, Int. J. Numer. Methods Eng. 102 (2015) 1192–1210. [Google Scholar]
  5. E. Celledoni and B. Owren, A class of intrinsic schemes for orthogonal integration. SIAM J. Numer. Anal. 40 (2002) 2069–2084. [CrossRef] [MathSciNet] [Google Scholar]
  6. S. Chaturantabut and D.C. Sorensen, Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32 (2010) 2737–2764. [Google Scholar]
  7. M. Cheng, T.Y. Hou and Z. Zhang, A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations II: adaptivity and generalizations. J. Comput. Phys. 242 (2013) 753–776. [CrossRef] [MathSciNet] [Google Scholar]
  8. M. Couplet, P. Sagaut and C. Basdevant, Intermodal energy transfers in a proper orthogonal decomposition–Galerkin representation of a turbulent separated flow. J. Fluid Mech. 491 (2003) 275–284. [CrossRef] [MathSciNet] [Google Scholar]
  9. V. Ehrlacher, D. Lombardi, O. Mula and F.-X. Vialard, Nonlinear model reduction on metric spaces. application to one-dimensional conservative PDEs in Wasserstein spaces. ESAIM: M2AN 54 (2020) 2159–2197. [CrossRef] [EDP Sciences] [Google Scholar]
  10. F. Feppon and P.F.J. Lermusiaux, A geometric approach to dynamical model order reduction. SIAM J. Matrix Anal. Appl. 39 (2018) 510–538. [CrossRef] [MathSciNet] [Google Scholar]
  11. E. Frénod, F. Salvarani and E. Sonnendrücker, Long time simulation of a beam in a periodic focusing channel via a two-scale pic-method. Math. Models Methods Appl. Sci. 19 (2009) 175–197. [CrossRef] [MathSciNet] [Google Scholar]
  12. A. George and J.W. Liu, Computer Solution of Large Sparse Positive Definite Systems. Prentice Hall Professional Technical Reference (1981). [Google Scholar]
  13. L. Giraud and J. Langou, A robust criterion for the modified Gram-Schmidt algorithm with selective reorthogonalization. SIAM J. Sci. Comput. 25 (2003) 417–441. [CrossRef] [MathSciNet] [Google Scholar]
  14. W. Givens, Computation of plane unitary rotations transforming a general matrix to triangular form. J. Soc. Ind. Appl. Math. 6 (1958) 26–50. [CrossRef] [Google Scholar]
  15. M.A. Grepl and A.T. Patera, A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: M2AN 39 (2005) 157–181. [CrossRef] [EDP Sciences] [Google Scholar]
  16. T.H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. 20 (1919) 292–296. [CrossRef] [MathSciNet] [Google Scholar]
  17. E. Hairer, C. Lubich and G. Wanner, Structure-preserving algorithms for ordinary differential equations. In: Geometric Numerical Integration, 2nd edition. Vol. 31 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (2006). [Google Scholar]
  18. E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems, 2nd edition. Vol. 8 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1993). [Google Scholar]
  19. S.A. Hirstoaga, Design and performant implementation of numerical methods for multiscale problems in plasma physics. Habilitation à diriger des recherches, Université de Strasbourg, IRMA UMR 7501 (April , 2019). [Google Scholar]
  20. A. Iollo and D. Lombardi, Advection modes by optimal mass transfer, Phys. Rev. E 89 (2014). [CrossRef] [Google Scholar]
  21. O. Koch and C. Lubich, Dynamical low-rank approximation. SIAM J. Matrix Anal. Appl. 29 (2007) 434–454. [Google Scholar]
  22. K. Lee and K.T. Carlberg, Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders. J. Comput. Phys. 404 (2020). [Google Scholar]
  23. M. Meyer and H.G. Matthies, Efficient model reduction in non-linear dynamics using the karhunen-loève expansion and dual-weighted-residual methods. Comput. Mech. 31 (2003) 179–191. [CrossRef] [Google Scholar]
  24. E. Musharbash and F. Nobile, Symplectic dynamical low rank approximation of wave equations with random parameters. Technical Report 18.2017, EPFL, Switzerland (2017). [Google Scholar]
  25. M. Ohlberger and S. Rave, Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing. C. R. Math. Acad. Sci. Paris 351 (2013) 901–906. [CrossRef] [MathSciNet] [Google Scholar]
  26. C. Pagliantini, Dynamical reduced basis methods for Hamiltonian systems. Numer. Math. 148 (2021) 409–448. [CrossRef] [MathSciNet] [Google Scholar]
  27. C. Paige and C. Van Loan, A Schur decomposition for Hamiltonian matrices. Linear Algebra Appl. 41 (1981) 11–32. [CrossRef] [MathSciNet] [Google Scholar]
  28. B. Peherstorfer and K. Willcox, Online adaptive model reduction for nonlinear systems via low-rank updates. SIAM J. Sci. Comput. 37 (2015) A2123–A2150. [CrossRef] [Google Scholar]
  29. L. Peng and K. Mohseni, Symplectic model reduction of Hamiltonian systems. SIAM J. Sci. Comput. 38 (2016) A1–A27. [Google Scholar]
  30. A. Quarteroni, A. Manzoni and F. Negri, Reduced Basis Methods for Partial Differential Equations: An Introduction. Vol. 92 of Unitext. Springer, Cham (2016) La Matematica per il 3+2. [Google Scholar]
  31. J. Reiss, P. Schulze, J. Sesterhenn and V. Mehrmann, The shifted proper orthogonal decomposition: a mode decomposition for multiple transport phenomena. SIAM J. Sci. Comput. 40 (2018) A1322–A1344. [Google Scholar]
  32. D. Rim, B. Peherstorfer and K.T. Mandli, Manifold approximations via transported subspaces: model reduction for transport-dominated problems. Preprint arXiv:1912.13024 (2019). [Google Scholar]
  33. A. Salam, On theoretical and numerical aspects of symplectic Gram–Schmidt-like algorithms. Numer. Algorithms 39 (2005) 437–462. [CrossRef] [MathSciNet] [Google Scholar]
  34. T.P. Sapsis and P.F.J. Lermusiaux, Dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty. Phys. D: Nonlinear Phenomena 241 (2012) 60–76. [CrossRef] [MathSciNet] [Google Scholar]
  35. A. Spantini, Preconditioning techniques for stochastic partial differential equations. Ph.D. thesis, Massachusetts Institute of Technology (2013). [Google Scholar]
  36. R. Ştefănescu, A. Sandu and I.M. Navon, Comparison of pod reduced order strategies for the nonlinear 2d shallow water equations. Int. J. Numer. Methods Fluids 76 (2014) 497–521. [CrossRef] [Google Scholar]
  37. S. Sultana and Z. Rahman, Hamiltonian formulation for water wave equation. Open J. Fluid Dyn. 03 (2013) 75–81. [CrossRef] [Google Scholar]
  38. T. Taddei, A registration method for model order reduction: data compression and geometry reduction. SIAM J. Sci. Comput. 42 (2020) A997–A1027. [CrossRef] [Google Scholar]
  39. K. Urban and A. Patera, An improved error bound for reduced basis approximation of linear parabolic problems. Math. Comput. 83 (2014) 1599–1615. [Google Scholar]
  40. C.F. Van Loan, A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix. Linear Algebra Appl. 61 (1984) 233–251. [CrossRef] [MathSciNet] [Google Scholar]
  41. G. Welper, Interpolation of functions with parameter dependent jumps by transformed snapshots. SIAM J. Sci. Comput. 39 (2017) A1225–A1250. [Google Scholar]

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