Open Access
Issue
ESAIM: M2AN
Volume 59, Number 1, January-February 2025
Page(s) 101 - 135
DOI https://doi.org/10.1051/m2an/2024044
Published online 14 January 2025
  1. C. Audouze, F. De Vuyst and P.B. Nair, Nonintrusive reduced-order modeling of parametrized time-dependent partial differential equations. Numer. Methods Part. Differ. Equ. 29 (2013) 1587–1628. [CrossRef] [Google Scholar]
  2. E. Bader, M. K¨archer, M.A. Grepl and K. Veroy, Certified reduced basis methods for parametrized distributed elliptic optimal control problems with control constraints. SIAM J. Sci. Comput. 38 (2016) A3921–A3946. [CrossRef] [Google Scholar]
  3. M. Barrault, C. Nguyen, A. Patera and Y. Maday, An “empirical interpolation” method: application to efficient reduced-basis discretization of partial differential equations. C. R. Acad. Sci. Sér. I Math. 339-9 (2004) 667–672. [Google Scholar]
  4. P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova and P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal. 43 (2011) 1457–1472. [Google Scholar]
  5. E. Borgonovo and E. Plischke, Sensitivity analysis: a review of recent advances. Eur. J. Oper. Res. 248 (2016) 869–887. [CrossRef] [Google Scholar]
  6. D.G. Cacuci, C.F. Weber, E.M. Oblow and J.H. Marable, Sensitivity theory for general systems of nonlinear equations. Nucl. Sci. Eng. 75 (1980) 88–110. [CrossRef] [Google Scholar]
  7. F. Casenave, A. Ern and T. Lelièvre, A nonintrusive reduced basis method applied to aeroacoustic simulations. Adv. Comput. Math. 41 (2014) 961–986. [Google Scholar]
  8. R. Chakir and Y. Maday, A two-grid finite-element/reduced basis scheme for the approximation of the solution of parametric dependent PDE, in 9e Colloque national en calcul des structures (2009). [Google Scholar]
  9. R. Chakir, B. Streichenberger and P. Chatellier, A non-intrusive reduced basis method for urban flows simulation, in ECCOMAS Congress 2020, 14th World Congress on Computational Mechanics (WCCM) (2021). [Google Scholar]
  10. G. Chavent, Identification of functional parameters in partial differential equations, in Joint Automatic Control Conference (1974) 155–156. [Google Scholar]
  11. L. Dede, Reduced basis method and a posteriori error estimation for parametrized linear-quadratic optimal control problems. SIAM J. Sci. Comput. 32 (2010) 997–1019. [CrossRef] [MathSciNet] [Google Scholar]
  12. L. Dede, Reduced basis method and error estimation for parametrized optimal control problems with control constraints. J. Sci. Comput. 50 (2012) 287–305. [CrossRef] [MathSciNet] [Google Scholar]
  13. Z. Ding, J. Shi, Q. Gao, Q. Huang and W-H. Liao, Design sensitivity analysis for transient responses of viscoelastically damped systems using model order reduction techniques. Struct. Multi. Optim. 64 (2021) 1501–1526. [CrossRef] [Google Scholar]
  14. D. Duvenaud, Automatic model construction with Gaussian processes. Ph.D. Thesis, University of Cambridge (2014). [Google Scholar]
  15. T.D. Economon, F. Palacios, S.R. Copeland, T.W. Lukaczyk and J.J. Alonso, Su2: an open-source suite for multi-physics simulation and design. AIAA J. 54 (2016) 828–846. [CrossRef] [Google Scholar]
  16. L.C. Evans, Partial Differential Equations. American Mathematical Society, Providence, RI (2010). [Google Scholar]
  17. R. Geelen, S. Wright and K. Willcox, Operator inference for non-intrusive model reduction with quadratic manifolds. Comput. Methods Appl. Mech. Eng. 403 (2023) 115717. [CrossRef] [Google Scholar]
  18. D. Givoli, A tutorial on the adjoint method for inverse problems. Comput. Methods Appl. Mech. Eng. 380 (2021) 113810. [CrossRef] [Google Scholar]
  19. E. Grosjean, Variations and further developments on the Non-Intrusive Reduced Basis two-grid method. Ph.D. Thesis, Sorbonne Universitè (2022). [Google Scholar]
  20. E. Grosjean and Y. Maday, Error estimate of the non-intrusive reduced basis method with finite volume schemes. ESAIM: Math. Model. Numer. Anal. 55 (2021) 1941–1961. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  21. E. Grosjean and Y. Maday, A doubly reduced approximation for the solution to PDE’s based on a domain truncation and a reduced basis method: application to Navier–Stokes equations. Preprint hal-03588508 (2022). [Google Scholar]
  22. E. Grosjean and Y. Maday, Error estimate of the Non-Intrusive Reduced Basis (NIRB) two-grid method with parabolic equations. SMAI J. Comput. Math. 9 (2023) 227–256. [CrossRef] [MathSciNet] [Google Scholar]
  23. E. Grosjean and B. Simeon, Python code for “The non-intrusive reduced basis two-grid method applied to sensitivity analysis” (2023). https://github.com/grosjean1/SensitivityAnalysisWithNIRBTwoGridMethod. [Google Scholar]
  24. M. Guo and J.S. Hesthaven, Reduced order modeling for nonlinear structural analysis using Gaussian process regression. Comput. Methods Appl. Mech. Eng. 341 (2018) 807–826. [CrossRef] [Google Scholar]
  25. M. Guo and J.S. Hesthaven, Data-driven reduced order modeling for time-dependent problems. Comput. Methods Appl. Mech. Eng. 345 (2019) 75–99. [CrossRef] [Google Scholar]
  26. B. Haasdonk, Convergence rates of the POD–greedy method. ESAIM: Math. Model. Numer. Anal. 47 (2013) 859–873. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  27. B. Haasdonk and M. Ohlberger, Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM: Math. Model. Numer. Anal. 42 (2008) 277–302. [CrossRef] [EDP Sciences] [Google Scholar]
  28. B. Haasdonk, H. Kleikamp, M. Ohlberger, F. Schindler and T. Wenzel, A new certified hierarchical and adaptive RB-ML-ROM surrogate model for parametrized PDEs. SIAM J. Sci. Comput. 45 (2023) A1039–A1065. [CrossRef] [Google Scholar]
  29. A. Hay, J. Borggaard and D. Pelletier, On the use of sensitivity analysis to improve reduced-order models, in 4th Flow Control Conference (2008) 4192. [Google Scholar]
  30. A. Hay, J.T. Borggaard and D. Pelletier, Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition. J. Fluid Mech. 629 (2009) 41–72. [CrossRef] [MathSciNet] [Google Scholar]
  31. F. Hecht, New development in FreeFem++. J. Numer. Math. 20 (2012) 251–266. [Google Scholar]
  32. J.S. Hesthaven and S. Ubbiali, Non-intrusive reduced order modeling of nonlinear problems using neural networks. J. Comput. Phys. 363 (2018) 55–78. [CrossRef] [MathSciNet] [Google Scholar]
  33. K. Ito and S.S. Ravindran, A reduced basis method for control problems governed by PDEs, in Control and Estimation of Distributed Parameter Systems: International Conference in Vorau, Austria. Springer (1998) 153–168. [CrossRef] [Google Scholar]
  34. A. Janon, M. Nodet and C. Prieur, Goal-oriented error estimation for the reduced basis method, with application to sensitivity analysis. J. Sci. Comput. 68 (2016) 21–41. [Google Scholar]
  35. M. K¨archer, Z. Tokoutsi, M.A. Grepl and K. Veroy, Certified reduced basis methods for parametrized elliptic optimal control problems with distributed controls. J. Sci. Comput. 75 (2018) 276–307. [CrossRef] [MathSciNet] [Google Scholar]
  36. U. Kirsch, Reduced basis approximations of structural displacements for optimal design. Am. Inst. Aeronaut. Astron. J. 29 (1991) 1751–1758. [CrossRef] [Google Scholar]
  37. U. Kirsch, Efficient sensitivity analysis for structural optimization. Comput. Methods Appl. Mech. Eng. 117 (1994) 143–156. [CrossRef] [Google Scholar]
  38. D.J. Knezevic and A.T. Patera, A certified reduced basis method for the Fokker–Planck equation of dilute polymeric fluids: FENE dumbbells in extensional flow. SIAM J. Sci. Comput. 32 (2010) 793–817. [CrossRef] [MathSciNet] [Google Scholar]
  39. A. Kolmogoroff, Über die beste Ann¨aherung von Funktionen einer gegebenen Funktionenklasse. Ann. Math. 37 (1936) 107–110. [CrossRef] [MathSciNet] [Google Scholar]
  40. O. Kramer, Scikit-Learn, in Machine Learning for Evolution Strategies. Springer (2016) 45–53. [Google Scholar]
  41. D. Kumar, M. Raisee and C. Lacor, An efficient non-intrusive reduced basis model for high dimensional stochastic problems in CFD. Comput. Fluids 138 (2016) 67–82. [CrossRef] [MathSciNet] [Google Scholar]
  42. J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes. II. Annales de l’Institut Fourier 11 (1961) 137–178. [CrossRef] [Google Scholar]
  43. F. Lu and H. Sun, Positive definite dot product kernels in learning theory. Adv. Comput. Math. 22 (2005) 181–198. [CrossRef] [MathSciNet] [Google Scholar]
  44. Y. Maday, Reduced basis method for the rapid and reliable solution of partial differential equations, in International Congress of Mathematicians, Madrid 2006. Volume III. Invited Lectures (2006) 1255–1270. [Google Scholar]
  45. Y. Maday and B. Stamm, Locally adaptive greedy approximations for anisotropic parameter reduced basis spaces. SIAM J. Sci. Comput. 35 (2013) A2417–A2441. [Google Scholar]
  46. R.C. Mittal and R. Jiwari, Numerical solution of two-dimensional reaction–diffusion Brusselator system. Appl. Math. Comput. 217 (2011) 5404–5415. [CrossRef] [MathSciNet] [Google Scholar]
  47. F. Negri, G. Rozza, A. Manzoni and A. Quarteroni, Reduced basis method for parametrized elliptic optimal control problems. SIAM J. Sci. Comput. 35 (2013) A2316–A2340. [CrossRef] [Google Scholar]
  48. N.C. Nguyen and J. Peraire, Gaussian functional regression for output prediction: model assimilation and experimental design. J. Comput. Phys. 309 (2016) 52–68. [CrossRef] [MathSciNet] [Google Scholar]
  49. A.K. Noor, J.A. Tanner and J.M. Peters, Reduced basis technique for evaluating sensitivity derivatives of the nonlinear response of the space shuttle orbiter nose-gear tire. Tire Sci. Technol. 21 (1993) 232–259. [CrossRef] [Google Scholar]
  50. J.S. Peterson, The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput. 10 (1989) 777–786. [CrossRef] [Google Scholar]
  51. I. Prigogine and G. Nicolis, Self-organisation in nonequilibrium systems: towards a dynamics of complexity. Bifurcation Anal. (1985) 3–12. [CrossRef] [Google Scholar]
  52. A. Quarteroni, G. Rozza and A. Quaini, Reduced basis methods for optimal control of advection-diffusion problems. J. Numer. Math. (2006) 1–24. [Google Scholar]
  53. A. Quarteroni, A. Manzoni and F. Negri, Reduced Basis Methods for Partial Differential Equations: An Introduction. Vol. 92. Springer (2015). [Google Scholar]
  54. S.S. Ravindran, A reduced-order approach for optimal control of fluids using proper orthogonal decomposition. Int. J. Numer. Methods Fluids 34 (2000) 425–448. [CrossRef] [Google Scholar]
  55. B. Streichenberger, Approches multi-fidélités pour la simulation rapide d’écoulements d’air en milieu urbain. Ph.D. Thesis, Université Gustave Eiffel (2021). [Google Scholar]
  56. J.F. Sykes and J.L. Wilson, Adjoint sensitivity theory for the finite element method, in Finite Elements in Water Resources. Springer (1984) 3–12. [Google Scholar]
  57. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics. Vol. 25. Springer Science & Business Media (2007). [Google Scholar]
  58. T. Tonn, K. Urban and S. Volkwein, Comparison of the reduced-basis and POD a posteriori error estimators for an elliptic linear-quadratic optimal control problem. Math. Comput. Model. Dyn. Syst. 17 (2011) 355–369. [CrossRef] [MathSciNet] [Google Scholar]
  59. B.C. Watson and A.K. Noor, Sensitivity analysis for large-deflection and postbuckling responses on distributed-memory computers. Comput. Methods Appl. Mech. Eng. 129 (1996) 393–409. [CrossRef] [Google Scholar]
  60. B. Wirthl, S. Brandstaeter, J. Nitzler, B.A. Schrefler and W.A. Wall, Global sensitivity analysis based on Gaussian-process metamodelling for complex biomechanical problems. Int. J. Numer. Methods Biomed. Eng. 39 (2022) e3675. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you