Open Access
Volume 56, Number 4, July-August 2022
Page(s) 1361 - 1400
Published online 27 June 2022
  1. E. Allgower and K. Georg, Introduction to Numerical Continuation Methods. Society for Industrial and Applied Mathematics (2003). [Google Scholar]
  2. A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis. Cambridge University Press, Cambridge (1993). [Google Scholar]
  3. E. Bader, M. Kärcher, M.A. Grepl and K. Veroy-Grepl, A certified reduced basis approach for parametrized linear-quadratic optimal control problems with control constraints. IFAC-PapersOnLine 48 (2015) 719–720. [CrossRef] [Google Scholar]
  4. E. Bader, M. Kärcher, 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]
  5. F. Ballarin, A. Manzoni, A. Quarteroni and G. Rozza, Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier-Stokes equations. Int. J. Numer. Methods Eng. 102 (2015) 1136–1161. [Google Scholar]
  6. M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An Empirical Interpolation Method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. 339 (2004) 667–672. [CrossRef] [MathSciNet] [Google Scholar]
  7. L. Bauer and E.L. Reiss, Nonlinear buckling of rectangular plates. J. Soc. Ind. Appl. Math. 13 (1965) 603–626. [CrossRef] [Google Scholar]
  8. M. Benzi and V. Simoncini, On the eigenvalues of a class of saddle point matrices. Numer. Math. 103 (2006) 173–196. [CrossRef] [MathSciNet] [Google Scholar]
  9. M. Benzi and A.J. Wathen, Some Preconditioning Techniques for Saddle Point Problems. Springer Berlin Heidelberg, Berlin, Heidelberg (2008) 195–211. [Google Scholar]
  10. M. Benzi, G.H. Golub and J. Liesen, Numerical solution of saddle point problems. Acta Numer. 14 (2005) 1–137. [CrossRef] [MathSciNet] [Google Scholar]
  11. M.S. Berger, On Von Kármán’s equations and the buckling of a thin elastic plate, I the clamped plate. Commun. Pure App. Math. 20 (1967) 687–719. [CrossRef] [Google Scholar]
  12. P.B. Bochev and M.D. Gunzburger, Least-Squares Finite Element Methods. Vol. 166. Springer-Verlag, New York (2009). [Google Scholar]
  13. J. Bramble, J. Pasciak and A. Vassilev, Uzawa type algorithms for nonsymmetric saddle point problems. Math. Comput. 69 (2000) 667–689. [Google Scholar]
  14. J. Burkardt, M. Gunzburger and H. Lee, POD and CVT-based reduced-order modeling of Navier-Stokes flows. Comput. Methods Appl. Mech. Eng. 196 (2006) 337–355. [CrossRef] [Google Scholar]
  15. G. Caloz and J. Rappaz, Numerical analysis for nonlinear and bifurcation problems. Handb. Numer. Anal. 5 (1997) 487–637. [Google Scholar]
  16. D. Chapelle, A. Gariah, P. Moireau and J. Sainte-Marie, A Galerkin strategy with proper orthogonal decomposition for parameter-dependent problems: Analysis, assessments and applications to parameter estimation. ESAIM: M2AN 47 (2013) 1821–1843. [CrossRef] [EDP Sciences] [Google Scholar]
  17. E. Charalampidis, P. Kevrekidis and P. Farrell, Computing stationary solutions of the two-dimensional Gross-Pitaevskii equation with deflated continuation. Commun. Nonlinear Sci. Numer. Simul. 54 (2018) 482–499. [CrossRef] [MathSciNet] [Google Scholar]
  18. P. Ciarlet, Linear and Nonlinear Functional Analysis with Applications. Other Titles in Applied Mathematics. Society for Industrial and Applied Mathematics (2013). [Google Scholar]
  19. L. Dedè, 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]
  20. A. Fursikov, M. Gunzburger and L. Hou, Boundary value problems and optimal boundary control for the Navier-Stokes system: the two-dimensional case. SIAM J. Control Optim. 36 (1998) 852–894. [CrossRef] [MathSciNet] [Google Scholar]
  21. A.L. Gerner and K. Veroy, Certified reduced basis methods for parametrized saddle point problems. SIAM J. Sci. Comput. 34 (2012) A2812–A2836. [CrossRef] [Google Scholar]
  22. M. Gunzburger, Adjoint equation-based methods for control problems in incompressible, viscous flows. Flow Turbul. Combust. 65 (2000) 249–272. [CrossRef] [MathSciNet] [Google Scholar]
  23. M.D. Gunzburger, Perspectives in Flow Control and Optimization. Vol. 5. SIAM, Philadelphia (2003). [Google Scholar]
  24. M. Hess, A. Alla, A. Quaini, G. Rozza and M. Gunzburger, A localized reduced-order modeling approach for PDEs with bifurcating solutions. Comput. Methods Appl. Mech. Eng. 351 (2019) 379–403. [CrossRef] [Google Scholar]
  25. M. Hess, A. Quaini and G. Rozza, Reduced basis model order reduction for Navier-Stokes equations in domains with walls of varying curvature. Int. J. Comput. Fluid Dyn. 34 (2020) 119–126. [CrossRef] [MathSciNet] [Google Scholar]
  26. J.S. Hesthaven, G. Rozza and B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Springer Briefs in Mathematics. Springer, Milano (2015). [Google Scholar]
  27. M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints. Vol. 23. Springer Science & Business Media, Antwerp (2008). [Google Scholar]
  28. M. Kärcher and M.A. Grepl, A certified reduced basis method for parametrized elliptic optimal control problems. ESAIM: COCV 20 (2014) 416–441. [CrossRef] [EDP Sciences] [Google Scholar]
  29. M. Kärcher, 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. [Google Scholar]
  30. P. Kevrekidis, D. Frantzeskakis and R. Carretero-González, The Defocusing Nonlinear Schrödinger Equation. Society for Industrial and Applied Mathematics, Philadelphia, PA (2015). [CrossRef] [Google Scholar]
  31. M. Khamlich, F. Pichi and G. Rozza, Model order reduction for bifurcating phenomena in fluid-structure interaction problems. Preprint arXiv:2110.06297 (2021). [Google Scholar]
  32. H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs. Applied Mathematical Sciences. Springer, New York (2006). [Google Scholar]
  33. Y. Kuznetsov, Elements of Applied Bifurcation Theory. Applied Mathematical Sciences. Springer, New York (2004). [CrossRef] [Google Scholar]
  34. G. Leugering, P. Benner, S. Engell, A. Griewank, H. Harbrecht, M. Hinze, R. Rannacher and S. Ulbrich, Trends in PDE Constrained Optimization. Springer, New York (2014). [Google Scholar]
  35. J.L. Lions, Optimal Control of System Governed by Partial Differential Equations. Vol. 170. Springer-Verlagr, Berlin and Heidelberg (1971). [CrossRef] [Google Scholar]
  36. A. Logg, K. Mardal and G. Wells, Automated Solution of Differential Equations by the Finite Element Method. Springer-Verlag, Berlin (2012). [CrossRef] [Google Scholar]
  37. S. Middelkamp, P. Kevrekidis, D. Frantzeskakis, R. Carretero-González and P. Schmelcher, Emergence and stability of vortex clusters in Bose-Einstein condensates: a bifurcation approach near the linear limit. Phys. D: Nonlinear Phenom. 240 (2011) 1449–1459. [CrossRef] [Google Scholar]
  38. H.K. Moffatt, Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18 (1964) 1–18. [CrossRef] [Google Scholar]
  39. 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]
  40. F. Negri, A. Manzoni and G. Rozza, Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations. Comput. Math. App. 69 (2015) 319–336. [Google Scholar]
  41. F. Pichi and G. Rozza, Reduced basis approaches for parametrized bifurcation problems held by non-linear Von Kármán equations. J. Sci. Comput. 81 (2019) 112–135. [CrossRef] [MathSciNet] [Google Scholar]
  42. F. Pichi, A. Quaini and G. Rozza, A reduced order modeling technique to study bifurcating phenomena: application to the Gross-Pitaevskii equation. SIAM J. Sci. Comput. 42 (2020) B1115–B1135. [CrossRef] [Google Scholar]
  43. F. Pichi, F. Ballarin, G. Rozza and J.S. Hesthaven, An artificial neural network approach to bifurcating phenomena in computational fluid dynamics. Preprint arXiv:2109.10765 (2021). [Google Scholar]
  44. M. Pintore, F. Pichi, M. Hess, G. Rozza and C. Canuto, Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method. Adv. Comput. Math. 47 (2021) 1–39. [CrossRef] [Google Scholar]
  45. G. Pitton and G. Rozza, On the application of reduced basis methods to bifurcation problems in incompressible fluid dynamics. J. Sci. Comput. 73 (2017) 157–177. [CrossRef] [MathSciNet] [Google Scholar]
  46. G. Pitton, A. Quaini and G. Rozza, Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: applications to Coanda effect in cardiology. J. Comput. Phys. 344 (2017) 534–557. [CrossRef] [MathSciNet] [Google Scholar]
  47. A. Quaini, R. Glowinski and S. Canic, Symmetry breaking and preliminary results about a Hopf bifurcation for incompressible viscous flow in an expansion channel. Int. J. Comput. Fluid Dyn. 30 (2016) 7–19. [CrossRef] [MathSciNet] [Google Scholar]
  48. A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Vol. 23. Springer Science & Business Media, Berlin and Heidelberg (2008). [Google Scholar]
  49. A. Quarteroni, G. Rozza and A. Quaini, Reduced basis methods for optimal control of advection-diffusion problems. In: Advances in Numerical Mathematics. CMCS-CONF-2006-003. RAS and University of Houston (2007) 193–216. [Google Scholar]
  50. RBniCS – reduced order modelling in FEniCS. (2015). [Google Scholar]
  51. G. Rozza and K. Veroy, On the stability of the reduced basis method for Stokes equations in parametrized domains. Comput. Methods Appl. Mech. Eng. 196 (2007) 1244–1260. [CrossRef] [Google Scholar]
  52. R. Seydel, Practical Bifurcation and Stability Analysis. Interdisciplinary Applied Mathematics. Springer, New York (2009). [Google Scholar]
  53. M. Stoll and A. Wathen, All-at-once solution of time-dependent Stokes control. J. Comput. Phys. 232 (2013) 498–515. [MathSciNet] [Google Scholar]
  54. M. Strazzullo, F. Ballarin, R. Mosetti and G. Rozza, Model reduction for parametrized optimal control problems in environmental marine sciences and engineering. SIAM J. Sci. Comput. 40 (2018) B1055–B1079. [CrossRef] [Google Scholar]
  55. M. Strazzullo, F. Ballarin and G. Rozza, POD-Galerkin model order reduction for parametrized nonlinear time dependent optimal flow control: an application to Shallow Water Equations. J. Numer. Math. 30 (2021) 63–84. [Google Scholar]
  56. M. Strazzullo, Z. Zainib, F. Ballarin and G. Rozza, Reduced order methods for parametrized nonlinear and time dependent optimal flow control problems: towards applications in biomedical and environmental sciences. In: Numerical Mathematics and Advanced Applications ENUMATH 2019 (2021). [Google Scholar]
  57. D.J. Tritton, Physical Fluid Dynamics. Springer Science & Business Media (2012). [Google Scholar]
  58. F. Tröltzsch, Optimal Control of Partial Differential Equations. Graduate Studies in mathematics. Vol. 112. American Mathimatical Society, Verlag, Wiesbad (2010). [Google Scholar]
  59. T. Von Kármán, Festigkeitsprobleme im Maschinenbau. Encyclopädie der Mathematischen Wissenschaften. Vol. 4 (1910). [Google Scholar]
  60. Z. Zainib, F. Ballarin, S. Fremes, P. Triverio, L. Jiménez-Juan and G. Rozza, Reduced order methods for parametric optimal flow control in coronary bypass grafts, toward patient-specific data assimilation. Int. J. Numer. Methods Biomed. Eng. 37 (2021) e3367. [CrossRef] [PubMed] [Google Scholar]

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