A Reduced Order technique to study bifurcating phenomena: application to the Gross-Pitaevskii equation

We propose a computationally efficient framework to treat nonlinear partial differential equations having bifurcating solutions as one or more physical control parameters are varied. Our focus is on steady bifurcations. Plotting a bifurcation diagram entails computing multiple solutions of a parametrized, nonlinear problem, which can be extremely expensive in terms of computational time. In order to reduce these demanding computational costs, our approach combines a continuation technique and Newton's method with a Reduced Order Modeling (ROM) technique, suitably supplemented with a hyper-reduction method. To demonstrate the effectiveness of our ROM approach, we trace the steady solution branches of a nonlinear Schrodinger equation, called Gross--Pitaevskii equation, as one or two physical parameters are varied. In the two parameter study, we show that our approach is 60 times faster in constructing a bifurcation diagram than a standard Full Order Method.

[1]  J. Hesthaven,et al.  Certified Reduced Basis Methods for Parametrized Partial Differential Equations , 2015 .

[2]  Yvon Maday,et al.  RB (Reduced basis) for RB (Rayleigh–Bénard) , 2013 .

[3]  Petros Boufounos,et al.  Sparse Sensing and DMD-Based Identification of Flow Regimes and Bifurcations in Complex Flows , 2015, SIAM J. Appl. Dyn. Syst..

[4]  A. Quarteroni,et al.  Reduced Basis Methods for Partial Differential Equations: An Introduction , 2015 .

[5]  Yuri S. Kivshar,et al.  Nonlinear modes of a macroscopic quantum oscillator , 1999 .

[6]  Dmitry A. Zezyulin,et al.  Nonlinear modes for the Gross–Pitaevskii equation—a demonstrative computation approach , 2007 .

[7]  Gianluigi Rozza,et al.  Reduced Order Methods for Modeling and Computational Reduction , 2013 .

[8]  Annalisa Quaini,et al.  Reduced basis model order reduction for Navier–Stokes equations in domains with walls of varying curvature , 2019, International Journal of Computational Fluid Dynamics.

[9]  N. Nguyen,et al.  An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations , 2004 .

[10]  Anthony T. Patera,et al.  A space–time variational approach to hydrodynamic stability theory , 2013, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[11]  Anders Logg,et al.  Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book , 2012 .

[12]  Gianluigi Rozza,et al.  Reduced Basis Approaches for Parametrized Bifurcation Problems held by Non-linear Von Kármán Equations , 2018, Journal of Scientific Computing.

[13]  Panayotis G. Kevrekidis,et al.  The Defocusing Nonlinear Schrödinger Equation - From Dark Solitons to Vortices and Vortex Rings , 2015 .

[14]  Ahmed K. Noor,et al.  On making large nonlinear problems small , 1982 .

[15]  D. E. Pelinovsky,et al.  Bifurcations of Multi-Vortex Configurations in Rotating Bose–Einstein Condensates , 2017, 1701.01494.

[16]  P. G. Ciarlet,et al.  Linear and Nonlinear Functional Analysis with Applications , 2013 .

[17]  Ricardo Carretero-González,et al.  Bifurcations, stability, and dynamics of multiple matter-wave vortex states , 2010 .

[18]  Danny C. Sorensen,et al.  Nonlinear Model Reduction via Discrete Empirical Interpolation , 2010, SIAM J. Sci. Comput..

[19]  Andres Contreras,et al.  Global bifurcation of vortex and dipole solutions in Bose-Einstein condensates , 2015, 1511.06843.

[20]  Gianluigi Rozza,et al.  On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics , 2017, J. Sci. Comput..

[21]  Ahmed K. Noor,et al.  Recent advances in reduction methods for instability analysis of structures , 1983 .

[22]  P. E. Farrell,et al.  Computing stationary solutions of the two-dimensional Gross-Pitaevskii equation with deflated continuation , 2016, Commun. Nonlinear Sci. Numer. Simul..

[23]  Vladimir V. Konotop,et al.  Dynamic generation of matter solitons from linear states via time-dependent scattering lengths , 2005 .

[24]  José M. Vega,et al.  On the use of POD-based ROMs to analyze bifurcations in some dissipative systems , 2012 .

[25]  Steven L. Brunton,et al.  Compressive Sensing and Low-Rank Libraries for Classification of Bifurcation Regimes in Nonlinear Dynamical Systems , 2013, SIAM J. Appl. Dyn. Syst..

[26]  H. Herrero,et al.  A flexible symmetry-preserving Galerkin/POD reduced order model applied to a convective instability problem , 2015 .

[27]  P. G. Kevrekidis,et al.  Bifurcation analysis of stationary solutions of two-dimensional coupled Gross-Pitaevskii equations using deflated continuation , 2020, Commun. Nonlinear Sci. Numer. Simul..

[28]  Annalisa Quaini,et al.  Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: Applications to Coanda effect in cardiology , 2017, J. Comput. Phys..

[29]  E. Allgower,et al.  Introduction to Numerical Continuation Methods , 1987 .

[30]  D. J. Frantzeskakis,et al.  Emergence and stability of vortex clusters in Bose–Einstein condensates: A bifurcation approach near the linear limit , 2010, 1012.1840.

[31]  P. G. Kevrekidis,et al.  Excited states in the large density limit: a variational approach , 2009, 0910.5249.

[32]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[33]  Annalisa Quaini,et al.  A localized reduced-order modeling approach for PDEs with bifurcating solutions , 2018, Computer Methods in Applied Mechanics and Engineering.

[34]  J. Hesthaven,et al.  Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations , 2007 .

[35]  C. Johnson,et al.  The FEniCS project , 2003 .

[36]  Ahmed K. Noor,et al.  Recent Advances and Applications of Reduction Methods , 1994 .