A localized reduced-order modeling approach for PDEs with bifurcating solutions

Abstract Reduced-order modeling (ROM) commonly refers to the construction, based on a few solutions (referred to as snapshots) of an expensive discretized partial differential equation (PDE), and the subsequent application of low-dimensional discretizations of partial differential equations (PDEs) that can be used to more efficiently treat problems in control and optimization, uncertainty quantification, and other settings that require multiple approximate PDE solutions. Although ROMs have been successfully used in many settings, ROMs built specifically for the efficient treatment of PDEs having solutions that bifurcate as the values of input parameters change have not received much attention. In such cases, the parameter domain can be subdivided into subregions, each of which corresponds to a different branch of solutions. Popular ROM approaches such as proper orthogonal decomposition (POD), results in a global low-dimensional basis that does not respect the often large differences in the PDE solutions corresponding to different subregions. In this work, we develop and test a new ROM approach specifically aimed at bifurcation problems. In the new method, the k-means algorithm is used to cluster snapshots so that within cluster snapshots are similar to each other and are dissimilar to those in other clusters. This is followed by the construction of local POD bases, one for each cluster. The method also can detect which cluster a new parameter point belongs to, after which the local basis corresponding to that cluster is used to determine a ROM approximation. Numerical experiments show the effectiveness of the method both for problems for which bifurcation cause continuous and discontinuous changes in the solution of the PDE.

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

[2]  A. Ambrosetti,et al.  A primer of nonlinear analysis , 1993 .

[3]  D. Drikakis.  Bifurcation phenomena in incompressible sudden expansion flows , 1997 .

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

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

[6]  Paul Houston,et al.  Adaptivity and a posteriori error control for bifurcation problems I: the Bratu problem , 2010 .

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

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

[9]  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..

[10]  Mario Ohlberger,et al.  True Error Control for the Localized Reduced Basis Method for Parabolic Problems , 2016, 1606.09216.

[11]  Mario Ohlberger,et al.  Localized Orthogonal Decomposition for two-scale Helmholtz-typeproblems , 2016, 1605.03410.

[12]  P. Drazin,et al.  Bifurcations of two-dimensional channel flows , 1986, Journal of Fluid Mechanics.

[13]  Samuele Rubino,et al.  Assessment of self-adapting local projection-based solvers for laminar and turbulent industrial flows , 2018, Journal of Mathematics in Industry.

[14]  Jens L. Eftang,et al.  An hp certified reduced basis method for parametrized parabolic partial differential equations , 2011 .

[15]  Susanne Ebersbach,et al.  Numerical Simulation Of Oscillatory Convection In Low Pr Fluids , 2016 .

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

[17]  David Amsallem,et al.  PEBL-ROM: Projection-error based local reduced-order models , 2016, Advanced Modeling and Simulation in Engineering Sciences.

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

[19]  Benjamin Peherstorfer,et al.  Localized Discrete Empirical Interpolation Method , 2014, SIAM J. Sci. Comput..

[20]  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.

[21]  K. Goda,et al.  A multistep technique with implicit difference schemes for calculating two- or three-dimensional cavity flows , 1979 .

[22]  Alfio Quarteroni,et al.  Numerical approximation of parametrized problems in cardiac electrophysiology by a local reduced basis method , 2018, Computer Methods in Applied Mechanics and Engineering.

[23]  David Amsallem,et al.  Fast local reduced basis updates for the efficient reduction of nonlinear systems with hyper-reduction , 2015, Adv. Comput. Math..

[24]  José M. Vega,et al.  Reduced order models based on local POD plus Galerkin projection , 2010, J. Comput. Phys..

[25]  G. Karniadakis,et al.  Spectral/hp Element Methods for CFD , 1999 .

[26]  Krishnamurthy Jayaraman,et al.  Asymmetric flows in planar symmetric channels with large expansion ratio , 2002 .

[27]  Steven L. Brunton,et al.  On dynamic mode decomposition: Theory and applications , 2013, 1312.0041.

[28]  Paul Houston,et al.  Adaptivity and a Posteriori Error Control for Bifurcation Problems III: Incompressible Fluid Flow in Open Systems with O(2) Symmetry , 2012, J. Sci. Comput..

[29]  Roger Pierre,et al.  Localization of Hopf bifurcations in fluid flow problems , 1997 .

[30]  Bernd R. Noack,et al.  Cluster-based reduced-order modelling of a mixing layer , 2013, Journal of Fluid Mechanics.

[31]  T. A. Zang,et al.  Spectral Methods: Fundamentals in Single Domains , 2010 .

[32]  Paul Houston,et al.  Adaptivity and a Posteriori Error Control for Bifurcation Problems II: Incompressible Fluid Flow in Open Systems with Z2 Symmetry , 2011, J. Sci. Comput..

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

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

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

[36]  Annalisa Quaini,et al.  Symmetry breaking and preliminary results about a Hopf bifurcation for incompressible viscous flow in an expansion channel , 2016 .

[37]  Anthony T. Patera,et al.  An "hp" Certified Reduced Basis Method for Parametrized Elliptic Partial Differential Equations , 2010, SIAM J. Sci. Comput..

[38]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[39]  Ahmed K. Noor,et al.  Multiple‐parameter reduced basis technique for bifurcation and post‐buckling analyses of composite plates , 1983 .

[40]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[41]  T. Mullin,et al.  Nonlinear flow phenomena in a symmetric sudden expansion , 1990, Journal of Fluid Mechanics.

[42]  Charbel Farhat,et al.  Nonlinear model order reduction based on local reduced‐order bases , 2012 .

[43]  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..

[44]  Zvi Rusak,et al.  The dynamics of a laminar flow in a symmetric channel with a sudden expansion , 2001, Journal of Fluid Mechanics.

[45]  Claudio Canuto,et al.  Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics (Scientific Computation) , 2007 .

[46]  Marco Fossati,et al.  Local Reduced-Order Modeling and Iterative Sampling for Parametric Analyses of Aero-Icing Problems , 2015 .

[47]  Sinisa Krajnovic,et al.  Cluster-based reduced-order modelling of the flow in the wake of a high speed train , 2015 .