zbMATH — the first resource for mathematics

A localized reduced-order modeling approach for PDEs with bifurcating solutions. (English) Zbl 1441.65082
Summary: 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.

65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
35B32 Bifurcations in context of PDEs
35Q30 Navier-Stokes equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65P30 Numerical bifurcation problems
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI
[1] Noor, A., On making large nonlinear problems small, Comput. Methods Appl. Mech. Engrg., 34, 1, 955-985 (1982) · Zbl 0478.65031
[2] Noor, A., Recent advances and applications of reduction methods, ASME. Appl. Mech. Rev., 5, 47, 125-146 (1994)
[3] Noor, A.; Peters, J., Multiple-parameter reduced basis technique for bifurcation and post-buckling analyses of composite materiale, Int. J. Numer. Methods Eng., 19, 1783-1803 (1983) · Zbl 0557.73070
[4] Noor, A.; Peters, J., Recent advances in reduction methods for instability analysis of structures, Comput. Struct., 16, 1, 67-80 (1983) · Zbl 0498.73094
[5] Terragni, F.; Vega, J., On the use of POD-based ROMs to analyze bifurcations in some dissipative systems, Physica D, 241, 17, 1393-1405 (2012) · Zbl 1251.65169
[6] Herrero, H.; Maday, Y.; Pla, F., RB (Reduced Basis) for RB (Rayleigh-Bénard), Comput. Methods Appl. Mech. Engrg., 261-262, 132-141 (2013) · Zbl 1286.76084
[7] Pla, F.; Herrero, H.; Vega, J., A flexible symmetry-preserving Galerkin/POD reduced order model applied to a convective instability problem, Comput. & Fluids, 119, 162-175 (2015) · Zbl 1390.76348
[8] Pitton, G.; Rozza, G., On the application of reduced basis methods to bifurcation problems in incompressible fluid dynamics, J. Sci. Comput., 73, 1, 157-177 (2017) · Zbl 1433.76122
[9] Pitton, G.; Quaini, A.; Rozza, G., Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: Applications to coanda effect in cardiology, J. Comput. Phys., 344, 534-557 (2017)
[10] Quaini, A.; Glowinski, R.; Canic, S., Symmetry breaking and preliminary results about a Hopf bifurcation for incompressible viscous flow in an expansion channel, Int. J. Comput. Fluid Dyn., 30, 1, 7-19 (2016)
[11] Cliffe, K.; Hall, E.; Houston, P.; Phipps, E.; Salinger, A., Adaptivity and a posteriori error control for bifurcation problems I: the Bratu problem, Commun. Comput. Phys., 8, 4, 845-865 (2010) · Zbl 1364.65246
[12] Cliffe, K.; Hall, E.; Houston, P.; Phipps, E.; Salinger, A., Adaptivity and a posteriori error control for bifurcation problems III: Incompressible fluid flow in open systems with Z(2) symmetry, J. Sci. Comput., 47, 3, 389-418 (2011) · Zbl 1311.76052
[13] Cliffe, K.; Hall, E.; Houston, P.; Phipps, E.; Salinger, A., Adaptivity and a posteriori error control for bifurcation problems III: Incompressible fluid flow in open systems with O(2) symmetry, J. Sci. Comput., 52, 1, 153-179 (2012) · Zbl 1311.76053
[14] Pichi, F.; Rozza, G., Reduced basis approaches for parametrized bifurcation problems held by non-linear Von Kármán equations (2018), arXiv:1804.02014 · Zbl 1427.35275
[15] Brunton, S. L.; Tu, J. H.; Bright, I.; Kutz, J. N., Compressive sensing and low-rank libraries for classification of bifurcation regimes in nonlinear dynamical systems, SIAM J. Appl. Dyn. Syst., 13, 4, 1716-1732 (2014) · Zbl 1354.37078
[16] Kramer, B.; Grover, P.; Boufounos, P.; Nabi, S.; Benosman, M., Sparse sensing and DMD-based identification of flow regimes and bifurcations in complex flows, SIAM J. Appl. Dyn. Syst., 16, 2, 1164-1196 (2017) · Zbl 1373.37185
[17] Rapun, M. L.; Vega, J., Reduced order models based on local POD plus Galerkin projection, J. Comput. Phys., 229, 3046-3063 (2010) · Zbl 1187.65111
[18] Amsallem, D.; Haasdonk, B., PEBL-ROM: Projection-error based local reduced-order models, Adv. Model. Simul. Eng. Sci., 3:6 (2016), 1-1
[19] Amsallem, D.; Zahr, M.; Farhat, C., Nonlinear model order reduction based on local reduced-order bases, Internat. J. Numer. Methods Engrg., 92, 891-916 (2012) · Zbl 1352.65212
[20] Amsallem, D.; Zahr, M.; Washabaugh, K., Fast local reduced basis updates for the efficient reduction of nonlinear systems with hyper-reduction, Adv. Comput. Math. Eng. Sci., 41, 1187-1230 (2015) · Zbl 1331.65094
[21] Kaiser, E.; Noack, B.; Cordier, L.; Spohn, A.; Segond, M.; Abel., M.; Daviller, G.; Östh, J.; Krajnovi’c, S., Cluster-based reduced-order modelling of a mixing layer, J. Fluid Mech., 754, 365-414 (2014) · Zbl 1329.76177
[22] Östh, J.; Kaiser, E.; Krajnovi’c, S.; Noack, B., Cluster-based reduced-order modelling of the flow in the wake of a high speed train, J. Wind Eng. Ind. Aerodyn., 145, 327-338 (2015)
[23] Peherstorfer, B.; Butnaru, D.; Willcox, K.; Bungartz, H.-J., Reduced order models based on local POD plus Galerkin projection, SIAM J. Sci. Comput., 36, A168-A192 (2014)
[24] Zhan, Z.; Habashi, W.; Fossati, M., Local reduced-order modeling and iterative sampling for parametric analyses of aero-icing problems, AIAA J., 53, 2174-2185 (2015)
[25] Pagani, S.; Manzoni, A.; Quarteroni, A., Numerical approximation of parametrized problems in cardiac electrophysiology by a local reduced basis method (2017), Lausanne
[26] Ohlberger, M.; Rave, S.; Schindler, F., True error control for the localized reduced basis method for parabolic problems, (Benner, P.; Ohlberger, M.; Patera, A.; Rozza, G.; Urban, K., Model Reduction of Parametrized Systems. MS&A (Modeling, Simulation and Applications), vol. 17 (2017), Springer: Springer Cham) · Zbl 1448.65170
[27] Ohlberger, M.; Verfurth, B., Localized orthogonal decomposition for two-scale Helmholtz-type problems, AIMS Math., 2, Math-02-00458, 458-478 (2017) · Zbl 1427.65374
[28] Ohlberger, M.; Schindler, F., Non-conforming localized model reduction with online enrichment: Towards optimal complexity in PDE constrained optimization, AIMS Math., 2, 458-478 (2017) · Zbl 1365.65174
[29] Eftang, J.; Patera, A.; Ronquist, E., An hp certified reduced basis method for parametrized parabolic partial differential equations, (Mathematical and Computer Modelling of Dynamical Systems, vol. 17 (2011), Taylor & Francis), 179-187 · Zbl 1216.65128
[30] Chacón Rebollo, T.; Delgado Ávila, E.; Gómez Mármol, M.; Rubino, S., Assessment of self-adapting local projection-based solvers for laminar and turbulent industrial flows, J. Math. Ind., 8, 1 (2018) · Zbl 1419.76436
[31] Tu, J. H.; Rowley, C. W.; Luchtenburg, D. M.; Brunton, S. L.; Kutz, J. N., On dynamic mode decomposition: Theory and applications, J. Comput. Dyn., 1, 391-421 (2014) · Zbl 1346.37064
[32] Gunzburger, M., (Finite element method for viscous incompressible flows: a guide to theory, practice and algorithms. Finite element method for viscous incompressible flows: a guide to theory, practice and algorithms, Computer Science and Scientific Computing (1989), Academic Press: Academic Press San Diego)
[33] Quarteroni, A.; Valli, A., Numerical Approximation of Partial Differential Equations (1994), Springer-Verlag · Zbl 0803.65088
[34] Brenner, S.; Scott, L., The Mathematical Theory of Finite Element Methods (1994), Springer-Verlag · Zbl 0804.65101
[35] Karniadakis, G.; Sherwin, S., (Spectral/hp Element Methods for CFD. Spectral/hp Element Methods for CFD, Numerical mathematics and scientific computation (2005), Oxford University Press)
[36] Canuto, C.; Hussaini, M.; Quarteroni, A.; Zhang, T., (Spectral Methods Fundamentals in Single Domains. Spectral Methods Fundamentals in Single Domains, Scientific Computation (2006), Springer) · Zbl 1093.76002
[37] Canuto, C.; Hussaini, M.; Quarteroni, A.; Zhang, T., (Spectral Methods Evolution to Complex Geometries and Applications to Fluid Dynamics. Spectral Methods Evolution to Complex Geometries and Applications to Fluid Dynamics, Scientific Computation (2007), Springer) · Zbl 1121.76001
[38] Hesthaven, J.; Rozza, G.; Stamm, B., Certified Reduced Basis Methods for Parametrized Partial Differential Equations (2015), Springer Briefs in Mathematics
[39] Quarteroni, A.; Manzoni, A.; Negri, F., (Reduced Basis Methods for Partial Differential Equations. Reduced Basis Methods for Partial Differential Equations, UNITEXT, 92 (2016), Springer) · Zbl 1337.65113
[40] Volkwein, S., Proper orthogonal decomposition: Theory and reduced-order modelling, (Lecture Notes (2013), University of Konstanz, Department of Mathematics and Statistics)
[41] Fearn, R.; Mullin, T.; Cliffe, K., Nonlinear flow phenomena in a symmetric sudden expansion, J. Fluid Mech., 211, 595-608 (1990)
[42] Drikakis, D., Bifurcation phenomena in incompressible sudden expansion flows, Phys. Fluids, 9, 1, 76-87 (1997)
[43] Hawa, T.; Rusak, Z., The dynamics of a laminar flow in a symmetric channel with a sudden expansion, J. Fluid Mech., 436, 283-320 (2001) · Zbl 1003.76029
[44] Mishra, S.; Jayaraman, K., Asymmetric flows in planar symmetric channels with large expansion ratios, Internat. J. Numer. Methods Fluids, 38, 945-962 (2002) · Zbl 1037.76019
[45] Fortin, A.; Jardak, M.; Gervais, J.; Pierre, R., Localization of Hopf bifurcations in fluid flow problems, Internat. J. Numer. Methods Fluids, 24, 1185-1210 (1997) · Zbl 0886.76042
[46] Sobey, I.; Drazin, P., Bifurcations of two-dimensional channel flows, J. Fluid Mech., 171, 263-287 (1986) · Zbl 0609.76050
[47] Ambrosetti, A.; Prodi, G., A Primer of Nonlinear Analysis (1993), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0781.47046
[48] Goda, K., A multistep technique with implicit difference schemes for calculating two- or three-dimensional cavity flows, J. Comput. Phys., 30, 76-95 (1979) · Zbl 0405.76017
[49] Yano, M.; Patera, A., A space-time variational approach to hydrodynamic stability theory, Proc. R. Soc. A, 469, 2155 (2013) · Zbl 1371.76054
[50] (Roux, B., Numerical Simulation of Oscillatory Convection in Low-Pr Fluids. Numerical Simulation of Oscillatory Convection in Low-Pr Fluids, Notes on Numerical Fluid Mechanics and Multidisciplinary Design, 27 (1990), Springer) · Zbl 0712.00022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.