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A greedy non-intrusive reduced order model for shallow water equations. (English) Zbl 07512322

Summary: In this work, we develop Non-Intrusive Reduced Order Models (NIROMs) that combine Proper Orthogonal Decomposition (POD) with a Radial Basis Function (RBF) interpolation method to construct efficient reduced order models for time-dependent problems arising in large scale environmental flow applications. The performance of the POD-RBF NIROM is compared with a traditional nonlinear POD (NPOD) model by evaluating the accuracy and robustness for test problems representative of riverine flows. Different greedy algorithms are studied in order to determine a near-optimal distribution of interpolation points for the RBF approximation. A new power-scaled residual greedy (psr-greedy) algorithm is proposed to address some of the primary drawbacks of the existing greedy approaches. The relative performances of these greedy algorithms are studied with numerical experiments using realistic two-dimensional (2D) shallow water flow applications involving coastal and riverine dynamics.

MSC:

41Axx Approximations and expansions
65Dxx Numerical approximation and computational geometry (primarily algorithms)
76Mxx Basic methods in fluid mechanics

Software:

Matlab; redbKIT
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Savant, G.; Berger, R.; McAlpin, T.; Tate, J., Efficient implicit finite-element hydrodynamic model for dam and levee breach, J. Hydraul. Eng., 137, 9, 1005-1018 (2011)
[2] Westerink, J. J.; Luettich, R. A.; Feyen, J. C.; Atkinson, J. H.; Dawson, C. N.; Roberts, H. J.; Powell, M. D.; Dunion, J. P.; Kubatko, E. J.; Pourtaheri, H., A basin- to channel-scale unstructured grid hurricane storm surge model applied to southern Louisiana, Mon. Weather Rev., 136, 833-864 (2008)
[3] Vreugdenhil, C., Numerical Methods for Shallow-Water Flow (1992), Kluwer Academic Publishers
[4] Quarteroni, A.; Manzoni, A.; Negri, F., Reduced Basis Methods for Partial Differential Equations (2016), Springer: Springer Cham · Zbl 1337.65113
[5] Razavi, S.; Tolson, B.; Burn, D., Review of surrogate modeling in water resources, Water Resour. Res., 48, 7, Article w07401 pp. (2012)
[6] Benner, P.; Gugercin, S.; Willcox, K., A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Rev., 57, 4, 483-531 (2015) · Zbl 1339.37089
[7] Sirovich, L., Turbulence and the dynamics of coherent structures. Part I: coherent structures, Q. Appl. Math., 45, 561-571 (1987) · Zbl 0676.76047
[8] Berkooz, G.; Holmes, P.; Lumley, J., The proper orthogonal decomposition in the analysis of turbulent flows, Annu. Rev. Fluid Mech., 25, 1, 539-575 (1993)
[9] Liang, Y.; Lee, H.; Lim, S.; Lin, W.; Lee, K.; Wu, C., Proper orthogonal decomposition and its applications-part I: theory, J. Sound Vib., 252, 3, 527-544 (2002) · Zbl 1237.65040
[10] Jolliffe, I., Principal Component Analysis (1986), Springer New York: Springer New York USA · Zbl 1011.62064
[11] Deheuvels, P.; Martynov, G., A Karhunen-Loeve decomposition of a Gaussian process generated by independent pairs of exponential random variables, J. Funct. Anal., 255, 9, 2363-2394 (2008) · Zbl 1155.60015
[12] Vermeulen, P.; Heemink, A., Model-reduced variational data assimilation, Mon. Weather Rev., 134, 2888-2899 (2006)
[13] Fang, F.; Zhang, T.; Pavlidis, D.; Pain, C.; Buchanan, A.; Navon, I., Reduced order modelling of an unstructured mesh air pollution model and application in 2D/3D urban street canyons, Atmos. Environ., 96, 96-106 (2014)
[14] San, O.; Borggaard, J., Principal interval decomposition framework for POD reduced-order modeling of convective Boussinesq flow, Int. J. Numer. Methods Fluids, 78, 1, 37-62 (2015)
[15] Bistrian, D.; Navon, I., An improved algorithm for the shallow water equations model reduction: dynamic mode decomposition vs POD, Int. J. Numer. Methods Fluids (2015)
[16] Stefanescu, R.; Sandu, A.; Navon, I., Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations, Int. J. Numer. Methods Fluids, 76, 8, 497-521 (2014)
[17] Lozovskiy, A.; Farthing, M.; Kees, C.; Gildin, E., POD-based model reduction for stabilized finite element approximations of shallow water flows, J. Comput. Appl. Math., 302, 50-70 (2016) · Zbl 1381.76189
[18] Lozovskiy, A.; Farthing, M.; Kees, C., Evaluation of Galerkin and Petrov-Galerkin model reduction for finite element approximations of the shallow water equations, Comput. Methods Appl. Mech. Eng., 318, 537-571 (2017) · Zbl 1439.76086
[19] Koopman, B., Hamiltonian systems and transformation in Hilbert space, Proc. Natl. Acad. Sci. USA, 17, 5, 315-318 (1931) · JFM 57.1010.02
[20] Rowley, C.; Mezić, I.; Bagheri, S.; Schlatter, P.; Henningson, D., Spectral analysis of nonlinear flows, J. Fluid Mech., 641, 115-127 (2009) · Zbl 1183.76833
[21] Schmid, P., Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656, 5-28 (2010) · Zbl 1197.76091
[22] Alekseev, A.; Bistrian, D.; Bondarev, A.; Navon, I., On linear and nonlinear aspects of dynamic mode decomposition, Int. J. Numer. Methods Fluids, 82, 6, 348-371 (2016)
[23] Bistrian, D.; Navon, I., Randomized dynamic mode decomposition for nonintrusive reduced order modelling, Int. J. Numer. Methods Eng., 112, 3-25 (2017)
[24] Amsallem, D.; Zahr, M.; Choi, Y.; Farhat, C., Design optimization using hyper-reduced-order models, Struct. Multidiscip. Optim., 51, 4, 919-940 (2015)
[25] Barrault, M.; Maday, Y.; Nguyen, N.; Patera, A., An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations, C. R. Math., 339, 9, 667-672 (2004) · Zbl 1061.65118
[26] Chaturantabut, S.; Sorensen, D., Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32, 5, 2737-2764 (2010) · Zbl 1217.65169
[27] Willcox, K., Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition, Comput. Fluids, 35, 2, 208-226 (2006) · Zbl 1160.76394
[28] Nguyen, N.; Peraire, J., An efficient reduced-order modeling approach for non-linear parametrized partial differential equations, Int. J. Numer. Methods Eng., 76, 27-55 (2008) · Zbl 1162.65407
[29] Xiao, D.; Fang, F.; Buchan, A.; Pain, C.; Navon, I.; Du, J.; Hu, G., Non-linear model reduction for the Navier-Stokes equations using residual DEIM method, J. Comput. Phys., 263, 1-18 (2014) · Zbl 1349.76288
[30] Ballarin, F.; Manzoni, A.; Quarteroni, A.; Rozza, G., Supremizer stabilization of POD-Galerkin approximation of parametrized steady incompressible Navier-Stokes equations, Int. J. Numer. Methods Eng., 102, 1136-1161 (2015) · Zbl 1352.76039
[31] Amsallem, D.; Farhat, C., Stabilization of projection-based reduced order models, Int. J. Numer. Methods Eng., 91, 358-377 (2012) · Zbl 1253.90184
[32] Xiao, D.; Fang, F.; Du, J.; Pain, C.; Navon, I.; Buchan, A.; ElSheikh, A.; Hu, G., Non-linear Petrov-Galerkin methods for reduced order modelling of the Navier-Stokes equations using a mixed finite element pair, Comput. Methods Appl. Mech. Eng., 255, 147-157 (2013) · Zbl 1297.76107
[33] Fang, F.; Pain, C.; Navon, I.; ElSheikh, A.; Du, J.; Xiao, D., Non-linear Petrov-Galerkin methods for reduced order hyperbolic equations and discontinuous finite element methods, J. Comput. Phys., 234, 540-559 (2013) · Zbl 1284.65132
[34] Carlberg, K.; Bou-Mosleh, C.; Farhat, C., Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations, Int. J. Numer. Methods Eng., 86, 155-181 (2011) · Zbl 1235.74351
[35] Hesthaven, J.; Ubbiali, S., Non-intrusive reduced order modeling of nonlinear problems using neural networks, J. Comput. Phys., 363, 55-78 (2018) · Zbl 1398.65330
[36] Dutta, S.; Rivera-Casillas, P.; Farthing, M., Neural ordinary differential equations for data-driven reduced order modeling of environmental hydrodynamics, (Proc. AAAI 2021 Spring Symp. Comb. Artif. Intell. Mach. Learn. with Phys. Sci.. Proc. AAAI 2021 Spring Symp. Comb. Artif. Intell. Mach. Learn. with Phys. Sci., CEUR-WS, Stanford, CA, USA (2021))
[37] Guo, M.; Hesthaven, J., Data-driven reduced order modeling for time-dependent problems, Comput. Methods Appl. Mech. Eng., 345, 75-99 (2019) · Zbl 1440.62346
[38] Narcowich, F.; Ward, J., Scattered-data interpolation on R̂n: error estimates for radial basis and band-limited functions, SIAM J. Math. Anal., 36, 1, 284-300 (2004) · Zbl 1081.41014
[39] Fornberg, B.; Flyer, N., Solving PDEs with radial basis functions, Acta Numer., 24, 215-258 (2015) · Zbl 1316.65073
[40] Wu, D.; Warwick, K.; Ma, Z.; Gasson, M.; Burgess, J.; Pan, S.; Aziz, T., Prediction of Parkinson’s disease tremor onset using a radial basis function neural network based on particle swarm optimization, Int. J. Neural Syst., 20, 109-116 (2010)
[41] Shahrokhabadi, M.; Neisy, A.; Perracchione, E.; Polato, M., Learning with subsampled kernel-based methods: environmental and financial applications, Dolomites Res. Notes Approx., 12, 1, 17-27 (2019)
[42] Xiao, D.; Fang, F.; Pain, C.; Hu, G., Non-intrusive reduced-order modelling of the Navier-Stokes equations based on RBF interpolation, Int. J. Numer. Methods Fluids, 79, 580-595 (2015) · Zbl 1455.76099
[43] Audouze, C.; De Vuyst, F.; Nair, P., Nonintrusive reduced-order modeling of parametrized time-dependent partial differential equations, Numer. Methods Partial Differ. Equ., 29, 5, 1587-1628 (2013) · Zbl 1274.65270
[44] Xiao, D.; Fang, F.; Pain, C.; Navon, I., A parameterized non-intrusive reduced order model and error analysis for general time-dependent nonlinear partial differential equations and its applications, Comput. Methods Appl. Mech. Eng., 317, 868-889 (2017) · Zbl 1439.65124
[45] Chen, W.; Hesthaven, J.; Junqiang, B.; Yang, Z.; Tihao, Y., A greedy non-intrusive reduced order model for fluid dynamics, AIAA J., 56, 12, 1-39 (2018)
[46] Iuliano, E.; Quagliarella, D., Aerodynamic shape optimization via non-intrusive POD-based surrogate modelling, (2013 IEEE Congr. Evol. Comput. CEC 2013 (2013), IEEE), 1467-1474
[47] McAlpin, T.; Sharp, J.; Scott, S.; Savant, G., Habitat restoration and flood control protection in the Kissimmee river, Wetlands, 33, 3, 551-560 (2013)
[48] Bova, S.; Carey, G., A symmetric formulation and SUPG scheme for the shallow-water equations, Adv. Water Resour., 19, 3, 123-131 (1996)
[49] Hughes, T.; Mallet, M., A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Eng., 58, 3, 305-328 (1986) · Zbl 0622.76075
[50] Trahan, C.; Savant, G.; Berger, R.; Farthing, M.; McAlpin, T.; Pettey, L.; Choudhary, G.; Dawson, C., Formulation and application of the adaptive hydraulics three-dimensional shallow water and transport models, J. Comput. Phys., 374, 47-90 (2018) · Zbl 1416.76130
[51] De Marchi, S.; Schaback, R.; Wendland, H., Near-optimal data-independent point locations for radial basis function interpolation, Adv. Comput. Math., 23, 3, 317-330 (2005) · Zbl 1070.65008
[52] Schaback, R.; Wendland, H., Adaptive greedy techniques for approximate solution of large RBF systems, Numer. Algorithms, 24, 3, 239-254 (2000) · Zbl 0957.65021
[53] Wirtz, D.; Haasdonk, B., A vectorial kernel orthogonal greedy algorithm, Dolomites Res. Notes Approx., 6, 83-100 (2013)
[54] Aizinger, V.; Dawson, C. N., A discontinuous Galerkin method for two-dimensional flow and transport in shallow water, Adv. Water Resour., 25, 1, 67-84 (2002)
[55] Berger, R.; Stockstill, R., Finite element model for high-velocity channels, J. Hydraul. Eng., 121, 10 (1995)
[56] Hughes, T.; Feijóo, G.; Mazzei, L.; Quincy, J.-B., The variational multiscale method – a paradigm for computational mechanics, Comput. Methods Appl. Mech. Eng., 166, 1, 3-24 (1998) · Zbl 1017.65525
[57] Antoulas, A.; Sorensen, D., Approximation of large-scale dynamical systems: an overview, Int. J. Appl. Math. Comput. Sci., 11, 5, 1093-1121 (2001) · Zbl 1024.93008
[58] Lassila, T.; Manzoni, A.; Quarteroni, A.; Rozza, G., Model Order Reduction in Fluid Dynamics: Challenges and Perspectives, 235-273 (2014), Springer International Publishing: Springer International Publishing Cham · Zbl 1395.76058
[59] Carlberg, K.; Farhat, C.; Cortial, J.; Amsallem, D., The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows, J. Comput. Phys., 242, 623-647 (2013) · Zbl 1299.76180
[60] Carlson, R.; Foley, T., The parameter R̂2 in multiquadric interpolation, Comput. Math. Appl., 21, 9, 29-42 (1991) · Zbl 0725.65009
[61] Fasshauer, G., Meshfree Approximation Methods with MATLAB, Interdisciplinary Mathematical Sciences, vol. 6 (2007), World Scientific Publishing Company · Zbl 1123.65001
[62] Wendland, H., Scattered Data Approximation (2005), Cambridge University Press · Zbl 1075.65021
[63] DeVore, R., Nonlinear approximation, Acta Numer., 7, 51-150 (1998) · Zbl 0931.65007
[64] Floater, M.; Iske, A., Thinning algorithms for scattered data interpolation, BIT Numer. Math., 38, 4, 705-720 (1998) · Zbl 0926.65013
[65] Lim, E.; Zainuddin, Z., An improved fast training algorithm for RBF networks using symmetry-based fuzzy C-means clustering, Matematika, 24, 2, 141-148 (2008)
[66] Liu, H.; He, J., The application of dynamic K-means clustering algorithm in the center selection of RBF neural networks, (3rd Int. Conf. Genet. Evol. Comput. WGEC 2009 (2009)), 488-491
[67] Sing, J.; Basu, D.; Nasipuri, M.; Kundu, M., Improved k-means algorithm in the design of RBF neural networks, (TENCON 2003. Conf. Converg. Technol. Asia-Pacific Reg. (2003)), 841-845
[68] Tropp, J.; Gilbert, A.; Strauss, M., Algorithms for simultaneous sparse approximation. Part I: greedy pursuit, Signal Process., 86, 3, 572-588 (2006) · Zbl 1163.94396
[69] Temlyakov, V., Greedy approximation, Acta Numer., 17, 235-409 (2008) · Zbl 1178.65050
[70] Santin, G.; Haasdonk, B., Convergence rate of the data-independent P-greedy algorithm in kernel-based approximation, Dolomites Res. Notes Approx., 10, 2, 68-78 (2017) · Zbl 1370.94401
[71] Leviatan, D.; Temlyakov, V., Simultaneous greedy approximation in Banach spaces, J. Complex., 21, 3, 275-293 (2005) · Zbl 1088.41019
[72] Schaback, R.; Werner, J., Linearly constrained reconstruction of functions by kernels with applications to machine learning, Adv. Comput. Math., 25, 1-3, 237-258 (2006) · Zbl 1106.65008
[73] Pazouki, M.; Schaback, R., Bases for kernel-based spaces, J. Comput. Appl. Math., 236, 4, 575-588 (2011) · Zbl 1234.41003
[74] Walton, S.; Hassan, O.; Morgan, K., Reduced order modelling for unsteady fluid flow using proper orthogonal decomposition and radial basis functions, Appl. Math. Model., 37, 20-21, 8930-8945 (2013) · Zbl 1426.76576
[75] Fasshauer, G.; Zhang, J., On choosing “optimal” shape parameters for RBF approximation, Numer. Algorithms, 45, 1-4, 345-368 (2007) · Zbl 1127.65009
[76] Xiao, D., Error estimation of the parametric non-intrusive reduced order model using machine learning, Comput. Methods Appl. Mech. Eng., 355, 513-534 (2019) · Zbl 1441.62081
[77] Bozzini, M.; Lenarduzzi, L.; Rossini, M.; Schaback, R., Interpolation with variably scaled kernels, IMA J. Numer. Anal., 35, 1, 199-219 (2015) · Zbl 1309.65015
[78] De Marchi, S.; Martínez, A.; Perracchione, E., Fast and stable rational RBF-based partition of unity interpolation, J. Comput. Appl. Math., 349, 331-343 (2019) · Zbl 1524.65075
[79] De Marchi, S.; Marchetti, F.; Perracchione, E., Jumping with variably scaled discontinuous kernels (VSDKs), BIT Numer. Math., 60, 2, 441-463 (2020) · Zbl 07209368
[80] Savant, G.; Berger, R., Adaptive time stepping-operator splitting strategy to couple implicit numerical hydrodynamic and water quality codes, J. Environ. Eng., 138, 9, 979-984 (2012)
[81] Tate, J.; Berger, R.; Stockstill, R., Refinement indicator for mesh adaption in shallow-water modeling, J. Hydraul. Eng., 132, 8, 854-857 (2006)
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