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Reduced basis methods for time-dependent problems

Published online by Cambridge University Press:  09 June 2022

Jan S. Hesthaven Open the ORCID record for Jan S. Hesthaven [Opens in a new window] ,
Cecilia Pagliantini  and
Gianluigi Rozza
Jan S. Hesthaven
Affiliation:
Ecole Polytechnique Federale de Lausanne (EPFL), CH-1015Lausanne, Switzerland E-mail: [email protected]
Cecilia Pagliantini
Affiliation:
Eindhoven University of Technology, 5600MBEindhoven, Netherlands E-mail: [email protected]
Gianluigi Rozza
Affiliation:
SISSA – International School for Advanced Studies, 34136Trieste, Italy E-mail: [email protected]
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Abstract

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Numerical simulation of parametrized differential equations is of crucial importance in the study of real-world phenomena in applied science and engineering. Computational methods for real-time and many-query simulation of such problems often require prohibitively high computational costs to achieve sufficiently accurate numerical solutions. During the last few decades, model order reduction has proved successful in providing low-complexity high-fidelity surrogate models that allow rapid and accurate simulations under parameter variation, thus enabling the numerical simulation of increasingly complex problems. However, many challenges remain to secure the robustness and efficiency needed for the numerical simulation of nonlinear time-dependent problems. The purpose of this article is to survey the state of the art of reduced basis methods for time-dependent problems and draw together recent advances in three main directions. First, we discuss structure-preserving reduced order models designed to retain key physical properties of the continuous problem. Second, we survey localized and adaptive methods based on nonlinear approximations of the solution space. Finally, we consider data-driven techniques based on non-intrusive reduced order models in which an approximation of the map between parameter space and coefficients of the reduced basis is learned. Within each class of methods, we describe different approaches and provide a comparative discussion that lends insights to advantages, disadvantages and potential open questions.

Type
Research Article
Information
Acta Numerica , Volume 31 , May 2022 , pp. 265 - 345
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2022. Published by Cambridge University Press

References

Ahmed, S. E., Pawar, S., San, O., Rasheed, A., Iliescu, T. and Noack, B. R. (2021), On closures for reduced order models: A spectrum of first-principle to machine-learned avenues, Phys. Fluids 33, 091301.CrossRefGoogle Scholar
Almroth, B. O., Stern, P. and Brogan, F. A. (1978), Automatic choice of global shape functions in structural analysis, AIAA J. 16, 525–528.CrossRefGoogle Scholar
Amsallem, D. and Haasdonk, B. (2016), PEBL-ROM: Projection-error based local reduced-order models, Adv. Model. Simul. Engrg Sci. 3, doi:10.1186/s40323-016-0059-7.Google Scholar
Amsallem, D., Zahr, M. and Farhat, C. (2012), Nonlinear model order reduction based on local reduced-order bases, Internat. J. Numer. Methods. Engrg 92, 891–916.CrossRefGoogle Scholar
Arnold, D. N., Falk, R. S. and Winther, R. (2006), Finite element exterior calculus, homological techniques, and applications, in Acta Numerica, Vol. 15, Cambridge University Press, pp. 1–155.Google Scholar
Arnol’d, V. I. (1966), On the topology of three-dimensional steady flows of an ideal fluid, J. Appl. Math. Mech. 30, 223–226.CrossRefGoogle Scholar
Astrid, P., Weiland, S., Willcox, K. and Backx, T. (2008), Missing point estimation in models described by proper orthogonal decomposition, IEEE Trans. Automat. Control 53, 2237–2251.CrossRefGoogle Scholar
Audouze, C., De Vuyst, F. and Nair, P. B. (2009), Reduced-order modeling of parameterized PDEs using time–space-parameter principal component analysis, Internat. J. Numer. Methods. Engrg 80, 1025–1057.CrossRefGoogle Scholar
Baesens, B. (2014), Analytics in a Big Data World: The Essential Guide to Data Science and its Applications, Wiley and SAS Business Series, Wiley.Google Scholar
Barrault, M., Maday, Y., Nguyen, N. C. and Patera, A. T. (2004), An ‘empirical interpolation’ method: Application to efficient reduced-basis discretization of partial differential equations, C. R. Math. Acad. Sci. Paris 339, 667–672.CrossRefGoogle Scholar
Barshan, E., Ghodsi, A., Azimifar, Z. and Zolghadri Jahromi, M. (2011), Supervised principal component analysis: Visualization, classification and regression on subspaces and submanifolds, Pattern Recognition 44, 1357–1371.CrossRefGoogle Scholar
Bashir, O., Willcox, K., Ghattas, O., Van, B., Waanders, B. and Hill, J. (2008), Hessian-based model reduction for large-scale systems with initial-condition inputs, Internat. J. Numer. Methods Engrg 73, 844–868.CrossRefGoogle Scholar
Beattie, C. and Gugercin, S. (2009), Interpolatory projection methods for structure-preserving model reduction, Systems Control Lett. 58, 225–232.CrossRefGoogle Scholar
Beattie, C. and Gugercin, S. (2011), Structure-preserving model reduction for nonlinear port-Hamiltonian systems, in 2011 50th IEEE Conference on Decision and Control and European Control Conference, pp. 6564–6569.Google Scholar
Beattie, C., Gugercin, S. and Mehrmann, V. (2019), Structure-preserving interpolatory model reduction for port-Hamiltonian differential-algebraic systems. Available at arXiv:1910.05674. To appear in a Festschrift to honour the 70th birthday of A. Antoulas.Google Scholar
Beck, M. H., Jäckle, A., Worth, G. A. and Meyer, H.-D. (2000), The multiconfiguration time-dependent Hartree (MCTDH) method: A highly efficient algorithm for propagating wavepackets, Phys. Rep. 324, 1–105.CrossRefGoogle Scholar
Belkin, M. and Niyogi, P. (2001), Laplacian eigenmaps and spectral techniques for embedding and clustering, in Advances in Neural Information Processing Systems 14 (Dietterich, T. et al., eds), MIT Press.Google Scholar
Benner, P. and Breiten, T. (2012), Interpolation-based ${H}_2$ -model reduction of bilinear control systems, SIAM J. Matrix Anal. Appl. 33, 859–885.CrossRefGoogle Scholar
Benner, P., Gugercin, S. and Willcox, K. (2015), A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Rev. 57, 483–531.CrossRefGoogle Scholar
Berzins, A., Helmig, J., Key, F. and Elgeti, S. (2020), Standardized non-intrusive reduced order modeling using different regression models with application to complex flow problems. Available at arXiv:2006.13706.Google Scholar
Beyn, W.-J. and Thümmler, V. (2004), Freezing solutions of equivariant evolution equations, SIAM J. Appl. Dyn. Syst. 3, 85–116.CrossRefGoogle Scholar
Bigoni, C. and Hesthaven, J. S. (2020), Simulation-based anomaly detection and damage localization: An application to structural health monitoring, Comput. Methods Appl. Mech. Engrg 363, 112896.CrossRefGoogle Scholar
Binev, P., Cohen, A., Dahmen, W., DeVore, R., Petrova, G. and Wojtaszczyk, P. (2011), Convergence rates for greedy algorithms in reduced basis methods , SIAM J. Math. Anal. 43, 1457–1472.CrossRefGoogle Scholar
Bistrian, D. A. and Navon, I. M. (2016), Randomized dynamic mode decomposition for nonintrusive reduced order modelling, Internat. J. Numer. Methods. Engrg 112, 3–25.CrossRefGoogle Scholar
Bonito, A., Cohen, A., DeVore, R., Guignard, D., Jantsch, P. and Petrova, G. (2021), Nonlinear methods for model reduction , ESAIM Math. Model. Numer. Anal. 55, 507–531.CrossRefGoogle Scholar
Brand, M. (2006), Fast low-rank modifications of the thin singular value decomposition, Linear Algebra Appl. 415, 20–30.CrossRefGoogle Scholar
Brunton, S. L., Noack, B. R. and Koumoutsakos, P. (2020), Machine learning for fluid mechanics, Annu. Rev. Fluid Mech. 52, 477–508.CrossRefGoogle Scholar
Brunton, S. L., Proctor, J. L. and Kutz, J. N. (2016), Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc. Natl. Acad. Sci. USA 113, 3932–3937.CrossRefGoogle ScholarPubMed
Buchfink, P., Bhatt, A. and Haasdonk, B. (2019), Symplectic model order reduction with non-orthonormal bases, Math. Comput. Appl. 24, 43.Google Scholar
Buffa, A., Maday, Y., Patera, A. T., C. Prud’homme and G. Turinici (2012), A priori convergence of the greedy algorithm for the parametrized reduced basis method , ESAIM Math. Model. Numer. Anal. 46, 595–603.CrossRefGoogle Scholar
Buhmann, M. and Dyn, N. (1993), Spectral convergence of multiquadric interpolation, Proc. Edinb. Math. Soc. 36, 319–333.CrossRefGoogle Scholar
Bui-Thanh, T., Damodaran, M. and Willcox, K. (2003), Proper orthogonal decomposition extensions for parametric applications in compressible aerodynamics, in 21st Applied Aerodynamics Conference. AIAA paper 2003-4213.Google Scholar
Bui-Thanh, T., Willcox, K., Ghattas, O. and van Bloemen Waanders, B. (2007), Goal-oriented, model-constrained optimization for reduction of large-scale systems, J. Comput. Phys. 224, 880–896.CrossRefGoogle Scholar
Cagniart, N., Maday, Y. and Stamm, B. (2019), Model order reduction for problems with large convection effects, in Contributions to Partial Differential Equations and Applications (Chetverushkin, B. et al., eds), Vol. 47 of Computational Methods in Applied Sciences, Springer, pp. 131–150.CrossRefGoogle Scholar
Carlberg, K. (2015), Adaptive $h$ -refinement for reduced-order models, Internat. J. Numer. Methods Engrg 102, 1192–1210.CrossRefGoogle Scholar
Carlberg, K., Choi, Y. and Sargsyan, S. (2018), Conservative model reduction for finite-volume models, J. Comput. Phys. 371, 280–314.CrossRefGoogle Scholar
Carlberg, K., Farhat, C., Cortial, J. and Amsallem, D. (2013), The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows, J. Comput. Phys. 242, 623–647.CrossRefGoogle Scholar
Carlberg, K., Guzzetti, S., Khalil, M. and Sargsyan, K. (2019), The network uncertainty quantification method for propagating uncertainties in component-based systems. Available at arXiv:1908.11476.Google Scholar
Carlberg, K., Tuminaro, R. and Boggs, P. (2015), Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics , SIAM J. Sci. Comput. 37, B153–B184.CrossRefGoogle Scholar
Chaturantabut, S. and Sorensen, D. C. (2010), Nonlinear model reduction via discrete empirical interpolation , SIAM J. Sci. Comput. 32, 2737–2764.CrossRefGoogle Scholar
Chaturantabut, S., Beattie, C. and Gugercin, S. (2016), Structure-preserving model reduction for nonlinear port-Hamiltonian systems , SIAM J. Sci. Comput. 38, B837–B865.CrossRefGoogle Scholar
Chen, K. K., Tu, J. H. and Rowley, C. W. (2012), Variants of dynamic mode decomposition: Boundary condition, Koopman, and Fourier analyses, J. Nonlin. Sci. 22, 887–915.CrossRefGoogle Scholar
Chen, W., Hesthaven, J. S., Junqiang, B., Qiu, Y., Yang, Z. and Tihao, Y. (2018), Greedy nonintrusive reduced order model for fluid dynamics, AIAA J. 56, 4927–4943.CrossRefGoogle Scholar
Chen, Y., Hesthaven, J. S., Maday, Y. and Rodríguez, J. (2009), Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2D Maxwell’s problem , ESAIM Math. Model. Numer. Anal. 43, 1099–1116.CrossRefGoogle Scholar
Chen, Y., Hesthaven, J. S., Maday, Y. and Rodríguez, J. (2010), Certified reduced basis methods and output bounds for the harmonic Maxwell’s equations , SIAM J. Sci. Comput. 32, 970–996.CrossRefGoogle Scholar
Christensen, E. A., Brøns, M. and Sørensen, J. N. (1999), Evaluation of proper orthogonal decomposition–based decomposition techniques applied to parameter-dependent nonturbulent flows , SIAM J. Sci. Comput. 21, 1419–1434.CrossRefGoogle Scholar
Christiansen, S. H., Munthe-Kaas, H. Z. and Owren, B. (2011), Topics in structure-preserving discretization, in Acta Numerica, Vol. 20, Cambridge University Press, pp. 1–119.Google Scholar
CISCO (2019), Cisco visual networking index: Global mobile data traffic forecast update, 2017–2022.Google Scholar
Constantine, P. G., Dow, E. and Wang, Q. (2014), Active subspace methods in theory and practice: Applications to kriging surfaces, SIAM J. Sci. Comput.Google Scholar
Ştefănescu, R., Sandu, A. and Navon, I. M. (2014), Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations, Internat. J. Numer. Methods Fluids 76, 497–521.CrossRefGoogle Scholar
Cuong, N. N., Veroy, K. and Patera, A. T. (2005), Certified real-time solution of parametrized partial differential equations, in Handbook of Materials Modeling (Yip, S., ed.), Springer, pp. 1529–1564.CrossRefGoogle Scholar
Cybenko, G. (1989), Approximation by superpositions of a sigmoidal function, Math. Control Signals Systems 2, 303–314.CrossRefGoogle Scholar
Darboux, G. (1882), Sur le problème de Pfaff, Bulletin des Sciences Mathématiques et Astronomiques 6, 14–36.Google Scholar
Demo, N., Tezzele, M. and Rozza, G. (2019), A non-intrusive approach for the reconstruction of POD modal coefficients through active subspaces, Comptes Rendus Mécanique 347, 873–881.CrossRefGoogle Scholar
DeVore, R. A. (2017), The theoretical foundation of reduced basis methods, in Model Reduction and Approximation: Theory and Algorithms, SIAM, chapter 3, pp. 137–168.CrossRefGoogle Scholar
DeVore, R. A., Petrova, G. and Wojtaszczyk, P. (2013), Greedy algorithms for reduced bases in Banach spaces, Constr. Approx. 37, 455–466.CrossRefGoogle Scholar
Dewdney, P. E., Hall, P. J., Schilizzi, R. T. and Lazio, T. J. L. W. (2009), The square kilometre array, Proc. IEEE 97, 1482–1496.CrossRefGoogle Scholar
Dirac, P. A. M. (1930), Note on exchange phenomena in the Thomas atom, Math. Proc. Cambridge Philos. Soc. 26, 376–385.CrossRefGoogle Scholar
Discacciati, N. and Hesthaven, J. S. (2021), Modeling synchronization in globally coupled oscillatory systems using model order reduction, Chaos 31, 053127.CrossRefGoogle ScholarPubMed
Dupuis, R., Jouhaud, J.-C. and Sagaut, P. (2018), Surrogate modeling of aerodynamic simulations for multiple operating conditions using machine learning, AIAA J. 56, 3622–3635.CrossRefGoogle Scholar
Eftang, J., Patera, A. and Rønquist, E. (2010), An ‘hp’ certified reduced basis method for parametrized elliptic partial differential equations, SIAM. J. Sci. Comput. 32, 3170–3200.CrossRefGoogle Scholar
Ehrlacher, V., Lombardi, D., Mula, O. and Vialard, F.-X. (2020), Nonlinear model reduction on metric spaces: Application to one-dimensional conservative PDEs in Wasserstein spaces , ESAIM Math. Model. Numer. Anal. 54, 2159–2197.CrossRefGoogle Scholar
Eldred, M. and Dunlavy, D. (2006), Formulations for surrogate-based optimization with data fit, multifidelity, and reduced-order models, in 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, pp. 1–20. AIAA paper 2006-7117.Google Scholar
Farhat, C., Chapman, T. and Avery, P. (2015), Structure-preserving, stability, and accuracy properties of the energy-conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models, Internat. J. Numer. Methods Engrg 102, 1077–1110.CrossRefGoogle Scholar
Feppon, F. and Lermusiaux, P. F. J. (2018), A geometric approach to dynamical model order reduction, SIAM J. Matrix Anal. Appl. 39, 510–538.CrossRefGoogle Scholar
Figotin, A. and Schenker, J. H. (2007), Hamiltonian structure for dispersive and dissipative dynamical systems, J. Statist. Phys. 128, 969–1056.CrossRefGoogle Scholar
Fink, J. P. and Rheinboldt, W. C. (1983), On the error behavior of the reduced basis technique for nonlinear finite element approximations, Z. Angew. Math. Mech. 63, 21–28.CrossRefGoogle Scholar
Fink, J. P. and Rheinboldt, W. C. (1984), Solution manifolds and submanifolds of parametrized equations and their discretization errors, Numer. Math. 45, 323–343.CrossRefGoogle Scholar
Fox, R. L. and Miura, H. (1971), An approximate analysis technique for design calculations, AIAA J. 9, 177–179.CrossRefGoogle Scholar
Fresca, S. and Manzoni, A. (2022), POD-DL-ROM: Enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition, Comput. Methods Appl. Mech. Engrg 388, 114181.CrossRefGoogle Scholar
Freund, R. W. (2003), Model reduction methods based on Krylov subspaces, in Acta Numerica, Vol. 12, Cambridge University Press, pp. 267–319.Google Scholar
Ganesh, M., Hesthaven, J. S. and Stamm, B. (2012), A reduced basis method for electromagnetic scattering by multiple particles in three dimensions, J. Comput. Phys. 231, 7756–7779.CrossRefGoogle Scholar
Gao, Z., Liu, Q., Hesthaven, J. S., Wang, B.-S., Don, W. S. and Wen, X. (2021), Non-intrusive reduced order modeling of convection dominated flows using artificial neural networks with application to Rayleigh–Taylor instability, Commun. Comput. Phys. 30, 97–123.CrossRefGoogle Scholar
Gerbeau, J.-F. and Lombardi, D. (2014), Approximated Lax pairs for the reduced order integration of nonlinear evolution equations, J. Comput. Phys. 265, 246–269.CrossRefGoogle Scholar
Gong, Y., Wang, Q. and Wang, Z. (2017), Structure-preserving Galerkin POD reduced-order modeling of Hamiltonian systems, Comput. Methods Appl. Mech. Engrg 315, 780–798.CrossRefGoogle Scholar
Goodfellow, I., Bengio, Y. and Courville, A. (2016), Deep Learning, MIT Press.Google Scholar
Gouasmi, A., Parish, E., and Duraisamy, K. (2017), A priori estimation of memory effects in reduced order modeling of nonlinear systems using the Mori–Zwanzig formalism, Proc. Royal Soc. Ser A 473, 20170385.CrossRefGoogle Scholar
Grepl, M. A. and Patera, A. T. (2005), A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations , ESAIM Math. Model. Numer. Anal. 39, 157–181.CrossRefGoogle Scholar
Grepl, M. A., Maday, Y., Nguyen, N. C. and Patera, A. T. (2007), Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations , M2AN Math. Model. Numer. Anal. 41, 575–605.CrossRefGoogle Scholar
Gugercin, S., Polyuga, R. V., Beattie, C. and van der Schaft, A. (2012), Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems , Automatica J. IFAC 48, 1963–1974.CrossRefGoogle Scholar
Gunzburger, M. D. (1989), Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms, Computer Science and Scientific Computing, Academic Press.Google Scholar
Gunzburger, M. D., Peterson, J. S. and Shadid, J. N. (2007), Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data, Comput. Methods Appl. Mech. Engrg 196, 1030–1047.CrossRefGoogle Scholar
Guo, M. and Hesthaven, J. S. (2018), Reduced order modeling for nonlinear structural analysis using Gaussian process regression, Comput. Methods Appl. Mech. Engrg 341, 807–826.CrossRefGoogle Scholar
Guo, M. and Hesthaven, J. S. (2019), Data-driven reduced order modeling for time-dependent problems, Comput. Methods Appl. Mech. Engrg 345, 75–99.CrossRefGoogle Scholar
Gupta, A. and Lermusiaux, P. F. J. (2021), Neural closure models for dynamical systems, Proc. R. Soc. A. 477, 20201004.CrossRefGoogle Scholar
Haasdonk, B. (2013), Convergence rates of the POD-greedy method , ESAIM Math. Model. Numer. Anal. 47, 859–873.CrossRefGoogle Scholar
Haasdonk, B. and Ohlberger, M. (2008), Reduced basis method for finite volume approximations of parametrized linear evolution equations, M2AN Math. Model. Numer. Anal. 42, 277–302.CrossRefGoogle Scholar
Haasdonk, B., Dihlmann, M. and Ohlberger, M. (2011), A training set and multiple bases generation approach for parameterized model reduction based on adaptive grids in parameter space, Math. Comput. Model. Dyn. Syst. 17, 423–442.CrossRefGoogle Scholar
Hairer, E., Lubich, C. and Wanner, G. (2006), Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Vol. 31 of Springer Series in Computational Mathematics, second edition, Springer.Google Scholar
Hess, M., Alla, A., Quaini, A., Rozza, G. and Gunzburger, M. (2019), A localized reduced-order modeling approach for PDEs with bifurcating solutions, Comput. Methods Appl. Mech. Engrg 351, 379–403.CrossRefGoogle Scholar
Hesthaven, J. S. and Pagliantini, C. (2021), Structure-preserving reduced basis methods for Poisson systems, Math. Comp. 90, 1701–1740.CrossRefGoogle Scholar
Hesthaven, J. S. and Ubbiali, S. (2018), Non-intrusive reduced order modeling of nonlinear problems using neural networks, J. Comput. Phys. 363, 55–78.CrossRefGoogle Scholar
Hesthaven, J. S., Pagliantini, C. and Ripamonti, N. (2022), Rank-adaptive structure-preserving model order reduction of Hamiltonian systems, ESAIM Math. Model. Numer. Anal. 56, 617–650.CrossRefGoogle Scholar
Hesthaven, J. S., Rozza, G. and Stamm, B. (2015), Certified Reduced Basis Methods for Parametrized Partial Differential Equations, Springer Briefs in Mathematics, Springer.Google Scholar
Hesthaven, J. S., Stamm, B. and Zhang, S. (2014), Efficient greedy algorithms for high-dimensional parameter spaces with applications to empirical interpolation and reduced basis methods , ESAIM Math. Model. Numer. Anal. 48, 259–283.CrossRefGoogle Scholar
Hijazi, S., Stabile, G., Mola, A. and Rozza, G. (2020), Data-driven POD–Galerkin reduced order model for turbulent flows, J. Comput. Phys. 416, 109513.CrossRefGoogle Scholar
Himpe, C. and Ohlberger, M. (2015), Data-driven combined state and parameter reduction for inverse problems, Adv. Comput. Math. 41, 1343–1364.CrossRefGoogle Scholar
Hiptmair, R. (2002), Finite elements in computational electromagnetism, in Acta Numerica, Vol. 11, Cambridge University Press, pp. 237–339.Google Scholar
Hotelling, H. (1933), Analysis of a complex of statistical variables into principal components, 24, 417–441.Google Scholar
Huynh, D. B. P., Knezevic, D. J. and Patera, A. T. (2011), A Laplace transform certified reduced basis method: Application to the heat equation and wave equation, C. R. Acad. Sci. Paris, Ser. I 349, 401–405.CrossRefGoogle Scholar
Iollo, A. and Lombardi, D. (2014), Advection modes by optimal mass transfer, Phys. Rev. E 89, 022923.CrossRefGoogle ScholarPubMed
Ionescu, T. C. and Astolfi, A. (2013), Moment matching for nonlinear port Hamiltonian and gradient systems, IFAC Proceedings Volumes 46, 395–399.CrossRefGoogle Scholar
Ionescu, T. C., Fujimoto, K. and Scherpen, J. M. A. (2010), Dissipativity preserving balancing for nonlinear systems: A Hankel operator approach, Systems Control Lett. 59, 180–194.CrossRefGoogle Scholar
Iserles, A., Munthe-Kaas, H. Z., Nørsett, S. P. and Zanna, A. (2000), Lie-group methods, in Acta Numerica, Vol. 9, Cambridge University Press, pp. 215–365.Google Scholar
Jolliffe, I. T. (1986), Principal Component Analysis, Springer Series in Statistics, Springer.CrossRefGoogle Scholar
Karasözen, B. and Uzunca, M. (2018), Energy preserving model order reduction of the nonlinear Schrödinger equation, Adv. Comput. Math. 44, 1769–1796.CrossRefGoogle Scholar
Karasözen, B., Yıldız, S. and Uzunca, M. (2021), Structure preserving model order reduction of shallow water equations, Math. Methods Appl. Sci. 44, 476–492.CrossRefGoogle Scholar
Karhunen, K. (1947), Über lineare Methoden in der Wahrscheinlichkeitsrechnung, Ann. Acad. Sci. Fenn. Ser. A. I. Math.-Phys. 1947, 79.Google Scholar
Koch, O. and Lubich, C. (2007), Dynamical low-rank approximation, SIAM J. Matrix Anal. Appl. 29, 434–454.CrossRefGoogle Scholar
Kostant, B. (1979), The solution to a generalized Toda lattice and representation theory, Adv. Math. 34, 195–338.CrossRefGoogle Scholar
Kramer, B. and Willcox, K. E. (2019), Nonlinear model order reduction via lifting transformations and proper orthogonal decomposition, AIAA J. 57, 2297–2307.CrossRefGoogle Scholar
Kunisch, K. and Volkwein, S. (2002), Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics , SIAM J. Numer. Anal. 40, 492–515.CrossRefGoogle Scholar
Kutyniok, G., Petersen, P., Raslan, M. and Schneider, R. (2022), A theoretical analysis of deep neural networks and parametric PDEs, Constr. Approx. 55, 73–125.CrossRefGoogle Scholar
Kutz, J. N., Brunton, S. L., Brunton, B. W. and Proctor, J. L. (2016a), Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems, SIAM.CrossRefGoogle Scholar
Kutz, J. N., Fu, X. and Brunton, S. L. (2016b), Multiresolution dynamic mode decomposition , SIAM J. Appl. Dyn. Syst. 15, 713–735.CrossRefGoogle Scholar
Lafon, S. and Lee, A. B. (2006), Diffusion maps and coarse-graining: A unified framework for dimensionality reduction, graph partitioning, and data set parameterization, IEEE Trans. Pattern Anal. Mach. Intel. 28, 1393–1403.CrossRefGoogle ScholarPubMed
Lall, S., Krysl, P. and Marsden, J. E. (2003), Structure-preserving model reduction for mechanical systems, Phys. D 184, 304–318.CrossRefGoogle Scholar
Lanczos, C. (1949), The Variational Principles of Mechanics, Vol. 4 of Mathematical Expositions, University of Toronto Press.CrossRefGoogle Scholar
Lazzaro, D. and Montefusco, L. B. (2002), Radial basis functions for the multivariate interpolation of large scattered data sets, J. Comput. Appl. Math. 140, 521–536.CrossRefGoogle Scholar
Lee, K. and Carlberg, K. T. (2020), Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders, J. Comput. Phys. 404, 108973.CrossRefGoogle Scholar
Lee, Y., Yang, H. and Yin, Z. (2017), PIV-DCNN: Cascaded deep convolutional neural networks for particle image velocimetry, Exp. Fluids 58, 171.CrossRefGoogle Scholar
LeGresley, P. A. and Alonso, J. J. (2000), Airfoil design optimization using reduced order models based on proper orthogonal decomposition, in Fluids 2000 Conference and Exhibit. AIAA paper 2000-2545.Google Scholar
Leimkuhler, B. and Reich, S. (2004), Simulating Hamiltonian Dynamics, Vol. 14 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press.Google Scholar
Li, K.-C. (1991), Sliced inverse regression for dimension reduction, J. Amer. Statist. Assoc. 86, 316–327.CrossRefGoogle Scholar
Lieberman, C., Willcox, K. and Ghattas, O. (2010), Parameter and state model reduction for large-scale statistical inverse problems , SIAM J. Sci. Comput. 32, 2523–2542.CrossRefGoogle Scholar
Loève, M. (1955), Probability Theory: Foundations, Random Sequences, Van Nostrand.Google Scholar
Lubich, C. (2008), From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis, Zurich Lectures in Advanced Mathematics, European Mathematical Society.CrossRefGoogle Scholar
Maboudi Afkham, B. and Hesthaven, J. S. (2017), Structure preserving model reduction of parametric Hamiltonian systems , SIAM J. Sci. Comput. 39, A2616–A2644.CrossRefGoogle Scholar
Maboudi Afkham, B. and Hesthaven, J. S. (2019), Structure-preserving model-reduction of dissipative Hamiltonian systems, J. Sci. Comput. 81, 3–21.CrossRefGoogle Scholar
Maboudi Afkham, B., Ripamonti, N., Wang, Q. and Hesthaven, J. S. (2020), Conservative model order reduction for fluid flow, in Quantification of Uncertainty: Improving Efficiency and Technology (D’Elia, M. et al., eds), Vol. 137 of Lecture Notes in Computational Science and Engineering, Springer, pp. 67–99.CrossRefGoogle Scholar
MacQueen, J. (1967), Some methods for classification and analysis of multivariate observations, in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability , Vol. 1 , Theory of Statistics (Lecam, L. and Neyman, J., eds), University of California Press, pp. 281–297.Google Scholar
Maday, Y. and Stamm, B. (2013), Locally adaptive greedy approximations for anisotropic parameter reduced basis spaces , SIAM J. Sci. Comput. 35, A2417–A2441.CrossRefGoogle Scholar
Marsden, J. E. and Ratiu, T. S. (1999), Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Vol. 17 of Texts in Applied Mathematics, second edition, Springer.CrossRefGoogle Scholar
Marsden, J. E., Weinstein, A., Ratiu, T., Schmid, R. and Spencer, R. G. (1983), Hamiltonian systems with symmetry, coadjoint orbits and plasma physics, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 117, 289–340.Google Scholar
Masci, J., Meier, U., Cireşan, D. and Schmidhuber, J. (2011), Stacked convolutional auto-encoders for hierarchical feature extraction, in Artificial Neural Networks and Machine Learning (ICANN 2011) (Honkela, T. et al., eds), Springer, pp. 52–59.Google Scholar
Miura, R. M., Gardner, C. S. and Kruskal, M. D. (1968), Korteweg–de Vries equation and generalizations, II: Existence of conservation laws and constants of motion, J. Math. Phys. 9, 1204–1209.CrossRefGoogle Scholar
Miyatake, Y. (2019), Structure-preserving model reduction for dynamical systems with a first integral, Japan. J. Ind. Appl. Math. 36, 1021–1037.CrossRefGoogle Scholar
Mojgani, R. and Balajewicz, M. (2017), Lagrangian basis method for dimensionality reduction of convection dominated nonlinear flows. Available at arXiv:1701.04343.Google Scholar
Montáns, F. J., Chinesta, F., Gómez-Bombarelli, R. and Kutz, J. N. (2019), Data-driven modeling and learning in science and engineering, Comptes Rendus Mécanique 347, 845–855.CrossRefGoogle Scholar
Morrison, P. J. (1980), The Maxwell–Vlasov equations as a continuous Hamiltonian system, Phys. Lett. A 80, 383–386.CrossRefGoogle Scholar
Musharbash, E. and Nobile, F. (2017), Symplectic dynamical low rank approximation of wave equations with random parameters. Technical report 18.2017, EPFL-SB-Institute of Mathematics-Mathicse.Google Scholar
Musharbash, E., Nobile, F. and Zhou, T. (2015), Error analysis of the dynamically orthogonal approximation of time dependent random PDEs , SIAM J. Sci. Comput. 37, A776–A810.CrossRefGoogle Scholar
Neal, R. M. (1996), Priors for infinite networks, in Bayesian Learning for Neural Networks, Vol. 118 of Lecture Notes in Statistics, Springer, pp. 29–53.CrossRefGoogle Scholar
Nguyen, N. C. and Peraire, J. (2016), Gaussian functional regression for output prediction: Model assimilation and experimental design, J. Comput. Phys. 309, 52–68.CrossRefGoogle Scholar
Nguyen, N.-C., Rozza, G. and Patera, A. T. (2009), Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation, Calcolo 46, 157–185.CrossRefGoogle Scholar
Noether, E. (1918), Invariante variationsprobleme, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1918, 235–257.Google Scholar
Noor, A. K. (1981), Recent advances in reduction methods for nonlinear problems, Computers & Structures 13, 31–44.CrossRefGoogle Scholar
Noor, A. K. (1982), On making large nonlinear problems small, Comput. Methods Appl. Mech. Engrg 34, 955–985.CrossRefGoogle Scholar
Noor, A. K. and Peters, J. M. (1980), Reduced basis technique for nonlinear analysis of structures, AIAA J. 18, 455–462.CrossRefGoogle Scholar
Noor, A. K., Balch, C. D. and Shibut, M. A. (1984), Reduction methods for nonlinear steady-state thermal analysis, Internat. J. Numer. Methods Engrg 20, 1323–1348.CrossRefGoogle Scholar
Novati, G., Mahadevan, L. and Koumoutsakos, P. (2019), Controlled gliding and perching through deep-reinforcement-learning, Phys. Rev. Fluids 4, 093902.CrossRefGoogle Scholar
Ohlberger, M. and Rave, S. (2013), Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing, C. R. Math. Acad. Sci. Paris 351, 901–906.CrossRefGoogle Scholar
Ortali, G., Demo, N. and Rozza, G. (2022), A Gaussian process regression approach within a data-driven POD framework for engineering problems in fluid dynamics, Math. Engrg 4, 1.Google Scholar
Pagliantini, C. (2021), Dynamical reduced basis methods for Hamiltonian systems, Numer. Math. 148, 409–448.CrossRefGoogle Scholar
Park, K. H., Jun, S. O., Baek, S. M., Cho, M. H., Yee, K. J. and Lee, D. H. (2013), Reduced-order model with an artificial neural network for aerostructural design optimization, J. Aircraft 50, 1106–1116.CrossRefGoogle Scholar
Patera, A. T. and Rozza, G. (2007), Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations, MIT-Pappalardo Graduate Monographs in Mechanical Engineering, Massachusetts Institute of Technology.Google Scholar
Peherstorfer, B. and Willcox, K. (2015a), Dynamic data-driven reduced-order models, Comput. Methods Appl. Mech. Engrg 291, 21–41.CrossRefGoogle Scholar
Peherstorfer, B. and Willcox, K. (2015b), Online adaptive model reduction for nonlinear systems via low-rank updates , SIAM J. Sci. Comput. 37, A2123–A2150.CrossRefGoogle Scholar
Peherstorfer, B. and Willcox, K. (2016a), Data-driven operator inference for nonintrusive projection-based model reduction, Comput. Methods Appl. Mech. Engrg 306, 196–215.CrossRefGoogle Scholar
Peherstorfer, B. and Willcox, K. (2016b), Dynamic data-driven model reduction: Adapting reduced models from incomplete data, Adv. Model. Simul. Engrg Sci. 3, 11.CrossRefGoogle Scholar
Peherstorfer, B., Butnaru, D., Willcox, K. and Bungartz, H.-J. (2014), Localized discrete empirical interpolation method , SIAM J. Sci. Comput. 36, A168–A192.CrossRefGoogle Scholar
Peng, L. and Mohseni, K. (2016a), Geometric model reduction of forced and dissipative Hamiltonian systems, in 2016 IEEE 55th Conference on Decision and Control (CDC), IEEE, pp. 7465–7470.CrossRefGoogle Scholar
Peng, L. and Mohseni, K. (2016b), Symplectic model reduction of Hamiltonian systems , SIAM J. Sci. Comput. 38, A1–A27.CrossRefGoogle Scholar
Peterson, J. S. (1989), The reduced basis method for incompressible viscous flow calculations , SIAM J. Sci. Statist. Comput. 10, 777–786.CrossRefGoogle Scholar
Pichi, F., Ballarin, F., Rozza, G. and Hesthaven, J. S. (2021), An artificial neural network approach to bifurcating phenomena in computational fluid dynamics. Available at arXiv:2109.10765.Google Scholar
Polyuga, R. V. and van der Schaft, A. (2010), Structure preserving model reduction of port-Hamiltonian systems by moment matching at infinity, Automatica J. IFAC 46, 665–672.Google Scholar
Polyuga, R. V. and van der Schaft, A. (2011), Structure preserving moment matching for port-Hamiltonian systems: Arnoldi and Lanczos, IEEE Trans. Automat. Control 56, 1458–1462.CrossRefGoogle Scholar
Porsching, T. A. and Lee, M. L. (1987), The reduced basis method for initial value problems , SIAM J. Numer. Anal. 24, 1277–1287.CrossRefGoogle Scholar
Prud’homme, C., Rovas, D. V., Veroy, K., Machiels, L., Maday, Y., Patera, A. T. and Turinici, G. (2002), Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods, J. Fluids Engrg 124, 70–80.CrossRefGoogle Scholar
Quarteroni, A., Manzoni, A. and Negri, F. (2016), Reduced Basis Methods for Partial Differential Equations, Springer.CrossRefGoogle Scholar
Quarteroni, A., Rozza, G. and Manzoni, A. (2011), Certified reduced basis approximation for parametrized PDE and applications, J. Math Industry 1, 3.CrossRefGoogle Scholar
Raissi, M., Perdikaris, P. and Karniadakis, G. E. (2019), Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 378, 686–707.CrossRefGoogle Scholar
Rasmussen, C. E. and Williams, C. K. I. (2005), Gaussian Processes for Machine Learning, Adaptive Computation and Machine Learning, MIT Press.CrossRefGoogle Scholar
Ravindran, S. S. (2000), A reduced-order approach for optimal control of fluids using proper orthogonal decomposition, Internat. J. Numer. Methods Fluids 34, 425–448.3.0.CO;2-W>CrossRefGoogle Scholar
Reis, T. and Stykel, T. (2008), A survey on model reduction of coupled systems, in Model Order Reduction: Theory, Research Aspects and Applications, Vol. 13 of Mathematics in Industry, Springer, pp. 133–155.CrossRefGoogle Scholar
Reiss, J., Schulze, P., Sesterhenn, J. and Mehrmann, V. (2018), The shifted proper orthogonal decomposition: A mode decomposition for multiple transport phenomena , SIAM J. Sci. Comput. 40, A1322–A1344.CrossRefGoogle Scholar
Rewienski, M. (2003), A trajectory piecewise-linear approach to model order reduction of nonlinear dynamical systems. PhD thesis, Massachusetts Institute of Technology.CrossRefGoogle Scholar
Rim, D., Moe, S. and LeVeque, R. J. (2018), Transport reversal for model reduction of hyperbolic partial differential equations , SIAM/ASA J. Uncertain. Quantif. 6, 118–150.CrossRefGoogle Scholar
Rosenblatt, F. (1958), The perceptron: A probabilistic model for information storage and organization in the brain, Psychol. Rev. 65 6, 386–408.CrossRefGoogle Scholar
Rowley, C. W. and Marsden, J. E. (2000), Reconstruction equations and the Karhunen–Loève expansion for systems with symmetry, Phys. D 142, 1–19.CrossRefGoogle Scholar
Rowley, C. W., Kevrekidis, I. G., Marsden, J. E. and Lust, K. (2003), Reduction and reconstruction for self-similar dynamical systems, Nonlinearity 16, 1257–1275.CrossRefGoogle Scholar
Rozza, G. (2005), Reduced-basis methods for elliptic equations in sub-domains with a posteriori error bounds and adaptivity, Appl. Numer. Math. 55, 403–424.CrossRefGoogle Scholar
Rozza, G. (2014), Fundamentals of reduced basis method for problems governed by parametrized PDEs and applications, in Separated Representation and PGD Based Model Reduction: Fundamentals and Applications (Chinesta, F. and Ladevèze, P., eds), Vol. 554 of CISM International Centre for Mechanical Sciences, Springer, pp. 153–227.Google Scholar
Rozza, G., Huynh, D. B. P. and Patera, A. T. (2008), Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: Application to transport and continuum mechanics, Arch. Comput. Methods Engrg 15, 229–275.CrossRefGoogle Scholar
Rozza, G., Huynh, D. B. P., Nguyen, N. C. and Patera, A. T. (2009), Real-time reliable simulation of heat transfer phenomena, in ASME 2009 Heat Transfer Summer Conference collocated with the InterPACK09 and 3rd Energy Sustainability Conferences, American Society of Mechanical Engineers, pp. 851–860.Google Scholar
Rozza, G., Malik, H., Demo, N., Tezzele, M., Girfoglio, M., Stabile, G. and Mola, A. (2018), Advances in reduced order methods for parametric industrial problems in computational fluid dynamics, in ECCOMAS ECFD 7: Proceedings of 6th European Conference on Computational Mechanics (ECCM 6) and 7th European Conference on Computational Fluid Dynamics (ECFD 7) (Owen, R. et al., eds), pp. 59–76.Google Scholar
Rumelhart, D. E., Hinton, G. E. and Williams, R. J. (1986), Learning representations by back-propagating errors, Nature 323, 533–536.CrossRefGoogle Scholar
Ryckelynck, D. (2009), Hyper-reduction of mechanical models involving internal variables, Internat. J. Numer. Methods Engrg 77, 75–89.CrossRefGoogle Scholar
Salam, A. (2005), On theoretical and numerical aspects of symplectic Gram–Schmidt-like algorithms, Numer. Algorithms 39, 437–462.CrossRefGoogle Scholar
Sammon, J. W. (1969), A nonlinear mapping for data structure analysis, IEEE Trans. Comput. C-18, 401–409.CrossRefGoogle Scholar
Sanderse, B. (2020), Non-linearly stable reduced-order models for incompressible flow with energy-conserving finite volume methods, J. Comput. Phys. 421, 109736.CrossRefGoogle Scholar
Sanz-Serna, J. M. and Calvo, M. P. (1994), Numerical Hamiltonian Problems, Vol. 7 of Applied Mathematics and Mathematical Computation, Chapman & Hall.CrossRefGoogle Scholar
Sapsis, T. P. and Lermusiaux, P. F. J. (2009), Dynamically orthogonal field equations for continuous stochastic dynamical systems, Phys. D 238, 2347–2360.CrossRefGoogle Scholar
Schein, A., Carlberg, K. T. and Zahr, M. J. (2021), Preserving general physical properties in model reduction of dynamical systems via constrained-optimization projection, Internat. J. Numer. Methods. Engrg 122, 3368–3399.CrossRefGoogle Scholar
Schmid, P. J. (2010), Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech. 656, 5–28.CrossRefGoogle Scholar
Schölkopf, B., Smola, A. and Müller, K.-R. (1997), Kernel principal component analysis, in Artificial Neural Networks (ICANN’97) (Gerstner, W. et al., eds), Springer, pp. 583–588.Google Scholar
Semeraro, O., Bellani, G. and Lundell, F. (2012), Analysis of time-resolved PIV measurements of a confined turbulent jet using POD and Koopman modes, Exp. Fluids 53, 1203–1220.CrossRefGoogle Scholar
Sesterhenn, J. and Shahirpour, A. (2019), A characteristic dynamic mode decomposition, Theoret. Comput. Fluid Dynamics 33, 281–305.CrossRefGoogle Scholar
Shashkov, M. (1996), Conservative Finite-Difference Methods on General Grids, Symbolic and Numeric Computation series, CRC Press.Google Scholar
Stegeman, P., Ooi, A. and Soria, J. (2015), Proper orthogonal decomposition and dynamic mode decomposition of under-expanded free-jets with varying nozzle pressure ratios, in Instability and Control of Massively Separated Flows (Theofilis, V. and Soria, J., eds), Vol. 107 of Fluid Mechanics and its Applications, Springer, pp. 85–90.CrossRefGoogle Scholar
Taddei, T. (2020), A registration method for model order reduction: Data compression and geometry reduction , SIAM J. Sci. Comput. 42, A997–A1027.CrossRefGoogle Scholar
Tenenbaum, J. B., de Silva, V. and Langford, J. C. (2000), A global geometric framework for nonlinear dimensionality reduction, Science 290, 2319–2323.CrossRefGoogle ScholarPubMed
Tezzele, M., Ballarin, F. and Rozza, G. (2018a), Combined parameter and model reduction of cardiovascular problems by means of active subspaces and POD-Galerkin methods, in Mathematical and Numerical Modeling of the Cardiovascular System and Applications, SEMA SIMAI Springer Series, Springer, pp. 185–207.CrossRefGoogle Scholar
Tezzele, M., Demo, N., Gadalla, M., Mola, A. and Rozza, G. (2018b), Model order reduction by means of active subspaces and dynamic mode decomposition for parametric hull shape design hydrodynamics, in Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research, IOS Press, pp. 569–576.Google Scholar
Uzunca, M., Karasözen, B. and Yıldız, S. (2021), Structure-preserving reduced-order modeling of Korteweg–de Vries equation, Math. Comput. Simul. 188, 193–211.CrossRefGoogle Scholar
van der Maaten, L. and Hinton, G. (2008), Visualizing data using t-SNE, J. Mach. Learning Res. 9, 2579–2605.Google Scholar
van der Schaft, A. (2006), Port-Hamiltonian systems: An introductory survey, in International Congress of Mathematicians 2006, European Mathematical Society, pp. 1339–1365.Google Scholar
Veroy, K., Prud’homme, C., Rovas, D. V. and Patera, A. T. (2003), A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations, in Proceedings of the 16th AIAA Computational Fluid Dynamics Conference. AIAA paper 2003-3847.Google Scholar
Walton, S., Hassan, O. and Morgan, K. (2013), Reduced order modelling for unsteady fluid flow using proper orthogonal decomposition and radial basis functions, Appl. Math. Model. 37, 8930–8945.CrossRefGoogle Scholar
Wang, Q., Hesthaven, J. S. and Ray, D. (2019), Non-intrusive reduced order modeling of unsteady flows using artificial neural networks with application to a combustion problem, J. Comput. Phys. 384, 289–307.CrossRefGoogle Scholar
Wang, Q., Ripamonti, N. and Hesthaven, J. S. (2020), Recurrent neural network closure of parametric POD-Galerkin reduced-order models based on the Mori–Zwanzig formalism, J. Comput. Phys. 410, 109402.CrossRefGoogle Scholar
Wang, Y., Yao, H. and Zhao, S. (2016), Auto-encoder based dimensionality reduction, Neurocomput. 184, 232–242.CrossRefGoogle Scholar
Weickum, G., Eldred, M. S. and Maute, K. (2008), A multi-point reduced-order modeling approach of transient structural dynamics with application to robust design optimization, Struct. Multidiscip. Optim. 38, 599.CrossRefGoogle Scholar
Weinberger, K. Q., Sha, F. and Saul, L. K. (2004), Learning a kernel matrix for nonlinear dimensionality reduction, in Proceedings of the Twenty-First International Conference on Machine Learning (ICML ’04), Association for Computing Machinery, p. 106.Google Scholar
Welper, G. (2017), Interpolation of functions with parameter dependent jumps by transformed snapshots , SIAM J. Sci. Comput. 39, A1225–A1250.CrossRefGoogle Scholar
Willcox, K. and Peraire, J. (2002), Balanced model reduction via the proper orthogonal decomposition, AIAA J. 40, 2323–2330.CrossRefGoogle Scholar
Wirtz, D., Sorensen, D. C. and Haasdonk, B. (2014), A posteriori error estimation for DEIM reduced nonlinear dynamical systems , SIAM J. Sci. Comput. 36, A311–A338.CrossRefGoogle Scholar
Wolf, T., Lohmann, B., Eid, R. and Kotyczka, P. (2010), Passivity and structure preserving order reduction of linear port-Hamiltonian systems using Krylov subspaces, Eur. J. Control 16, 401–406.CrossRefGoogle Scholar
Xiao, D., Fang, F., Pain, C. and Hu, G. (2015), Non-intrusive reduced order modelling of the Navier–Stokes equations based on RBF interpolation, Internat. J. Numer. Methods Fluids 79, 580–595.CrossRefGoogle Scholar
Xu, H. (2003), An SVD-like matrix decomposition and its applications, Linear Algebra Appl. 368, 1–24.CrossRefGoogle Scholar
Yano, M., Patera, A. T. and Urban, K. (2014), A space-time hp-interpolation-based certified reduced basis method for Burgers’ equation, Math. Models Methods Appl. Sci. 24, 1903–1935.CrossRefGoogle Scholar
Yu, J., Yan, C. and Guo, M. (2019), Non-intrusive reduced-order modeling for fluid problems: A brief review , Proc. Inst. Mech. Engrs G. J. Aerosp. Engrg 233, 5896–5912.CrossRefGoogle Scholar
Zahm, O., Constantine, P., Prieur, C. and Marzouk, Y. (2020), Gradient-based dimension reduction of multivariate vector-valued functions , SIAM J. Sci. Comput. 42, A534–A558.CrossRefGoogle Scholar
Zancanaro, M., Mrosek, M., Stabile, G., Othmer, C. and Rozza, G. (2021), Hybrid neural network reduced order modelling for turbulent flows with geometric parameters, Fluids 6, 296.CrossRefGoogle Scholar
Zhang, Z. and Zha, H. (2004), Principal manifolds and nonlinear dimension reduction via local tangent space alignment , SIAM J. Sci. Comput. 26, 313–338.CrossRefGoogle Scholar
Zimmermann, R., Peherstorfer, B. and Willcox, K. (2018), Geometric subspace updates with applications to online adaptive nonlinear model reduction, SIAM J. Matrix Anal. Appl. 39, 234–261.CrossRefGoogle Scholar
Zou, X., Conti, M., Díez, P. and Auricchio, F. (2018), A non-intrusive proper generalized decomposition scheme with application in biomechanics, Internat. J. Numer. Methods. Engrg 113, 230–251.CrossRefGoogle Scholar