We propose a model reduction procedure for rapid and reliable solution of parameterized hyperbolic partial differential equations. Due to the presence of parameter-dependent shock waves and contact discontinuities, these problems are extremely challenging for traditional model reduction approaches based on linear approximation spaces. The main ingredients of the proposed approach are (i) an adaptive space-time registration-based data compression procedure to align local features in a fixed reference domain, (ii) a space-time Petrov–Galerkin (minimum residual) formulation for the computation of the mapped solution, and (iii) a hyper-reduction procedure to speed up online computations. We present numerical results for a Burgers model problem and a shallow water model problem, to empirically demonstrate the potential of the method.
Keywords: Parameterized hyperbolic partial differential equations, model order reduction, data compression
@article{M2AN_2021__55_1_99_0,
author = {Taddei, Tommaso and Zhang, Lei},
title = {Space-time registration-based model reduction of parameterized one-dimensional hyperbolic {PDEs}},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {99--130},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {1},
doi = {10.1051/m2an/2020073},
mrnumber = {4216836},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2020073/}
}
TY - JOUR AU - Taddei, Tommaso AU - Zhang, Lei TI - Space-time registration-based model reduction of parameterized one-dimensional hyperbolic PDEs JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 99 EP - 130 VL - 55 IS - 1 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2020073/ DO - 10.1051/m2an/2020073 LA - en ID - M2AN_2021__55_1_99_0 ER -
%0 Journal Article %A Taddei, Tommaso %A Zhang, Lei %T Space-time registration-based model reduction of parameterized one-dimensional hyperbolic PDEs %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 99-130 %V 55 %N 1 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2020073/ %R 10.1051/m2an/2020073 %G en %F M2AN_2021__55_1_99_0
Taddei, Tommaso; Zhang, Lei. Space-time registration-based model reduction of parameterized one-dimensional hyperbolic PDEs. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 1, pp. 99-130. doi: 10.1051/m2an/2020073
[1] , , and , Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749–1779. | MR | Zbl | DOI
[2] , , and , Discontinuous Galerkin solution of the Reynolds-averaged Navier-Stokes and turbulence model equations–. Comput. Fluids 34 (2005) 507–540. | Zbl | DOI
[3] and , A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131 (1997) 267–279. | MR | Zbl | DOI
[4] , and , The proper orthogonal decomposition in the analysis of turbulent flows. Ann. Rev. Fluid Mech. 25 (1993) 539–575. | MR | DOI
[5] , , , , and , Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal. 43 (2011) 1457–1472. | MR | Zbl | DOI
[6] , and , Finite dimensional approximation of nonlinear problems. Numer. Math. 36 (1980) 1–25. | MR | Zbl | DOI
[7] , and , (Parametrized) first order transport equations: realization of optimally stable Petrov-Galerkin methods. SIAM J. Sci. Comput. 41 (2019) A592–A621. | MR | DOI
[8] , and , Proper orthogonal decomposition extensions for parametric applications in compressible aerodynamics. In: 21st AIAA Applied Aerodynamics Conference (2003) 4213.
[9] and , Numerical analysis for nonlinear and bifurcation problems. Handbook Numer. Anal. 5 (1997) 487–637. | MR
[10] , , and , The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows. J. Comput. Phys. 242 (2013) 623–647. | MR | Zbl | DOI
[11] and , Une méthode combinée d’éléments finis à deux grilles/bases réduites pour l’approximation des solutions d’une EDP paramétrique. C. R. Math. 347 (2009) 435–440. | MR | Zbl | DOI
[12] , and , Petrov-Galerkin projection-based model reduction with an optimized test space. In: AIAA Scitech 2020 Forum (2020) 1562.
[13] , and , Double greedy algorithms: reduced basis methods for transport dominated problems. ESAIM: M2AN 48 (2014) 623–663. | MR | Zbl | Numdam | DOI
[14] , , and , Arbitrary Lagrangian-Eulerian methods. Encycl. Comput. Mech. Second Edition (2017) 1–23.
[15] and , Karhunen-Loeve procedure for gappy dat. JOSA A 12 (1995) 1657–1664. | DOI
[16] , and , Structure-preserving, stability, and accuracy properties of the energy-conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models. Int. J. Numer. Methods Eng. 102 (2015) 1077–1110. | MR | DOI
[17] , , , and , Reduced basis’ acquisition by a learning process for rapid on-line approximation of solution to PDE’s: laminar flow past a backstep. Arch. Comput. Methods Eng. 25 (2018) 131–141. | MR | DOI
[18] , and , A reduced basis method for the wave equation. Int. J. Comput. Fluid Dyn. 34 (2020) 139–146. | MR | DOI
[19] , and , On the stability of projection-based model order reduction for convection-dominated laminar and turbulent flows. J. Comput. Phys. 419 (2020) 109681. | MR | DOI
[20] and , Reduced order modeling for nonlinear structural analysis using gaussian process regression. Comput. Methods Appl. Mech. Eng. 341 (2018) 807–826. | MR | DOI
[21] , and , Certified Reduced Basis Methods for Parametrized Partial Differential Equations. SpringerBriefs in Mathematics. Springer, New York (2016). | MR
[22] and , Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Vol. 54 of Texts in Applied Mathematics Springer, New York (2007). | MR | Zbl
[23] , and , An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J. Comput. Phys. 14 (1974) 227–253. | MR | Zbl | DOI
[24] and , Advection modes by optimal mass transfer. Phys. Rev. E 89 (2014) 022923. | DOI
[25] and , Solving Least Squares Problems. Siam 161 (1974). | MR | Zbl
[26] and , Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders. J. Comput. Phys. 404 (2020) 108973. | MR | DOI
[27] , Numerical Methods for Conservation Laws. Vol. 3 of Lectures in Mathematics. ETH Zürich. Springer, Basel (1992). | MR | Zbl
[28] , and , A blackbox reduced-basis output bound method for noncoercive linear problems, edited by and . In: Studies in Mathematics and its Applications. Elsevier Science B. V.(2001) 533–569. | MR | Zbl
[29] and , Mathematical Foundations of Elasticity. Courier Corporation (1994). | MR
[30] , , , and , Dimensionality reduction and reduced-order modeling for traveling wave physics. Theor. Comput. Fluid Dyn. 34 (2020) 385–400. | MR | DOI
[31] and , Arbitrary Lagrangian Eulerian framework for efficient projection-based reduction of convection dominated nonlinear flows. In: APS Division of Fluid Dynamics Meeting Abstracts (2017).
[32] and , Model order reduction for stochastic dynamical systems with continuous symmetries. SIAM J. Sci. Comput. 40 (2018) A1669–A1695. | MR | DOI
[33] and , Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing. C. R. Math. 351 (2013) 901–906. | MR | Zbl | DOI
[34] and , Reduced basis methods: success, limitations and future challenges. Preprint (2015). | arXiv
[35] and , An LP empirical quadrature procedure for parametrized functions. C. R. Math. 355 (2017) 1161–1167. | MR | DOI
[36] , Model reduction for transport-dominated problems via online adaptive bases and adaptive sampling. SIAM J. Sci. Comput. 42 (2020) A2803–A2836. | MR | DOI
[37] and , Sub-cell shock capturing for discontinuous Galerkin methods. In: 44th AIAA Aerospace Sciences Meeting and Exhibit. (2006) 112.
[38] , and , Reduced Basis Methods for Partial Differential Equations: An Introduction. Vol. 92 of La Matematica per il 3+2. Springer, Cham (2016). | MR | Zbl
[39] , , and , The shifted proper orthogonal decomposition: a mode decomposition for multiple transport phenomena. SIAM J. Sci. Comput. 40 (2018) A1322–A1344. | MR | DOI
[40] , Mathematical Statistics and Data Analysis. Cengage Learning (2006). | Zbl
[41] , and , Manifold approximations via transported subspaces: model reduction for transport-dominated problems. Preprint (2019). | arXiv | MR
[42] and , Reconstruction equations and the Karhunen-Loève expansion for systems with symmetry. Phys. D: Nonlinear Phenom. 142 (2000) 1–19. | MR | Zbl | DOI
[43] , Turbulence and the dynamics of coherent structures. I. Coherent structures. Q. Appl. Math. 45 (1987) 561–571. | MR | Zbl | DOI
[44] , The many proofs of an identity on the norm of oblique projections. Numer. Algorithms 42 (2006) 309–323. | MR | Zbl | DOI
[45] , An offline/online procedure for dual norm calculations of parameterized functionals: empirical quadrature and empirical test spaces. Adv. Comput. Math. 45 (2019) 2429–2462. | MR | DOI
[46] , A registration method for model order reduction: data compression and geometry reduction. SIAM J. Sci. Comput. 42 (2020) A997–A10274. | MR | DOI
[47] , and , Reduced basis techniques for nonlinear conservation laws. ESAIM:M2AN 49 (2015) 787–814. | MR | Numdam | DOI
[48] and , An improved error bound for reduced basis approximation of linear parabolic problems. Math. Comput. 83 (2014) 1599–1615. | MR | Zbl | DOI
[49] , Model reduction using proper orthogonal decomposition. Lecture Notes, Institute of Mathematics and Scientific Computing, University of Graz. see math.uni-konstanz.de/numerik/personen/volkwein/teaching/POD-Vorlesung.pdf (2011) 1025.
[50] , Interpolation of functions with parameter dependent jumps by transformed snapshots. SIAM J. Sci. Comput. 39 (2017) A1225–A1250. | MR | Zbl | DOI
[51] , Scattered Data Approximation. Cambridge University Press (2004) 17. | MR | Zbl
[52] and , Some observations on Babuška and Brezzi theories. Numer. Math. 94 (2003) 195–202. | MR | Zbl | DOI
[53] , A space-time Petrov-Galerkin certified reduced basis method: application to the Boussinesq equations. SIAM J. Sci. Comput. 36 (2014) A232–A266. | MR | Zbl | DOI
[54] , Discontinuous Galerkin reduced basis empirical quadrature procedure for model reduction of parametrized nonlinear conservation laws. Adv. Comput. Math. 45 (2019) 2287–2320. | MR | Zbl | DOI
[55] and , An optimization-based approach for high-order accurate discretization of conservation laws with discontinuous solutions. J. Comput. Phys. 365 (2018) 105–134. | MR | Zbl | DOI
[56] , and , Implicit shock tracking using an optimization-based high-order discontinuous galerkin method. J. Comput. Phys. 410 (2020) 109385. | MR | Zbl | DOI
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