Space-time registration-based model reduction of parameterized one-dimensional hyperbolic PDEs

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.

[1]  Marco Luciano Savini,et al.  Discontinuous Galerkin solution of the Reynolds-averaged Navier–Stokes and k–ω turbulence model equations , 2005 .

[2]  J. Marsden,et al.  Reconstruction equations and the Karhunen—Loéve expansion for systems with symmetry , 2000 .

[3]  Charles L. Lawson,et al.  Solving least squares problems , 1976, Classics in applied mathematics.

[4]  Mario Ohlberger,et al.  Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing , 2013 .

[5]  Benjamin Peherstorfer,et al.  Model reduction for transport-dominated problems via online adaptive bases and adaptive sampling , 2018, SIAM J. Sci. Comput..

[6]  Per-Olof Persson,et al.  An optimization-based approach for high-order accurate discretization of conservation laws with discontinuous solutions , 2017, J. Comput. Phys..

[7]  Anthony T. Patera,et al.  An LP empirical quadrature procedure for parametrized functions , 2017 .

[8]  Patrick Gallinari,et al.  Reduced Basis’ Acquisition by a Learning Process for Rapid On-line Approximation of Solution to PDE’s: Laminar Flow Past a Backstep , 2017 .

[9]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[10]  A. Quarteroni,et al.  Reduced Basis Techniques For Nonlinear Conservation Laws , 2015 .

[11]  Masayuki Yano,et al.  A Space-Time Petrov-Galerkin Certified Reduced Basis Method: Application to the Boussinesq Equations , 2014, SIAM J. Sci. Comput..

[12]  Gerrit Welper,et al.  Interpolation of Functions with Parameter Dependent Jumps by Transformed Snapshots , 2017, SIAM J. Sci. Comput..

[13]  P. Holmes,et al.  The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows , 1993 .

[14]  Volker Mehrmann,et al.  The Shifted Proper Orthogonal Decomposition: A Mode Decomposition for Multiple Transport Phenomena , 2015, SIAM J. Sci. Comput..

[15]  Karsten Urban,et al.  An improved error bound for reduced basis approximation of linear parabolic problems , 2013, Math. Comput..

[16]  Ludmil T. Zikatanov,et al.  Some observations on Babu\vs}ka and Brezzi theories , 2003, Numerische Mathematik.

[17]  Finite dimensional approximation of nonlinear problems , 1980 .

[18]  L. Sirovich.  Turbulence and the dynamics of coherent structures. I. Coherent structures , 1987 .

[19]  Y. Maday,et al.  Une méthode combinée d'éléments finis à deux grilles/bases réduites pour l'approximation des solutions d'une E.D.P. paramétrique , 2009 .

[20]  Karen Willcox,et al.  Proper orthogonal decomposition extensions for parametric applications in compressible aerodynamics , 2003 .

[21]  Maciej Balajewicz,et al.  Arbitrary Lagrangian Eulerian framework for efficient projection-based reduction of convection dominated nonlinear flows , 2017 .

[22]  Tommaso Taddei,et al.  An offline/online procedure for dual norm calculations of parameterized functionals: empirical quadrature and empirical test spaces , 2018, Advances in Computational Mathematics.

[23]  J. Hesthaven,et al.  Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , 2007 .

[24]  Tommaso Taddei,et al.  A registration method for model order reduction: data compression and geometry reduction , 2019, SIAM J. Sci. Comput..

[25]  Mario Ohlberger,et al.  Reduced Basis Methods: Success, Limitations and Future Challenges , 2015, 1511.02021.

[26]  L. Sirovich.  Turbulence and the dynamics of coherent structures. II. Symmetries and transformations , 1987 .

[27]  Jan S. Hesthaven,et al.  Reduced order modeling for nonlinear structural analysis using Gaussian process regression , 2018, Computer Methods in Applied Mechanics and Engineering.

[28]  Steven L. Brunton,et al.  Dimensionality reduction and reduced-order modeling for traveling wave physics , 2019, 1911.00565.

[29]  Angelo Iollo,et al.  Advection modes by optimal mass transfer. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  D. Rovas,et al.  A blackbox reduced-basis output bound method for noncoercive linear problems , 2002 .

[31]  Benjamin Peherstorfer,et al.  Manifold Approximations via Transported Subspaces: Model reduction for transport-dominated problems , 2019, ArXiv.

[32]  C. W. Hirt,et al.  An Arbitrary Lagrangian-Eulerian Computing Method for All Flow Speeds , 1997 .

[33]  J. Peraire,et al.  Sub-Cell Shock Capturing for Discontinuous Galerkin Methods , 2006 .

[34]  Daniel B. Szyld,et al.  The many proofs of an identity on the norm of oblique projections , 2006, Numerical Algorithms.

[35]  Lawrence Sirovich,et al.  Karhunen–Loève procedure for gappy data , 1995 .

[36]  Karsten Urban,et al.  A reduced basis method for the wave equation , 2019, International Journal of Computational Fluid Dynamics.

[37]  Masayuki Yano,et al.  Discontinuous Galerkin reduced basis empirical quadrature procedure for model reduction of parametrized nonlinear conservation laws , 2019, Advances in Computational Mathematics.

[38]  Karsten Urban,et al.  (Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov-Galerkin Methods , 2018, SIAM J. Sci. Comput..

[39]  Wolfgang Dahmen,et al.  Convergence Rates for Greedy Algorithms in Reduced Basis Methods , 2010, SIAM J. Math. Anal..

[40]  F. Brezzi,et al.  Finite dimensional approximation of nonlinear problems , 1981 .

[41]  S. Rebay,et al.  A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations , 1997 .

[42]  Wolfgang Dahmen,et al.  DOUBLE GREEDY ALGORITHMS: REDUCED BASIS METHODS FOR TRANSPORT DOMINATED PROBLEMS ∗ , 2013, 1302.5072.

[43]  Charbel Farhat,et al.  The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows , 2012, J. Comput. Phys..

[44]  J. Rice.  Mathematical Statistics and Data Analysis , 1988 .

[45]  J. Rappaz,et al.  Numerical analysis for nonlinear and bifurcation problems , 1997 .

[46]  S. Volkwein,et al.  MODEL REDUCTION USING PROPER ORTHOGONAL DECOMPOSITION , 2008 .

[47]  Themistoklis P. Sapsis,et al.  Model Order Reduction for Stochastic Dynamical Systems with Continuous Symmetries , 2017, SIAM J. Sci. Comput..

[48]  Per-Olof Persson,et al.  Implicit shock tracking using an optimization-based, r-adaptive, high-order discontinuous Galerkin method , 2020, J. Comput. Phys..

[49]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

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

[51]  Kookjin Lee,et al.  Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders , 2018, J. Comput. Phys..

[52]  A. Huerta,et al.  Arbitrary Lagrangian–Eulerian Methods , 2004 .

[53]  C. Farhat,et al.  Structure‐preserving, stability, and accuracy properties of the energy‐conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models , 2015 .

[54]  M. Urner.  Scattered Data Approximation , 2016 .

[55]  C. Cesnik,et al.  Petrov-Galerkin Projection-Based Model Reduction with an Optimized Test Space , 2020 .

[56]  T. R. Hughes,et al.  Mathematical foundations of elasticity , 1982 .

[57]  A. Quarteroni,et al.  Reduced Basis Methods for Partial Differential Equations: An Introduction , 2015 .