The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows

By Kevin T. Carlberg, Charbel Farhat, Julien Cortial, David Amsallem. Journal of Computational Physics, 2012

656 citations recorded by the source corpus.

The Gauss-Newton with approximated tensors (GNAT) method is a nonlinear model-reduction method that operates on fully discretized computational models. It achieves dimension reduction by a Petrov-Galerkin projection associated with residual minimization; it delivers computational efficiency by a hyper-reduction procedure based on the 'gappy POD' technique. Originally presented in Ref. [1], where it was applied to implicit nonlinear structural-dynamics models, this method is further developed here and applied to the solution of a benchmark turbulent viscous flow problem. To begin, this paper develops global state-space error bounds that justify the method's design and highlight its advantages in terms of minimizing components of these error bounds. Next, the paper introduces a 'sample mesh' concept that enables a distributed, computationally efficient implementation of the GNAT method in finite-volume-based computational-fluid-dynamics (CFD) codes. The suitability of GNAT for parameterized problems is highlighted with the solution of an academic problem featuring moving discontinuities. Finally, the capability of this method to reduce by orders of magnitude the core-hours required for large-scale CFD computations, while preserving accuracy, is demonstrated with the simulation of turbulent flow over the Ahmed body. For an instance of this benchmark problem with over 17 million degrees of freedom, GNAT outperforms several other nonlinear model-reduction methods, reduces the required computational resources by more than two orders of magnitude, and delivers a solution that differs by less than 1% from its high-dimensional counterpart.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/45a249ff9a5e6074ab4eacd71bf53653b576fa3c

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Citations in the corpus

POD-DEIM model order reduction for strain softening viscoplasticity (2017, methodology): “To compute the derivative of the approximated force vector (25) with respect to the displacement vector, several methods are suggested in [27].”
A New Selection Operator for the Discrete Empirical Interpolation Method - Improved A Priori Error Bound and Extensions (2015, neutral): “, [21, 5, 3, 39, 14].”
Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations (2015, methodology): “We remark, however, that such (dense) third-order tensor may entail high storage costs; recent alternative approaches make use of a DEIM [39] or hyper-reduction techniques, such as gappy POD [40], or again compressive tensor approximations [41] to alleviate this problem.”
Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy‐based mesh sampling and weighting for computational efficiency (2014, methodology): “Other related approaches include discrete analogs to the empirical and best points interpolation methods [23, 24] and the Gauss–Newton with approximated tensors method [25], which was shown in [26] to speed up HDM simulations of turbulent flows by orders of magnitude while delivering ROM solutions t”
Compressive Sensing and Low-Rank Libraries for Classification of Bifurcation Regimes in Nonlinear Dynamical Systems (2013, methodology): “Including additional modes, such as the shift mode [32], to capture transients between qualitatively diﬀerent ﬂow regimes has resulted in additional methods such as double POD [42], and the GaussNewton with approximated tensors (GNAT) method [13] and trust-region POD [18, 6].”
Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders (2018, methodology): “As in previous works that considered applying LSPG on affine trial subspaces [13, 16, 12], we propose to solve the nonlinear least-squares problem (3.”
On closures for reduced order models—A spectrum of first-principle to machine-learned avenues (2021, neutral): “63,216,217 Examples of hyperreduction include the empirical interpolation method (EIM), 86 its discrete version (DEIM), 90,218 the gappy POD, 100,111,219 and the missing point estimation (MPE), 108,220 where the approximation is performed using sampling techniques.”
Interpolation-based reduced-order modelling for steady transonic flows via manifold learning (2014, methodology): “State-of-the-art methods belonging to this class of ROMs are the Gauss–Newton with approximated tensors (GNAT) method (Carlberg et al. 2013) and discrete empirical interpolation (DEIM) (Chaturantabut and Sorensen 2010).”
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, methodology): “Related or similar approaches have been developed specifically for hyper reduction at the semidiscrete [22, 23] and fully discrete [10, 11, 24, 25] levels.”
POD-Galerkin reduced order methods for CFD using Finite Volume Discretisation: vortex shedding around a circular cylinder (2017, methodology): “Yet, if richer reduced spaces are used, possible alternatives to the present approach could be using EIM-DEIM [43, 4] approaches or Gappy-POD [8].”
A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems (2015, neutral): “Among this class of meth-ods, the missing point estimation [19] and Gauss Newton with approximated tensors (GNAT) [66] methods both build upon the gappy POD interpolation method [87]; the empirical interpolation method (EIM) [26] and its discrete variant, the discrete empirical interpolation method ”