Certified real‐time solution of the parametrized steady incompressible Navier–Stokes equations: rigorous reduced‐basis a posteriori error bounds

By Karen Veroy, Anthony T. Patera. 2005

403 citations recorded by the source corpus.

Subfields: Computational Mechanics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/f4e0d6e468d800fce48fc8fcfaa0f742563eba22

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In Computational Mechanics

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Citations in the corpus

On the stability of the reduced basis method for Stokes equations in parametrized domains (2007, background): “See [31] for Navier–Stokes reduced basis formulation and [30,2] for non-affine parametric dependence in elliptic problems.”
REDUCED BASIS METHOD FOR FINITE VOLUME APPROXIMATIONS OF PARAMETRIZED LINEAR EVOLUTION EQUATIONS (2008, methodology): “The RB-methodology is also applicable to systems of equations such as the incompressible Navier-Stokes equations [40].”
Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations (2007, background): “For the Navier-Stokes equations (and for nonlinear equations more generally) we can not appeal to any simple monotonicity arguments; our focus is thus on the computational (quantitative) realization of the general Brezzi-Rappaz-Raviart (“BRR”) a posteriori theory [28, 34] — and development of associ”
Reduced basis methods for partial differential equations with stochastic influences (2013, methodology): “error analysis of state and linear output functional is very similar to [96] and [18], whereas the analysis of quadratic and statistical outputs is strongly based upon the results of chapter 5 and [45].”
Survey of multifidelity methods in uncertainty propagation, inference, and optimization (2018, background): “In some settings, error bounds or error estimates are available for the reduced-model outputs [182, 209, 86], and it may be possible to translate these error estimates on the model outputs into error estimates on the outer-loop result.”
Certified Real-Time Solution of Parametrized Partial Differential Equations (2005, background): “The first issue is that, to evaluate our error bound N (μ), we must appeal to our inf–sup lower bound; however, in the nonlinear case, this inf–sup lower bound, β̃N (μ), is defined with respect to the linearized state uNmax(μ) [22].”
The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows (2012, neutral): “These include linear time-invariant systems [2, 3], linear stationary systems whose operators are affine functions of the input parameters [4, 5], and a class of nonlinear systems characterized by quadratic nonlinearities [6, 7, 8].”
Hessian‐based model reduction for large‐scale systems with initial‐condition inputs (2008, methodology): “The approach taken in [21, 22] is to replace these constraints with error estimators, so that the full-scale model does not need to be invoked during solution of the optimization problem.”
Computational Reduction for Parametrized PDEs: Strategies and Applications (2012, neutral): “in [14,34,44,50,54].”
An efficient reduced‐order modeling approach for non‐linear parametrized partial differential equations (2008, methodology): “In particular, the reduced-basis method has been successfully developed for (a) parametrized linear elliptic and parabolic PDEs that are affine† in the parameter [7, 15, 16, 17, 18, 19] and (b) nonlinear PDEs that are at most quadratically nonlinear in the field variable [20, 21, 22].”
Bayesian Model and Dimension Reduction for Uncertainty Propagation: Applications in Random Media (2017, background): “A prominent role in these efforts, at least in the context of PDE-based models, is held by reduced-basis techniques [56, 63, 29] which are based on the identification of a low-dimensional linear subspace in the solution vector space on which a Galerkin projection of the governing equations is attemp”
An efficient computational framework for reduced basis approximation and a posteriori error estimation of parametrized Navier–Stokes flows (2014, methodology): “1 of [31], by considering the operator g(W,V ; μ) = Ã(W,V ; μ)− F̃ (V ; μ), and remarking that dg(Z;W,V ; μ) = dÃ(Z;W,V ; μ).”
Numerical solution of parametrized Navier–Stokes equations by reduced basis methods (2007, methodology): “In [34] they developed rigorous a posteriori error estimation and bounds for real time computation based on the Brezzi-Rappaz-Raviart theory.”