Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations

By Gianluigi Rozza, Dinh Bao Phuong Huynh, Anthony T. Patera. 2007

478 citations recorded by the source corpus.

Subfields: Computational Mechanics, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/2358d597964c830ae853d9055d9e07ed9e3ee6dd

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In Computational Mechanics

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In Statistical and Nonlinear Physics

Citations in the corpus

The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows (2012, unknown): “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].”
Survey of multifidelity methods in uncertainty propagation, inference, and optimization (2018, methodology): “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.”
Model Order Reduction for Linear and Nonlinear Systems: A System-Theoretic Perspective (2014, background): “This complements recent surveys on generating reduced-order models for parameter-dependent problems [37, 53, 169] which we do not consider here.”
Reduced basis methods for partial differential equations with stochastic influences (2013, background): “Besides linear elliptic [73, 77] and parabolic equations [40, 76], also more complicated quadratically nonlinear problems have been reduced [24, 86, 96].”
A PRIORI CONVERGENCE OF THE GREEDY ALGORITHM FOR THE PARAMETRIZED REDUCED BASIS METHOD (2012, background): “We briefly introduce such an estimator and refer to [7] for further details.”
POD-Galerkin reduced order methods for CFD using Finite Volume Discretisation: vortex shedding around a circular cylinder (2017, methodology): “For more details about the different methods the reader may see [33, 9, 16, 30, 10, 12].”
Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems (2011, methodology): “We ﬁrst recall the reduced basis (RB) recipe for second-order coercive elliptic PDEs (see [4] for a recent review and further references): Given μ ∈ D ⊂ R P , we determine y e ≡ y e ( μ ) We next introduce a truth ﬁnite element approximation space Y ⊂ Y of very large dimension N ; note that Y shall…”
A new error bound for reduced basis approximation of parabolic partial differential equations (2012, methodology): “(The offline stage for computation of the inf-sup lower bound by the Successive Constraint Method [5,8] does require special treatment, in particular due to the norm ||| · |||X ,δ.”
Reduced basis approximation and a posteriori error bounds for 4D-Var data assimilation (2018, unknown): “terial parameters often naturally satisfy an affine parameter dependence, and even geometric parameters can often be treated after introducing suitable affine mappings onto a reference domain [47].”
Multipole Graph Neural Operator for Parametric Partial Differential Equations (2020, methodology): “Machine learning based methods enable this since they are much faster to evaluate and require only observational data to train, in stark contrast to traditional Galerkin methods [45] and classical reduced order models[35].”
REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS (2011, methodology): “Once the reduced basis spaces are deﬁned we can then construct our SCM lower bound for the stability factor.”