On the stability of the reduced basis method for Stokes equations in parametrized domains

By Gianluigi Rozza, Karen Veroy. 2007

302 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/935e235263f37f3228597be47006cd56940add55

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Statistical and Nonlinear Physics

In Statistics and Probability

Citations in the corpus

Stabilized reduced basis method for parametrized advection-diffusion PDEs (2014, methodology): “Finally, we want to say that the SCM can be efficiently applied also to compute the inf-sup lower bound for non-coercive problems [45, 52].”
Reduced basis methods for Stokes equations in domains with non-affine parameter dependence (2009, methodology): “At this step we have adopted the second alternative option introduced in [ 30 ], where all the velocity reduced basis functions are µ (on-line) independent (they depend only on samples µ j used to store”
POD-Galerkin reduced order methods for CFD using Finite Volume Discretisation: vortex shedding around a circular cylinder (2017, methodology): “Among the methods using also pressure modes, some work directly on the system composed by the continuity and momentum equation and use ad hoc stabilization techniques to enforce the well posedness of the problem [3, 35, 7, 32]; other methods work with the momentum equation, without the gradient of p”
An efficient computational framework for reduced basis approximation and a posteriori error estimation of parametrized Navier–Stokes flows (2014, methodology): “[28] for the proof): βBr,N (μ) ≥ βBr,N (μ) > 0 ∀μ ∈ D, (5.”
Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations (2007, background): “Much work focuses on the stationary incompressible (quadratically nonlinear) Navier-Stokes equations [29,50,57] of incompressible fluid flow: suitable stable approximations are considered in [57,67, 114,123,136, 139 ]; rigorous a posteriori error estimation — within the general Brezzi-Rappaz-Raviart”
Certified reduced basis approximation for parametrized partial differential equations and applications (2011, background): “An interesting case of noncoercive problems is given by Stokes problems where approximation stability is guaranteed by the fullfillment of an equivalent inf-sup stability condition on the pressure term with RB approximation spaces properly enriched [41, 42].”
POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition (2021, background): “…minutes, required to achieve in the We remark that ensuring ROM stability in the classical POD-Galerkin framework usually requires additional computational eﬀorts, such as a suitable enrichment of the velocity reduced basis, and a consequent increase of the size of the ROM; see, e.g., [6, 7, 58].”
Surrogate and reduced-order modeling: a comparison of approaches for large-scale statistical inverse problems [Chapter 7] (2010, background): “Other choices forΨ are also possible, such as one that minimizes a least-squares weighted-residual (Bui-Thanh et al. 2008; Maday et al. 2002; Rovas 2003; Rozza and Veroy 2006), or one that includes output information through use of adjoint solutions (Lall et al. 2002; Willcox and Peraire 2002).”
Survey of multifidelity methods in uncertainty propagation, inference, and optimization (2018, methodology): “Another basis generation approach is the reducedbasis method [182, 86, 183, 181], where orthogonalized carefully selected snapshots”
Model Reduction for Large-Scale Systems with High-Dimensional Parametric Input Space (2008, methodology): “Recently, the minimum-residual approach has been successfully used in the reduced-basis context [23, 27, 28].”
Model Order Reduction in Fluid Dynamics: Challenges and Perspectives (2014, methodology): “[87, 100, 109] for their complete derivation.”
Shape optimization for viscous flows by reduced basis methods and free‐form deformation (2012, methodology): “The RB framework applied to the Stokes problem approximation has been used to study a simple bypass design problem in presence of either affine or nonaffine shape parametrization, treating some shape geometrical dimensions as design parameters (see [34, 35, 40]).”
Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations (2007, background): “(We also address here the constuction of div-stable [26, 27] RB (Navier)-Stokes approximations [122, 136].”
Numerical solution of parametrized Navier–Stokes equations by reduced basis methods (2007, background): “Then it has been proven in [24] that βN (μ) ≥ βh(μ) ≥ β0 > 0,∀μ ∈ D .”