REDUCED BASIS METHOD FOR FINITE VOLUME APPROXIMATIONS OF PARAMETRIZED LINEAR EVOLUTION EQUATIONS

By Bernard Haasdonk, Mario Ohlberger. 2008

445 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/d9bdb184607c2482933e12a3f64ba529ec40bc6b

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In Computational Mechanics

In Numerical Analysis

In Statistical and Nonlinear Physics

Citations in the corpus

Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Parabolic PDEs: Application to Real‐Time Bayesian Parameter Estimation (2010, methodology): “In the ﬁrst part we present a rather general RB formulation for linear parabolic equations: our development combines earlier work in the parabolic case (Grepl and Patera 2005; Haasdonk and Ohlberger 2008) and elliptic context (Prud’homme et al .”
CERTIFIED REDUCED BASIS METHODS FOR NONAFFINE LINEAR TIME-VARYING AND NONLINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS (2012, methodology): “We therefore propose a novel more efficient approach to generate the collateral reduced basis space which is based on a POD(in time)/Greedy(in parameter space) search.(20) A large number of model order reduction (MOR) techniques(2,8,9,29,34,39,44,50) have been developed to treat (nonlinear) time-dep”
Computational Reduction for Parametrized PDEs: Strategies and Applications (2012, unknown): “The reader interested in time-dependent problems can instead refer to [19, 21] or to the more recent review provided in [45].”
Survey of multifidelity methods in uncertainty propagation, inference, and optimization (2018, methodology): “Depending on the problem of interest, these reduced-basis models can be equipped with cheap a posteriori error estimators for the reduced-model outputs [107, 86, 92, 182, 206, 215].”
Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing (2013, unknown): “For linear problems, the POD-Greedy algorithm [5] is by now an established reduction approach which has recently been proved to be optimal in the sense that exponential or algebraic convergence rates of the Kolmogorov n-width are maintained by the algorithm [3].”
Model Order Reduction in Fluid Dynamics: Challenges and Perspectives (2014, methodology): “For sampling in time POD-greedy strategies have been proposed for linear evolution equations in [56], the viscous nonlinear Burgers’ equation in [93], and the Navier-Stokes equations in [127].”
REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS (2011, methodology): “The certified reduced basis method — which yields reduced order models equipped with rigorous error bounds — is well developed for linear parametrized parabolic partial differential equations [20,22,26,52].”
Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation (2009, methodology): “We follow [13] and combine the POD (Proper Orthogonal Decomposition) in tk — to capture the causality associated with our evolution equation — with the Greedy procedure [10,34,31] in ν — to treat efficiently the higher dimensions and more extensive ranges of parameter variation.”
Comparison of the reduced-basis and POD a posteriori error estimators for an elliptic linear-quadratic optimal control problem (2011, background): “1, but the basis functions themselves are obtained by performing a POD (in time) for this identified parameter (see [15,16]).”
A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems (2015, unknown): “It has since been applied in conjunction with POD methods [61, 122, 230] and rational interpolation methods [79].”
Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations (2007, background): “The exploration of the “parameter + time” framework in the important context of affine linear (stable) parabolic PDEs — such as the heat equation and the passive scalar convection-diffusion equation (also the Black-Scholes equation of derivative theory [118]) — is carried out in [52,55, 60 ,133]; ma”
Reduced basis methods for partial differential equations with stochastic influences (2013, unknown): “This procedure is denoted as POD-Greedy [42].”
Stabilized reduced basis method for parametrized advection-diffusion PDEs (2014, methodology): “More precisely, we construct the reduced basis in the time-dependent case following the so called POD-Greedy approach [11, 21, 22, 25].”
Data-driven reduced order modeling for time-dependent problems (2019, methodology): “For time-dependent problems, required to capture the causality with the evolution over time, a combined version has been proposed as the POD-greedy approach [16, 18].”
Certified reduced basis approximation for parametrized partial differential equations and applications (2011, methodology): “However, spectral element approaches [27, 28], finite volume [22], and other traditional discretization methods may be considered too.”
Convergence Rates of the POD–Greedy Method (2013, methodology): “As a practical and wellperforming alternative, we suggested and introduced the POD-Greedy algorithm in our work [6].”
The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows (2012, methodology): “that greedy sampling methods tailored for model reduction [38, 6, 39, 40, 33] could alternatively be used to determine Dtrain.”