Reduced basis method for finite volume approximations of parametrized linear evolution equations

Haasdonk, Bernard; Ohlberger, Mario

doi:10.1051/m2an:2008001

Cited by 355 publications

(479 citation statements)

References 34 publications

(47 reference statements)

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“…We can now define our error bounds in terms of these two ingredients as it can readily be proven [52,57] that for all µ ∈ P,…”

Section: A Posteriori Error Bounds For the Parabolic Casementioning

confidence: 99%

“…A purely greedy approach [52] may encounter difficulties best treated by including elements of the proper orthogonal decomposition selection process [57]. Hence, to capture the causality associated with the evolution, the sampling method combines the proper orthogonal decomposition, see Section 3.2.1, for the time-trajectories, with the greedy procedure in the parameter space [52,156,144] to enable the efficient treatment of the higher dimensions and extensive ranges of parameter variation.…”

mentioning

confidence: 99%

“…A slightly different version of the POD-greedy algorithm was proposed in [57] where the error trajectories u k δ (µ n ) − u k rb (µ n ) are added instead of u k δ (µ n ) to avoid the second POD-compression.…”

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confidence: 99%

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Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Hesthaven¹,

Rozza²,

Stamm³

2016

SpringerBriefs in Mathematics

841

1,221

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The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-22470-1International audienceThis book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples

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“…We can now define our error bounds in terms of these two ingredients as it can readily be proven [52,57] that for all µ ∈ P,…”

Section: A Posteriori Error Bounds For the Parabolic Casementioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Hesthaven¹,

Rozza²,

Stamm³

2016

SpringerBriefs in Mathematics

841

1,221

View full text Add to dashboard Cite

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“…It was further developed in reduced basis models for elliptic PDEs in [218], the steady incompressible Navier-Stokes equations in [217], and parabolic PDEs in [111,112]. It has since been applied in conjunction with POD methods [61,122,230] and rational interpolation methods [79]. The key advantage of the greedy approach is that the search over the parameter space embodies the structure of the problem, so that the underlying system dynamics guide the selection of appropriate parameter samples.…”

Section: Adaptive Parameter Sampling Via Greedy Searchmentioning

confidence: 99%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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Abstract. Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for which the equations governing the system behavior depend on a set of parameters. Examples include parameterized partial differential equations and large-scale systems of parameterized ordinary differential equations. The goal of parametric model reduction is to generate low-cost but accurate models that characterize system response for different values of the parameters. This paper surveys state-of-the-art methods in projection-based parametric model reduction, describing the different approaches within each class of methods for handling parametric variation and providing a comparative discussion that lends insights to potential advantages and disadvantages in applying each of the methods. We highlight the important role played by parametric model reduction in design, control, optimization, and uncertainty quantification-settings that require repeated model evaluations over different parameter values.

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“…The resulting reduced space X N aims to represent a suitable space of approximation for both values of α. In the parametrized case, by using greedy enrichment we aim at saving a part of the offline computational costs: indeed, in a standard POD-Greedy procedure (see [39]), each new evaluation of the parameter α requires the computation of the associated finite element solutions for each time instant n ∈ N T . In our application, this would require about 8 h on 256 processors.…”

Section: Application Of the Greedy Enriched Algorithm With Perturbed mentioning

confidence: 99%

Reduced Numerical Approximation of Reduced Fluid-Structure Interaction Problems With Applications in Hemodynamics

Colciago

Deparis

2018

Front. Appl. Math. Stat.

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This paper deals with fast simulations of the hemodynamics in large arteries by considering a reduced model of the associated fluid-structure interaction problem, which in turn allows an additional reduction in terms of the numerical discretisation. The resulting method is both accurate and computationally cheap. This goal is achieved by means of two levels of reduction: first, we describe the model equations with a reduced mathematical formulation which allows to write the fluid-structure interaction problem as a Navier-Stokes system with non-standard boundary conditions; second, we employ numerical reduction techniques to further and drastically lower the computational costs. The non standard boundary condition is of a generalized Robin type, with a boundary mass and boundary stiffness terms accounting for the arterial wall compliance. The numerical reduction is obtained coupling two well-known techniques: the proper orthogonal decomposition and the reduced basis method, in particular the greedy algorithm. We start by reducing the numerical dimension of the problem at hand with a proper orthogonal decomposition and we measure the system energy with specific norms; this allows to take into account the different orders of magnitude of the state variables, the velocity and the pressure. Then, we introduce a strategy based on a greedy procedure which aims at enriching the reduced discretization space with low offline computational costs. As application, we consider a realistic hemodynamics problem with a perturbation in the boundary conditions and we show the good performances of the reduction techniques presented in the paper. The results obtained with the numerical reduction algorithm are compared with the one obtained by a standard finite element method.The gains obtained in term of CPU time are of three orders of magnitude.

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Reduced basis method for finite volume approximations of parametrized linear evolution equations

Cited by 355 publications

References 34 publications

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

Reduced Numerical Approximation of Reduced Fluid-Structure Interaction Problems With Applications in Hemodynamics

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