Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods

Prud'Homme, Christophe; Rovas, Dimitrios; Veroy, Karen; Machiels, L.; Maday, Yvon; Patera, Anthony T.; Turinici, Gabriel

doi:10.1115/1.1448332

Cited by 490 publications

(547 citation statements)

References 23 publications

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“…Substantial recent efforts have been devoted to the development of techniques to formulate a posteriori error estimation procedures and rigorous error bounds for outputs of interest [127]. These a posteriori error bounds subsequently allow for the certification of the output of the reduced basis model for any parameter value.…”

Section: Historical Background and Perspectivesmentioning

confidence: 99%

“…This text does not seek to replace review articles on the topics (such as [127,144,129,142,143]) but aims to widen the perspectives on reduced basis methods and at providing an integration presentation. This text begins with a basic setting to introduce the general elements of certified reduced basis methods for elliptic affine coercive problems with linear compliant outputs and then gradually widens the field with extensions to non-affine, non-compliant, non-coercive operators, geometrical parametrization and time dependent problems.…”

Section: Prefacementioning

confidence: 99%

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Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Hesthaven¹,

Rozza²,

Stamm³

2016

SpringerBriefs in Mathematics

865

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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Section: Historical Background and Perspectivesmentioning

confidence: 99%

Section: Prefacementioning

confidence: 99%

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Hesthaven¹,

Rozza²,

Stamm³

2016

SpringerBriefs in Mathematics

865

1,221

View full text Add to dashboard Cite

show abstract

“…Thus we have nonlinear parameter dependence, which is fundamentally different from most reduced basis applications. The equations are not affine in their parameter dependence either; see Prud'homme et al (2002) for affine, linear parameter dependence, and Barrault et al (2004) for non-affine parameter dependence.…”

Section: Parameterizing the Geometriesmentioning

confidence: 99%

“…To get an estimate of how good our solution is we need a posteriori error estimation. Based on the theory developed in Prud'homme et al (2002), and following the strategy of Rovas (2002), the lower and upper output bounds, s Ϫ (u N ) and s ϩ (u N ), for the compliant output…”

Section: A Posteriori Error Estimationmentioning

confidence: 99%

“…The method takes its roots in domain decomposition methods and reduced basis discretizations (Fink & Rheinboldt (1983), Noor & Peters (1980), Prud'homme et al (2002)), and its applications extend to, for example, control and optimization problems. The basic idea is to first decompose the computational domain into a series of subdomains that are similar to a few reference domains (or generic computational parts).…”

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confidence: 99%

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A reduced basis element method for the steady Stokes problem: Application to hierarchical flow systems

Løvgren¹,

Maday²,

Rønquist³

2006

MIC

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The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations. The method takes its roots in domain decomposition methods and reduced basis discretizations (Fink & Rheinboldt (1983), Noor & Peters (1980), Prud'homme et al. (2002)), and its applications extend to, for example, control and optimization problems. The basic idea is to first decompose the computational domain into a series of subdomains that are similar to a few reference domains (or generic computational parts). Associated with each reference domain are precomputed solutions corresponding to the same governing partial differential equation, but solved for different choices of some underlying parameter. In this work, the parameters are representing the geometric shape associated with a computational part. The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mapped from the reference domain for the part to the actual domain. We extend earlier work (Maday & Rønquist (2002), ) in this direction to solve incompressible fluid flow problems governed by the steady Stokes equations. Particular focus is given to constructing the basis functions, to the mapping of the velocity fields, to satisfying the inf-sup condition, and to "gluing" the local solutions together in the multidomain case (Belgacem et al. (2000)). We also demonstrate an algorithm for choosing the most efficient precomputed solutions. Two-dimensional examples are presented for pipes, bifurcations, and couplings of pipes and bifurcations in order to simulate hierarchical flow systems.

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Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Parabolic PDEs: Application to Real‐Time Bayesian Parameter Estimation

Nguyen

Rozza

Huynh

et al. 2010

Large‐Scale Inverse Problems and Quantification of Uncertainty

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In this chapter we consider reduced basis approximation and a posteriori error estimation for linear functional outputs of affinely parametrized linear and non-linear parabolic partial differential equations. The essential ingredients are Galerkin projection onto a low-dimensional space associated with a smooth "parametric manifold" -dimension reduction; efficient and effective Greedy and POD-Greedy sampling methods for identification of optimal and numerically stable approximations -rapid convergence; rigorous and sharp a posteriori error bounds (and associated stability factors) for the linear-functional outputs of interest -certainty; and Offline-Online computational decomposition strategies -minimum marginal cost for high performance in the real-time/embedded (e.g., parameter estimation, control) and many-query (e.g., design optimization, uncertainty quantification Boyaval et al. (2008)

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Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods

Cited by 490 publications

References 23 publications

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

A reduced basis element method for the steady Stokes problem: Application to hierarchical flow systems

Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Parabolic PDEs: Application to Real‐Time Bayesian Parameter Estimation

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