A successive constraint linear optimization method for lower bounds of parametric coercivity and inf–sup stability constants

Huynh, Dinh Bao Phuong; Rozza, Gianluigi; Sen, Subhabrata; Patera, Anthony T.

doi:10.1016/j.crma.2007.09.019

Cited by 233 publications

(326 citation statements)

References 7 publications

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“…The online part is then reduced to provide a lower bound α LB (µ) of the coercivity constant α δ (µ) for each new parameter value µ ∈ P with an operation count that is independent of the dimension N δ . The original algorithm [67] was subsequently refined, extended to non-coercive problems and generalized to complex matrices in [25,26,27,58,61]. …”

Section: The Successive Constraint Methods (Scm)mentioning

confidence: 99%

“…It can be shown that Y n UB ⊂ Y ⊂ Y n LB (µ) (see [67] for the proof). Consequently, Y n UB , Y and Y n LB (µ) are nested as…”

Section: Lbmentioning

confidence: 96%

“…To address the challenges associated with the need to estimate α LB (µ) for more general problems, [67] proposed a local minimization approach, known as the successive constraint method (SCM). As for the multi-parameter Min-θ-approach, the SCM is an offline/online procedure where generalized eigenvalue problems of size N δ need to be solved during the offline phase.…”

Section: The Successive Constraint Methods (Scm)mentioning

confidence: 99%

See 2 more Smart Citations

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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Section: The Successive Constraint Methods (Scm)mentioning

confidence: 99%

“…It can be shown that Y n UB ⊂ Y ⊂ Y n LB (µ) (see [67] for the proof). Consequently, Y n UB , Y and Y n LB (µ) are nested as…”

Section: Lbmentioning

confidence: 96%

Section: The Successive Constraint Methods (Scm)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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“…and β LB (µ) is a computable lower bound for the inf-sup constant β(µ) [22]. An alternative expression of the error estimator (28) is given by…”

Section: Reduced Basis Space Assembling and Error Estimationmentioning

confidence: 99%

“…Moreover, also the dual norm of the residual appearing in (28) can be computed efficiently by using the Offline-Online procedure [31,37]. Since the map T (·; µ) (22) obtained by FFD method is a polynomial map, the tensor ν T (x; µ) is not affinely parametrized in the sense of (31). Hence, an intermediate step is necessary in order to recover the affinity assumption and thus the possibility of computing the RB solution through an Offline/Online decomposition.…”

Section: Offline-online Computational Strategymentioning

confidence: 99%

Shape optimization for viscous flows by reduced basis methods and free‐form deformation

Manzoni

Quarteroni

Rozza

2011

Numerical Methods in Fluids

107

108

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SUMMARYIn this paper we further develop an approach previosuly introduced in [23] for shape optimization that combines a suitable low-dimensional parametrization of the geometry (yielding a geometrical reduction) with reduced basis methods (yielding a reduction of computational complexity). More precisely, freeform deformation techniques are considered for the geometry description and its parametrization, while reduced basis methods are used upon a finite element discretization to solve systems of parametrized partial differential equations. This allows an efficient flow field computation and cost functional evaluation during the iterative optimization procedure, resulting in effective computational savings with respect to usual shape optimization strategies. This approach is very general and can be applied to a broad variety of problems. In this paper we apply it to find the optimal shape of aorto-coronaric bypass anastomoses based on vorticity minimization in the down-field region. Blood flows in the coronary arteries are modelled using Stokes equations; afterwards, results have been verified in feedback using Navier-Stokes equations.

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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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A successive constraint linear optimization method for lower bounds of parametric coercivity and inf–sup stability constants

Cited by 233 publications

References 7 publications

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Shape optimization for viscous flows by reduced basis methods and free‐form deformation

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

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