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On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition

Toni LassilaAndrea ManzoniGianluigi Rozza — 2012

ESAIM: Mathematical Modelling and Numerical Analysis

A new approach for computationally efficient estimation of stability factors for parametric partial differential equations is presented. The general parametric bilinear form of the problem is approximated by two affinely parametrized bilinear forms at different levels of accuracy (after an empirical interpolation procedure). The successive constraint method is applied on the coarse level to obtain a lower bound for the stability factors, and this...

On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition

Toni LassilaAndrea ManzoniGianluigi Rozza — 2012

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

A new approach for computationally efficient estimation of stability factors for parametric partial differential equations is presented. The general parametric bilinear form of the problem is approximated by two affinely parametrized bilinear forms at different levels of accuracy (after an empirical interpolation procedure). The successive constraint method is applied on the coarse level to obtain a lower bound for the stability factors, and this bound is extended to the fine level by adding a proper...

On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition

Toni LassilaAndrea ManzoniGianluigi Rozza — 2012

ESAIM: Mathematical Modelling and Numerical Analysis

A new approach for computationally efficient estimation of stability factors for parametric partial differential equations is presented. The general parametric bilinear form of the problem is approximated by two affinely parametrized bilinear forms at different levels of accuracy (after an empirical interpolation procedure). The successive constraint method is applied on the coarse level to obtain a lower bound for the stability factors, and this...

Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty

Toni LassilaAndrea ManzoniAlfio QuarteroniGianluigi Rozza — 2013

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We review the optimal design of an arterial bypass graft following either a (i) boundary optimal control approach, or a (ii) shape optimization formulation. The main focus is quantifying and treating the uncertainty in the residual flow when the hosting artery is not completely occluded, for which the worst-case in terms of recirculation effects is inferred to correspond to a strong orifice flow through near-complete occlusion.A worst-case optimal control approach is applied to the steady Navier-Stokes...

Generalized Reduced Basis Methods and n-width Estimates for the Approximation of the Solution Manifold of Parametric PDEs

Toni LassilaAndrea ManzoniAlfio QuarteroniGianluigi Rozza — 2013

Bollettino dell'Unione Matematica Italiana

The set of solutions of a parameter-dependent linear partial differential equation with smooth coefficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold. We focus on operators showing an affine parametric dependence, expressed as a linear combination of parameter-independent operators through some smooth, parameter-dependent scalar functions....

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