%0 Journal Article
%D 2014
%T Model Order Reduction in Fluid Dynamics: Challenges and Perspectives
%A Toni Lassila
%A Andrea Manzoni
%A Alfio Quarteroni
%A Gianluigi Rozza
%X This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities - which are mainly related either to nonlinear convection terms and/or some geometric variability - that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and-in the unsteady case - long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.
%I Springer
%G en
%1 34923
%2 Mathematics
%4 1
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%R 10.1007/978-3-319-02090-7_9
%0 Report
%D 2013
%T A Reduced Computational and Geometrical Framework for Inverse Problems in Haemodynamics
%A Toni Lassila
%A Andrea Manzoni
%A Alfio Quarteroni
%A Gianluigi Rozza
%I SISSA
%G en
%1 6571
%2 Mathematics
%4 1
%# MAT/08 ANALISI NUMERICA
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%0 Report
%D 2013
%T A reduced-order strategy for solving inverse Bayesian identification problems in physiological flows
%A Toni Lassila
%A Andrea Manzoni
%A Alfio Quarteroni
%A Gianluigi Rozza
%I SISSA
%G en
%1 6555
%2 Mathematics
%4 1
%# MAT/08 ANALISI NUMERICA
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%0 Report
%D 2013
%T Reduction Strategies for Shape Dependent Inverse Problems in Haemodynamics
%A Toni Lassila
%A Andrea Manzoni
%A Gianluigi Rozza
%I SISSA
%G en
%1 6554
%2 Mathematics
%4 1
%# MAT/08 ANALISI NUMERICA
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%0 Journal Article
%J Mathematical Modelling and Numerical Analysis, in press, 2012-13
%D 2012
%T Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty
%A Toni Lassila
%A Andrea Manzoni
%A Alfio Quarteroni
%A Gianluigi Rozza
%K shape optimization
%X 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,\\r\\nfor which the worst-case in terms of recirculation e ffects is inferred to correspond to a strong ori fice flow through near-complete occlusion. A worst-case optimal control approach is applied to the steady\\r\\nNavier-Stokes equations in 2D to identify an anastomosis angle and a cu ed shape that are robust with respect to a possible range of residual \\r\\nflows. We also consider a reduced order modelling framework\\r\\nbased on reduced basis methods in order to make the robust design problem computationally feasible. The results obtained in 2D are compared with simulations in a 3D geometry but without model\\r\\nreduction or the robust framework.
%B Mathematical Modelling and Numerical Analysis, in press, 2012-13
%I Cambridge University Press
%G en
%U http://hdl.handle.net/1963/6337
%1 6267
%2 Mathematics
%4 1
%# MAT/08 ANALISI NUMERICA
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%0 Book Section
%B Springer, Indam Series, Vol. 4, 2012
%D 2012
%T Generalized reduced basis methods and n-width estimates for the approximation of the solution manifold of parametric PDEs
%A Toni Lassila
%A Andrea Manzoni
%A Alfio Quarteroni
%A Gianluigi Rozza
%K solution manifold
%X The set of solutions of a parameter-dependent linear partial di fferential equation with smooth coe fficients 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 affi ne parametric dependence, expressed as a linear combination of parameter-independent operators through some smooth, parameter-dependent scalar functions. In the case that the parameter-dependent operator has a dominant term in its affi ne expansion, one can prove the existence of exponentially convergent uniform approximation spaces for the entire solution manifold. These spaces can be constructed without any assumptions on the parametric regularity of the manifold \\r\\nonly spatial regularity of the solutions is required. The exponential convergence rate is then inherited by the generalized reduced basis method. We provide a numerical example related to parametrized elliptic\\r\\nequations con rming the predicted convergence rates.
%B Springer, Indam Series, Vol. 4, 2012
%I Springer
%G en
%U http://hdl.handle.net/1963/6340
%1 6270
%2 Mathematics
%4 1
%# MAT/08 ANALISI NUMERICA
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