The introduction of computational models in cardiovascular sciences has been progressively bringing new and unique tools for the investigation of the physiopathology. Together with the dramatic improvement of imaging and measuring devices on one side, and of computational architectures on the other one, mathematical and numerical models have provided a new, clearly noninvasive, approach for understanding not only basic mechanisms but also patient-specific conditions, and for supporting the design and the development of new therapeutic options. The terminology in silico is, nowadays, commonly accepted for indicating this new source of knowledge added to traditional in vitro and in vivo investigations. The advantages of in silico methodologies are basically the low cost in terms of infrastructures and facilities, the reduced invasiveness and, in general, the intrinsic predictive capabilities based on the use of mathematical models. The disadvantages are generally identified in the distance between the real cases and their virtual counterpart required by the conceptual modeling that can be detrimental for the reliability of numerical simulations.

%B Cardiovascular Mechanics %I CRC Press %P 54 %G eng %U https://www.taylorfrancis.com/books/e/9781315280288/chapters/10.1201%2Fb21917-5 %& Computational methods in cardiovascular mechanics %0 Book Section %B Spectral and High Order Methods for Partial Differential Equations %D 2017 %T Certified Reduced Basis Method for Affinely Parametric Isogeometric Analysis NURBS Approximation %A Denis Devaud %A Gianluigi Rozza %XIn this work we apply reduced basis methods for parametric PDEs to an isogeometric formulation based on

NURBS. The motivation for this work is an integrated and complete work pipeline from CAD to parametrization

of domain geometry, then from full order to certified reduced basis solution. IsoGeometric Analysis

(IGA) is a growing research theme in scientic computing and computational mechanics, as well as reduced

basis methods for parametric PDEs. Their combination enhances the solution of some class of problems,

especially the ones characterized by parametrized geometries we introduced in this work. For a general

overview on Reduced Basis (RB) methods we recall [7, 15] and on IGA [3]. This work wants to demonstrate

that it is also possible for some class of problems to deal with ane geometrical parametrization combined

with a NURBS IGA formulation. This is what this work brings as original ingredients with respect to other

works dealing with reduced order methods and IGA (set in a non-affine formulation, and using a POD [2]

sampling without certication: see for example for potential flows [12] and for Stokes flows [17]). In this work

we show a certication of accuracy and a complete integration between IGA formulation and parametric

certified greedy RB formulation. Section 2 recalls the abstract setting for parametrized PDEs, Section 3

recalls IGA setting, Section 4 deals with RB formulation, and Section 5 illustrates two numerical examples in heat transfer with different parametrization.

%B Spectral and High Order Methods for Partial Differential Equations %7 Bittencourt, Dumont, Hesthaven. (Eds). %I Springer %C Heildeberg %V 119 %@ 978-3-319-65869-8 %G eng %0 Journal Article %J SIAM Journal on Numerical Analysis %D 2017 %T On a certified smagorinsky reduced basis turbulence model %A Rebollo, T.C. %A E.D. Ávila %A Marmol, M.G. %A Francesco Ballarin %A Gianluigi Rozza %B SIAM Journal on Numerical Analysis %V 55 %P 3047-3067 %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85039928218&doi=10.1137%2f17M1118233&partnerID=40&md5=221d9cd2bcc74121fcef93efd9d3d76c %R 10.1137/17M1118233 %0 Journal Article %J Journal of Computational Physics %D 2017 %T Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: Applications to Coanda effect in cardiology %A Giuseppe Pitton %A Annalisa Quaini %A Gianluigi Rozza %K Parametrized Navier–Stokes equations %K Reduced basis method %K Stability of flows %K Symmetry breaking bifurcation %XWe focus on reducing the computational costs associated with the hydrodynamic stability of solutions of the incompressible Navier–Stokes equations for a Newtonian and viscous fluid in contraction–expansion channels. In particular, we are interested in studying steady bifurcations, occurring when non-unique stable solutions appear as physical and/or geometric control parameters are varied. The formulation of the stability problem requires solving an eigenvalue problem for a partial differential operator. An alternative to this approach is the direct simulation of the flow to characterize the asymptotic behavior of the solution. Both approaches can be extremely expensive in terms of computational time. We propose to apply Reduced Order Modeling (ROM) techniques to reduce the demanding computational costs associated with the detection of a type of steady bifurcations in fluid dynamics. The application that motivated the present study is the onset of asymmetries (i.e., symmetry breaking bifurcation) in blood flow through a regurgitant mitral valve, depending on the Reynolds number and the regurgitant mitral valve orifice shape.

This 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.

%B Springer Briefs in Mathematics %7 1 %I Springer %C Switzerland %P 135 %@ 978-3-319-22469-5 %G eng %6 1 %0 Journal Article %D 2014 %T Comparison between reduced basis and stochastic collocation methods for elliptic problems %A Peng Chen %A Alfio Quarteroni %A Gianluigi Rozza %X The stochastic collocation method (Babuška et al. in SIAM J Numer Anal 45(3):1005-1034, 2007; Nobile et al. in SIAM J Numer Anal 46(5):2411-2442, 2008a; SIAM J Numer Anal 46(5):2309-2345, 2008b; Xiu and Hesthaven in SIAM J Sci Comput 27(3):1118-1139, 2005) has recently been applied to stochastic problems that can be transformed into parametric systems. Meanwhile, the reduced basis method (Maday et al. in Comptes Rendus Mathematique 335(3):289-294, 2002; Patera and Rozza in Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations Version 1.0. Copyright MIT, http://augustine.mit.edu, 2007; Rozza et al. in Arch Comput Methods Eng 15(3):229-275, 2008), primarily developed for solving parametric systems, has been recently used to deal with stochastic problems (Boyaval et al. in Comput Methods Appl Mech Eng 198(41-44):3187-3206, 2009; Arch Comput Methods Eng 17:435-454, 2010). In this work, we aim at comparing the performance of the two methods when applied to the solution of linear stochastic elliptic problems. Two important comparison criteria are considered: (1), convergence results of the approximation error; (2), computational costs for both offline construction and online evaluation. Numerical experiments are performed for problems from low dimensions O (1) to moderate dimensions O (10) and to high dimensions O (100). The main result stemming from our comparison is that the reduced basis method converges better in theory and faster in practice than the stochastic collocation method for smooth problems, and is more suitable for large scale and high dimensional stochastic problems when considering computational costs. %I Springer %G en %U http://urania.sissa.it/xmlui/handle/1963/34727 %1 34916 %2 Mathematics %4 1 %$ Approved for entry into archive by Maria Pia Calandra (calapia@sissa.it) on 2015-10-26T11:39:14Z (GMT) No. of bitstreams: 0 %R 10.1007/s10915-013-9764-2 %0 Journal Article %J Annals of Nuclear Energy %D 2014 %T Comparison of a Modal Method and a Proper Orthogonal Decomposition approach for multi-group time-dependent reactor spatial kinetics %A Alberto Sartori %A Davide Baroli %A Antonio Cammi %A Davide Chiesa %A Lelio Luzzi %A Roberto R. Ponciroli %A Ezio Previtali %A Marco E. Ricotti %A Gianluigi Rozza %A Monica Sisti %XIn this paper, two modelling approaches based on a Modal Method (MM) and on the Proper Orthogonal Decomposition (POD) technique, for developing a control-oriented model of nuclear reactor spatial kinetics, are presented and compared. Both these methods allow developing neutronics description by means of a set of ordinary differential equations. The comparison of the outcomes provided by the two approaches focuses on the capability of evaluating the reactivity and the neutron flux shape in different reactor configurations, with reference to a TRIGA Mark II reactor. The results given by the POD-based approach are higher-fidelity with respect to the reference solution than those computed according to the MM-based approach, in particular when the perturbation concerns a reduced region of the core. If the perturbation is homogeneous throughout the core, the two approaches allow obtaining comparable accuracy results on the quantities of interest. As far as the computational burden is concerned, the POD approach ensures a better efficiency rather than direct Modal Method, thanks to the ability of performing a longer computation in the preprocessing that leads to a faster evaluation during the on-line phase.

%B Annals of Nuclear Energy %I Elsevier %V 71 %P 229 %8 09/2014 %G en %U http://urania.sissa.it/xmlui/handle/1963/35039 %1 35270 %2 Physics %4 1 %$ Approved for entry into archive by Maria Pia Calandra (calapia@sissa.it) on 2015-11-18T12:07:02Z (GMT) No. of bitstreams: 0 %& 217 %R 10.1016/j.anucene.2014.03.043 %0 Journal Article %J Comptes Rendus Mathematique. Volume 351, Issue 15-16, August 2013, Pages 593-598 %D 2013 %T A combination between the reduced basis method and the ANOVA expansion: On the computation of sensitivity indices %A Denis Devaud %A Andrea Manzoni %A Gianluigi Rozza %K Partial differential equations %XWe consider a method to efficiently evaluate in a real-time context an output based on the numerical solution of a partial differential equation depending on a large number of parameters. We state a result allowing to improve the computational performance of a three-step RB-ANOVA-RB method. This is a combination of the reduced basis (RB) method and the analysis of variations (ANOVA) expansion, aiming at compressing the parameter space without affecting the accuracy of the output. The idea of this method is to compute a first (coarse) RB approximation of the output of interest involving all the parameter components, but with a large tolerance on the a posteriori error estimate; then, we evaluate the ANOVA expansion of the output and freeze the least important parameter components; finally, considering a restricted model involving just the retained parameter components, we compute a second (fine) RB approximation with a smaller tolerance on the a posteriori error estimate. The fine RB approximation entails lower computational costs than the coarse one, because of the reduction of parameter dimensionality. Our result provides a criterion to avoid the computation of those terms in the ANOVA expansion that are related to the interaction between parameters in the bilinear form, thus making the RB-ANOVA-RB procedure computationally more feasible.

%B Comptes Rendus Mathematique. Volume 351, Issue 15-16, August 2013, Pages 593-598 %I Elsevier %G en %U http://hdl.handle.net/1963/7389 %1 7434 %2 Mathematics %4 1 %# MAT/05 ANALISI MATEMATICA %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2014-06-19T08:56:09Z No. of bitstreams: 1 Devaud_Manzoni_Rozza_2013.pdf: 564002 bytes, checksum: 4c93e74468534915513e6805d440dee9 (MD5) %R 10.1016/j.crma.2013.07.023