%0 Journal Article %J Biomechanics and Modeling in Mechanobiology %D 2017 %T Numerical modeling of hemodynamics scenarios of patient-specific coronary artery bypass grafts %A F. Ballarin %A Elena Faggiano %A Andrea Manzoni %A Alfio Quarteroni %A Gianluigi Rozza %A Sonia Ippolito %A Roberto Scrofani %B Biomechanics and Modeling in Mechanobiology %V 16 %P 1373-1399 %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85015065851&doi=10.1007%2fs10237-017-0893-7&partnerID=40&md5=c388f20bd5de14187bad9ed7d9affbd0 %R 10.1007/s10237-017-0893-7 %0 Journal Article %J SIAM/ASA Journal on Uncertainty Quantification %D 2017 %T Reduced Basis Methods for Uncertainty Quantification %A Peng Chen %A Alfio Quarteroni %A Gianluigi Rozza %X

In this work we review a reduced basis method for the solution of uncertainty quantification problems. Based on the basic setting of an elliptic partial differential equation with random input, we introduce the key ingredients of the reduced basis method, including proper orthogonal decomposition and greedy algorithms for the construction of the reduced basis functions, a priori and a posteriori error estimates for the reduced basis approximations, as well as its computational advantages and weaknesses in comparison with a stochastic collocation method [I. Babuška, F. Nobile, and R. Tempone, SIAM Rev., 52 (2010), pp. 317--355]. We demonstrate its computational efficiency and accuracy for a benchmark problem with parameters ranging from a few to a few hundred dimensions. Generalizations to more complex models and applications to uncertainty quantification problems in risk prediction, evaluation of statistical moments, Bayesian inversion, and optimal control under uncertainty are also presented to illustrate how to use the reduced basis method in practice. Further challenges, advancements, and research opportunities are outlined.


Read More: http://epubs.siam.org/doi/abs/10.1137/151004550

%B SIAM/ASA Journal on Uncertainty Quantification %V 5 %P 869 %8 08/2017 %G eng %N 1 %9 reviewed %& 813 %0 Report %D 2016 %T A fast virtual surgery platform for many scenarios haemodynamics of patient-specific coronary artery bypass grafts %A F. Ballarin %A Elena Faggiano %A Andrea Manzoni %A Gianluigi Rozza %A Alfio Quarteroni %A Sonia Ippolito %A Roberto Scrofani %A Carlo Antona %X A fast computational framework is devised to the study of several configurations of patient-specific coronary artery bypass grafts. This is especially useful to perform a sensitivity analysis of the haemodynamics for different flow conditions occurring in native coronary arteries and bypass grafts, the investigation of the progression of the coronary artery disease and the choice of the most appropriate surgical procedure. A complete pipeline, from the acquisition of patientspecific medical images to fast parametrized computational simulations, is proposed. Complex surgical configurations employed in the clinical practice, such as Y-grafts and sequential grafts, are studied. A virtual surgery platform based on model reduction of unsteady Navier Stokes equations for blood dynamics is proposed to carry out sensitivity analyses in a very rapid and reliable way. A specialized geometrical parametrization is employed to compare the effect of stenosis and anastomosis variation on the outcome of the surgery in several relevant cases. %I Submitted %G en %U http://urania.sissa.it/xmlui/handle/1963/35240 %1 35545 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2016-10-14T23:07:40Z No. of bitstreams: 1 BMMB_SISSA_report.pdf: 16374062 bytes, checksum: 7ee82fd9d989ed91bf9cc721ae2114a0 (MD5) %0 Journal Article %J Computers and Mathematics with Applications %D 2016 %T Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries %A Laura Iapichino %A Alfio Quarteroni %A Gianluigi Rozza %X The aim of this work is to solve parametrized partial differential equations in computational domains represented by networks of repetitive geometries by combining reduced basis and domain decomposition techniques. The main idea behind this approach is to compute once, locally and for few reference shapes, some representative finite element solutions for different values of the parameters and with a set of different suitable boundary conditions on the boundaries: these functions will represent the basis of a reduced space where the global solution is sought for. The continuity of the latter is assured by a classical domain decomposition approach. Test results on Poisson problem show the flexibility of the proposed method in which accuracy and computational time may be tuned by varying the number of reduced basis functions employed, or the set of boundary conditions used for defining locally the basis functions. The proposed approach simplifies the pre-computation of the reduced basis space by splitting the global problem into smaller local subproblems. Thanks to this feature, it allows dealing with arbitrarily complex network and features more flexibility than a classical global reduced basis approximation where the topology of the geometry is fixed. %B Computers and Mathematics with Applications %I Elsevier %V 71 %P 430 %8 01/2016 %G en %N 1 %1 35187 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2015-11-05T14:59:46Z No. of bitstreams: 1 RDF2_reduced.pdf: 3254431 bytes, checksum: 508b8775e27dfb868f1b39c270cb4a6f (MD5) %& 408 %0 Report %D 2015 %T Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization %A F. Ballarin %A Elena Faggiano %A Sonia Ippolito %A Andrea Manzoni %A Alfio Quarteroni %A Gianluigi Rozza %A Roberto Scrofani %X In this work a reduced-order computational framework for the study of haemodynamics in three-dimensional patient-specific configurations of coronary artery bypass grafts dealing with a wide range of scenarios is proposed. We combine several efficient algorithms to face at the same time both the geometrical complexity involved in the description of the vascular network and the huge computational cost entailed by time dependent patient-specific flow simulations. Medical imaging procedures allow to reconstruct patient-specific configurations from clinical data. A centerlines-based parametrization is proposed to efficiently handle geometrical variations. POD–Galerkin reduced-order models are employed to cut down large computational costs. This computational framework allows to characterize blood flows for different physical and geometrical variations relevant in the clinical practice, such as stenosis factors and anastomosis variations, in a rapid and reliable way. Several numerical results are discussed, highlighting the computational performance of the proposed framework, as well as its capability to perform sensitivity analysis studies, so far out of reach. %G en %U http://urania.sissa.it/xmlui/handle/1963/34623 %1 34824 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2015-10-07T00:30:27Z No. of bitstreams: 1 REPORT.pdf: 10426315 bytes, checksum: 6e5ddf4eb4cacdc7e803c2db1a540fc9 (MD5) %0 Journal Article %J Numerische Mathematik, (2015), 36 p. Article in Press %D 2015 %T Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations %A Gianluigi Rozza %A Peng Chen %A Alfio Quarteroni %X In this paper we develop and analyze a multilevel weighted reduced basis method for solving stochastic optimal control problems constrained by Stokes equations. We prove the analytic regularity of the optimal solution in the probability space under certain assumptions on the random input data. The finite element method and the stochastic collocation method are employed for the numerical approximation of the problem in the deterministic space and the probability space, respectively, resulting in many large-scale optimality systems to solve. In order to reduce the unaffordable computational effort, we propose a reduced basis method using a multilevel greedy algorithm in combination with isotropic and anisotropic sparse-grid techniques. A weighted a posteriori error bound highlights the contribution stemming from each method. Numerical tests on stochastic dimensions ranging from 10 to 100 demonstrate that our method is very efficient, especially for solving high-dimensional and large-scale optimization problems. %B Numerische Mathematik, (2015), 36 p. Article in Press %I Springer %G en %U http://urania.sissa.it/xmlui/handle/1963/34491 %1 34680 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2015-08-25T23:32:48Z No. of bitstreams: 1 06August2014_wRBM4SOC.pdf: 531409 bytes, checksum: d1a2f18b0de17872919c430779f7180c (MD5) %R 10.1007/s00211-015-0743-4 %0 Journal Article %D 2015 %T Supremizer stabilization of POD-Galerkin approximation of parametrized Navier-Stokes equations %A F. Ballarin %A Andrea Manzoni %A Alfio Quarteroni %A Gianluigi Rozza %X In this work, we present a stable proper orthogonal decomposition–Galerkin approximation for parametrized steady incompressible Navier–Stokes equations with low Reynolds number. %I Wiley %G en %U http://urania.sissa.it/xmlui/handle/1963/34701 %1 34915 %2 Mathematics %4 1 %# MAT/08 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2015-10-22T11:39:58Z No. of bitstreams: 1 IJNME_2014.pdf: 6761966 bytes, checksum: ad65f2c4d2dbd30a4a1590ff42ee49a0 (MD5) %R 10.1002/nme.4772 %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 %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 %$ Submitted by gfeltrin@sissa.it (gfeltrin@sissa.it) on 2015-10-22T15:11:51Z No. of bitstreams: 1 preprint2014.pdf: 287014 bytes, checksum: b195410aa3f63643829ed25f1adb6520 (MD5) %R 10.1007/978-3-319-02090-7_9 %0 Conference Paper %B ECMI 2014 proceedings %D 2014 %T Reduced basis method for the Stokes equations in decomposable domains using greedy optimization %A Laura Iapichino %A Alfio Quarteroni %A Gianluigi Rozza %A Volkwein, Stefan %B ECMI 2014 proceedings %P 1–7 %G eng %0 Book %B MS&A %D 2014 %T Reduced Order Methods for Modeling and Computational Reduction %A Alfio Quarteroni %A Gianluigi Rozza %K reduced order methods, MOR, ROM, POD, RB, greedy, CFD, Numerical Analysis %X

This monograph addresses the state of the art of reduced order methods for modeling and computational reduction of complex parametrized systems, governed by ordinary and/or partial differential equations, with a special emphasis on real time computing techniques and applications in computational mechanics, bioengineering and computer graphics.

Several topics are covered, including: design, optimization, and control theory in real-time with applications in engineering; data assimilation, geometry registration, and parameter estimation with special attention to real-time computing in biomedical engineering and computational physics; real-time visualization of physics-based simulations in computer science; the treatment of high-dimensional problems in state space, physical space, or parameter space; the interactions between different model reduction and dimensionality reduction approaches; the development of general error estimation frameworks which take into account both model and discretization effects.

This book is primarily addressed to computational scientists interested in computational reduction techniques for large scale differential problems.

 

%B MS&A %7 1 %I Springer %C Milano %V 9 %P 334 %G eng %6 1 %R 10.1007/978-3-319-02090-7 %0 Journal Article %D 2014 %T A weighted empirical interpolation method: A priori convergence analysis and applications %A Peng Chen %A Alfio Quarteroni %A Gianluigi Rozza %X We extend the classical empirical interpolation method [M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Compt. Rend. Math. Anal. Num. 339 (2004) 667-672] to a weighted empirical interpolation method in order to approximate nonlinear parametric functions with weighted parameters, e.g. random variables obeying various probability distributions. A priori convergence analysis is provided for the proposed method and the error bound by Kolmogorov N-width is improved from the recent work [Y. Maday, N.C. Nguyen, A.T. Patera and G.S.H. Pau, A general, multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2009) 383-404]. We apply our method to geometric Brownian motion, exponential Karhunen-Loève expansion and reduced basis approximation of non-affine stochastic elliptic equations. We demonstrate its improved accuracy and efficiency over the empirical interpolation method, as well as sparse grid stochastic collocation method. %I EDP Sciences %G en %U http://urania.sissa.it/xmlui/handle/1963/35021 %1 35253 %2 Mathematics %4 1 %# MAT/05 %$ Approved for entry into archive by Lucio Lubiana (lubiana@sissa.it) on 2015-11-17T10:26:06Z (GMT) No. of bitstreams: 0 %R 10.1051/m2an/2013128 %0 Journal Article %J Communications in Applied and Industrial Mathematics %D 2013 %T Free Form Deformation Techniques Applied to 3D Shape Optimization Problems %A Anwar Koshakji %A Alfio Quarteroni %A Gianluigi Rozza %X The purpose of this work is to analyse and study an efficient parametrization technique for a 3D shape optimization problem. After a brief review of the techniques and approaches already available in literature, we recall the Free Form Deformation parametrization, a technique which proved to be efficient and at the same time versatile, allowing to manage complex shapes even with few parameters. We tested and studied the FFD technique by establishing a path, from the geometry definition, to the method implementation, and finally to the simulation and to the optimization of the shape. In particular, we have studied a bulb and a rudder of a race sailing boat as model applications, where we have tested a complete procedure from Computer-Aided-Design to build the geometrical model to discretization and mesh generation. %B Communications in Applied and Industrial Mathematics %G eng %R 10.1685/journal.caim.452 %0 Journal Article %J SIAM Journal on Scientific Computing %D 2013 %T Reduced basis method for parametrized elliptic optimal control problems %A Federico Negri %A Gianluigi Rozza %A Andrea Manzoni %A Alfio Quarteroni %X We propose a suitable model reduction paradigm-the certified reduced basis method (RB)-for the rapid and reliable solution of parametrized optimal control problems governed by partial differential equations. In particular, we develop the methodology for parametrized quadratic optimization problems with elliptic equations as a constraint and infinite-dimensional control variable. First, we recast the optimal control problem in the framework of saddle-point problems in order to take advantage of the already developed RB theory for Stokes-type problems. Then, the usual ingredients of the RB methodology are called into play: a Galerkin projection onto a low-dimensional space of basis functions properly selected by an adaptive procedure; an affine parametric dependence enabling one to perform competitive offline-online splitting in the computational procedure; and an efficient and rigorous a posteriori error estimate on the state, control, and adjoint variables as well as on the cost functional. Finally, we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique. %B SIAM Journal on Scientific Computing %V 35 %P A2316–A2340 %G eng %R 10.1137/120894737 %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 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2013-04-30T14:47:07Z No. of bitstreams: 1 LMQR_inverse_problems_Haemo.pdf: 2913600 bytes, checksum: 9bff594072e6c7b4a664bdade361a1a9 (MD5) %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 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2013-04-12T10:16:42Z\nNo. of bitstreams: 1\nHanoi_RQML_2012_REVISED.pdf: 1360222 bytes, checksum: 4cb9ffaaf9fd67e3eacf4d9c060d0940 (MD5) %0 Journal Article %J SIAM Journal on Numerical Analysis %D 2013 %T Stochastic optimal robin boundary control problems of advection-dominated elliptic equations %A Peng Chen %A Alfio Quarteroni %A Gianluigi Rozza %X In this work we deal with a stochastic optimal Robin boundary control problem constrained by an advection-diffusion-reaction elliptic equation with advection-dominated term. We assume that the uncertainty comes from the advection field and consider a stochastic Robin boundary condition as control function. A stochastic saddle point system is formulated and proved to be equivalent to the first order optimality system for the optimal control problem, based on which we provide the existence and uniqueness of the optimal solution as well as some results on stochastic regularity with respect to the random variables. Stabilized finite element approximations in physical space and collocation approximations in stochastic space are applied to discretize the optimality system. A global error estimate in the product of physical space and stochastic space for the numerical approximation is derived. Illustrative numerical experiments are provided. %B SIAM Journal on Numerical Analysis %V 51 %P 2700–2722 %G eng %R 10.1137/120884158 %0 Journal Article %J SIAM Journal on Numerical Analysis %D 2013 %T A weighted reduced basis method for elliptic partial differential equations with random input data %A Peng Chen %A Alfio Quarteroni %A Gianluigi Rozza %X In this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equations (PDEs) with random input data. The PDEs are first transformed into a weighted parametric elliptic problem depending on a finite number of parameters. Distinctive importance of the solution at different values of the parameters is taken into account by assigning different weights to the samples in the greedy sampling procedure. A priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters. Numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and the stochastic collocation method in both univariate and multivariate stochastic problems. %B SIAM Journal on Numerical Analysis %V 51 %P 3163–3185 %G eng %R 10.1137/130905253 %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 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2012-12-13T15:38:41Z\\nNo. of bitstreams: 1\\nLMQR_M2AN_Special_SISSAreport.pdf: 5702019 bytes, checksum: 037e51bde713582eff1ee9766b1b4559 (MD5) %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 %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2012-12-13T17:47:45Z\\nNo. of bitstreams: 1\\nqlmr-bumi_FINAL_SISSAreport.pdf: 377397 bytes, checksum: ecaf5713afa6a6f68992f6331631aff4 (MD5) %0 Journal Article %J International Journal Numerical Methods Biomedical Engineering %D 2012 %T Simulation-based uncertainty quantification of human arterial network hemodynamics %A Peng Chen %A Alfio Quarteroni %A Gianluigi Rozza %K uncertainty quantification, mathematical modelling of the cardiovascular system, fluid-structure interaction %X This work aims at identifying and quantifying uncertainties from various sources in human cardiovascular\r\nsystem based on stochastic simulation of a one dimensional arterial network. A general analysis of\r\ndifferent uncertainties and probability characterization with log-normal distribution of these uncertainties\r\nis introduced. Deriving from a deterministic one dimensional fluid structure interaction model, we establish\r\nthe stochastic model as a coupled hyperbolic system incorporated with parametric uncertainties to describe\r\nthe blood flow and pressure wave propagation in the arterial network. By applying a stochastic collocation\r\nmethod with sparse grid technique, we study systemically the statistics and sensitivity of the solution with\r\nrespect to many different uncertainties in a relatively complete arterial network with potential physiological\r\nand pathological implications for the first time. %B International Journal Numerical Methods Biomedical Engineering %I Wiley %G en %1 6467 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2013-03-07T12:33:02Z\nNo. of bitstreams: 1\nreport.pdf: 1605405 bytes, checksum: bb12cf074ce32a80567a0cde0c0861db (MD5)