Publications

Combined Parameter and Model Reduction of Cardiovascular Problems by Means of Active Subspaces and POD-Galerkin Methods

Journal: 

Mathematical and Numerical Modeling of the Cardiovascular System and Applications

Date: 

2018

Authors: 

M. Tezzele, F. Ballarin, and G. Rozza

In this chapter we introduce a combined parameter and model reduction methodology and present its application to the efficient numerical estimation of a pressure drop in a set of deformed carotids. The aim is to simulate a wide range of possible occlusions after the bifurcation of the carotid. A parametric description of the admissible deformations, based on radial basis functions interpolation, is introduced. Since the parameter space may be very large, the first step in the combined reduction technique is to look for active subspaces in order to reduce the parameter space dimension.

Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier–Stokes equations

Journal: 

Computers & Fluids

Date: 

2018

Authors: 

G. Stabile and G. Rozza

In this work a stabilised and reduced Galerkin projection of the incompressible unsteady Navier–Stokes equations for moderate Reynolds number is presented. The full-order model, on which the Galerkin projection is applied, is based on a finite volumes approximation. The reduced basis spaces are constructed with a POD approach. Two different pressure stabilisation strategies are proposed and compared: the former one is based on the supremizer enrichment of the velocity space, and the latter one is based on a pressure Poisson equation approach.

A Spectral Element Reduced Basis Method in Parametric CFD

Journal: 

Numerical Mathematics and Advanced Applications – ENUMATH 2017, Springer, vol. 126

Date: 

2018

Authors: 

M. W. Hess and G. Rozza

We consider the Navier-Stokes equations in a channel with varying Reynolds numbers. The model is discretized with high-order spectral element ansatz functions, resulting in 14 259 degrees of freedom. The steady-state snapshot solu- tions define a reduced order space, which allows to accurately evaluate the steady- state solutions for varying Reynolds number with a reduced order model within a fixed-point iteration. In particular, we compare different aspects of implementing the reduced order model with respect to the use of a spectral element discretization.

Model Reduction for Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering

Journal: 

SIAM Journal on Scientific Computing, 40(4), p. pp. B1055-B1079

Date: 

2018

Authors: 

M. Strazzullo, F. Ballarin, R. Mosetti, and G. Rozza

We propose reduced order methods as a suitable approach to face parametrized optimal control problems governed by partial differential equations, with applications in en- vironmental marine sciences and engineering. Environmental parametrized optimal control problems are usually studied for different configurations described by several physical and/or geometrical parameters representing different phenomena and structures. The solution of parametrized problems requires a demanding computational effort.

Computational methods in cardiovascular mechanics

Journal: 

Cardiovascular Mechanics, M. F. Labrosse (ed.), CRC Press, p. pp. 54

Date: 

2018

Authors: 

F. Auricchio, M. Conti, A. Lefieux, S. Morganti, A. Reali, G. Rozza, and A. Veneziani

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.

Shape Optimization by means of Proper Orthogonal Decomposition and Dynamic Mode Decomposition

Journal: 

Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research, 2018, p. pp. 212–219

Date: 

2018

Authors: 

N. Demo, M. Tezzele, G. Gustin, G. Lavini, and G. Rozza

Shape optimization is a challenging task in many engineering fields, since the numerical solutions of parametric system may be computationally expensive. This work presents a novel optimization procedure based on reduced order modeling, applied to a naval hull design problem. The advantage introduced by this method is that the solution for a specific parameter can be expressed as the combination of few numerical solutions computed at properly chosen parametric points. The reduced model is built using the proper orthogonal decomposition with interpolation (PODI) method.

Model order reduction by means of active subspaces and dynamic mode decomposition for parametric hull shape design hydrodynamics

Journal: 

Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship & Maritime Research, 2018, p. pp. 569–576

Date: 

2018

Authors: 

M. Tezzele, N. Demo, M. Gadalla, A. Mola, and G. Rozza

We present the results of the application of a parameter space reduction methodology based on active subspaces (AS) to the hull hydrodynamic design problem. Several parametric deformations of an initial hull shape are considered to assess the influence of the shape parameters on the hull wave resistance. Such problem is relevant at the preliminary stages of the ship design, when several flow simulations are carried out by the engineers to establish a certain sensibility with respect to the parameters, which might result in a high number of time consuming hydrodynamic simulations.

Certified Reduced Basis Approximation for the Coupling of Viscous and Inviscid Parametrized Flow Models

Journal: 

Journal of Scientific Computing, 74, pp. 197-219

Date: 

2018

Authors: 

I. Martini, B. Haasdonk, and G. Rozza

We present a model order reduction approach for parametrized laminar flow problems including viscous boundary layers. The viscous effects are captured by the incompressible Navier–Stokes equations in the vicinity of the boundary layer, whereas a potential flow model is used in the outer region. By this, we provide an accurate model that avoids imposing the Kutta condition for potential flows as well as an expensive numerical solution of a global viscous model. To account for the parametrized nature of the problem, we apply the reduced basis method.

An efficient shape parametrisation by free-form deformation enhanced by active subspace for hull hydrodynamic ship design problems in open source environment

Journal: 

The 28th International Ocean and Polar Engineering Conference

Date: 

2018

Authors: 

N. Demo, M. Tezzele, A. Mola, and G. Rozza

In this contribution, we present the results of the application of a parameter space reduction methodology based on active subspaces to the hull hydrodynamic design problem. Several parametric deformations of an initial hull shape are considered to assess the influence of the shape parameters considered on the hull total drag. The hull resistance is typically computed by means of numerical simulations of the hydrodynamic flow past the ship.

Model Reduction Methods

Journal: 

Encyclopedia of Computational Mechanics Second Edition), John Wiley & Sons, pp. 1-36

Date: 

2017

Authors: 

F. Chinesta, A. Huerta, G. Rozza, and K. Willcox

Reduced-order semi-implicit schemes for fluid-structure interaction problems

Journal: 

Model Reduction of Parametrized Systems, P. Benner, M. Ohlberger, A. Patera, G. Rozza, and K. Urban (eds.), Springer International Publishing, p. pp. 149–167

Date: 

2017

Authors: 

F. Ballarin, G. Rozza, and Y. Maday

POD–Galerkin reduced-order models (ROMs) for fluid-structure interaction problems (incompressible fluid and thin structure) are proposed in this paper. Both the high-fidelity and reduced-order methods are based on a Chorin-Temam operator-splitting approach. Two different reduced-order methods are proposed, which differ on velocity continuity condition, imposed weakly or strongly, respectively. The resulting ROMs are tested and compared on a representative haemodynamics test case characterized by wave propagation, in order to assess the capabilities of the proposed strategies.

Numerical modeling of hemodynamics scenarios of patient-specific coronary artery bypass grafts

Journal: 

Biomechanics and Modeling in Mechanobiology, 16(4), p. pp. 1373–1399

Date: 

2017

Authors: 

F. Ballarin, E. Faggiano, A. Manzoni, A. Quarteroni, G. Rozza, S. Ippolito, C. Antona, and R. Scrofani

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.

A reduced order model for investigating the dynamics of the Gen-IV LFR coolant pool

Journal: 

Applied Mathematical Modelling, 46, pp. 263-284

Date: 

2017

Authors: 

S. Lorenzi, A. Cammi, L. Luzzi, and G. Rozza

In the control field, the study of the system dynamics is usually carried out relying on lumped-parameter or one-dimensional modelling. Even if these approaches are well suited for control purposes since they provide fast-running simulations and are easy to linearize, they may not be sufficient to deeply assess the complexity of the systems, in particular where spatial phenomena have a significant impact on dynamics. Reduced Order Methods (ROM) can offer the proper trade-off between computational cost and solution accuracy.

Pod-Galerkin Reduced Order Methods for CFD Using Finite Volume Discretisation: Vortex Shedding Around a Circular Cylinder

Journal: 

Communication in Applied Industrial Mathematics, 8(1), p. pp. 210–236

Date: 

2017

Authors: 

G. Stabile, S. N. Hijazi, S. Lorenzi, A. Mola, and G. Rozza

Vortex shedding around circular cylinders is a well known and studied phenomenon that appears in many engineering fields. In this work a Reduced Order Model (ROM) of the incompressible flow around a circular cylinder, built performing a Galerkin projection of the governing equations onto a lower dimensional space is presented. The reduced basis space is generated using a Proper Orthogonal Decomposition (POD) approach.

On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics

Journal: 

Journal of Scientific Computing

Date: 

2017

Authors: 

G. Pitton and G. Rozza

In this paper we apply a reduced basis framework for the computation of flow bifurcation (and stability) problems in fluid dynamics. The proposed method aims at reducing the complexity and the computational time required for the construction of bifurcation and stability diagrams. The method is quite general since it can in principle be specialized to a wide class of nonlinear problems, but in this work we focus on an application in incompressible fluid dynamics at low Reynolds numbers.

Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: Applications to Coanda effect in cardiology

Journal: 

Journal of Computational Physics, 344, p. pp. 557

Date: 

2017

Authors: 

G. Pitton, A. Quaini, and G. Rozza

We 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.

Certified Reduced Basis Method for Affinely Parametric Isogeometric Analysis NURBS Approximation

Journal: 

Spectral and High Order Methods for Partial Differential Equations), Springer, vol. 119

Date: 

2017

Authors: 

D. Devaud and G. Rozza

In 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 scientific computing and computational mechanics, as well as reduced basis methods for parametric PDEs.

Reduced Basis Methods for Uncertainty Quantification

Journal: 

SIAM/ASA Journal on Uncertainty Quantification, 5, p. pp. 813–869

Date: 

2017

Authors: 

P. Chen, A. Quarteroni, and G. Rozza

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. Babuska, F. Nobile, and R.

On a certified Smagorinsky reduced basis turbulence model

Journal: 

SIAM Journal on Numerical Analysis, 55(6), p. pp. 3047–3067

Date: 

2017

Authors: 

T. Chacón Rebollo, E. Delgado Ávila, M. Gómez Mármol, F. Ballarin, and G. Rozza

In this work we present a reduced basis Smagorinsky turbulence model for steady flows. We approximate the non-linear eddy diffusion term using the Empirical Interpolation Method, and the velocity-pressure unknowns by an independent reduced-basis procedure. This model is based upon an a posteriori error estimation for Smagorinsky turbulence model. The theoretical development of the a posteriori error estimation is based on previous works, according to the Brezzi-Rappaz-Raviart stability theory, and adapted for the non-linear eddy diffusion term.

Isogeometric analysis-based reduced order modelling for incompressible linear viscous flows in parametrized shapes

Journal: 

Advanced Modeling and Simulation in Engineering Sciences, 3(1), p. pp. 21

Date: 

2016

Authors: 

F. Salmoiraghi, F. Ballarin, L. Heltai, and G. Rozza

In this work we provide a combination of isogeometric analysis with reduced order modelling techniques, based on proper orthogonal decomposition, to guarantee computational reduction for the numerical model, and with free-form deformation, for versatile geometrical parametrization. We apply it to computational fluid dynamics problems considering a Stokes flow model. The proposed reduced order model combines efficient shape deformation and accurate and stable velocity and pressure approximation for incompressible viscous flows, computed with a reduced order method.

Advances in geometrical parametrization and reduced order models and methods for computational fluid dynamics problems in applied sciences and engineering: overview and perspectives

Journal: 

Proceedings of the ECCOMAS Congress 2016, VII European Conference on Computational Methods in Applied Sciences and Engineering

Date: 

2016

Authors: 

F. Salmoiraghi, F. Ballarin, G. Corsi, A. Mola, M. Tezzele, and G. Rozza

Several problems in applied sciences and engineering require reduction techniques in order to allow computational tools to be employed in the daily practice, especially in iterative procedures such as optimization or sensitivity analysis. Reduced order methods need to face increasingly complex problems in computational mechanics, especially into a multiphysics setting.

POD-Galerkin Method for Finite Volume Approximation of Navier-Stokes and RANS Equations

Journal: 

Computer Methods in Applied Mechanics and Engineering, 311, pp. 151-179

Date: 

2016

Authors: 

S. Lorenzi, A. Cammi, L. Luzzi, and G. Rozza

Numerical simulation of fluid flows requires important computational efforts but it is essential in engineering applications. Reduced Order Model (ROM) can be employed whenever fast simulations are required, or in general, whenever a trade-off between computational cost and solution accuracy is a preeminent issue as in process optimization and control.

Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries

Journal: 

Computers and Mathematics with Applications, 71(1), p. pp. 408–430

Date: 

2016

Authors: 

L. Iapichino, A. Quarteroni, and G. Rozza

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.

Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization

Journal: 

Journal of Computational Physics, 315, p. pp. 609–628

Date: 

2016

Authors: 

F. Ballarin, E. Faggiano, S. Ippolito, A. Manzoni, A. Quarteroni, G. Rozza, and R. Scrofani

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.

Reduced basis approaches in time-dependent non-coercive settings for modelling the movement of nuclear reactor control rods

Journal: 

Communications in Computational Physics, 20(1), p. pp. 23–59

Date: 

2016

Authors: 

A. Sartori, A. Cammi, L. Luzzi, and G. Rozza

In this work, two approaches, based on the certified Reduced Basis method, have been developed for simulating the movement of nuclear reactor control rods, in time-dependent non-coercive settings featuring a 3D geometrical framework. In particular, in a first approach, a piece-wise affine transformation based on subdomains division has been implemented for modelling the movement of one control rod. In the second approach, a staircase strategy has been adopted for simulating the movement of all the three rods featured by the nuclear reactor chosen as case study.

POD–Galerkin monolithic reduced order models for parametrized fluid-structure interaction problems

Journal: 

International Journal for Numerical Methods in Fluids, 82(12), p. pp. 1010–1034

Date: 

2016

Authors: 

F. Ballarin and G. Rozza

In this paper we propose a monolithic approach for reduced order modelling of parametrized fluid-structure interaction problems based on a proper orthogonal decomposition (POD)–Galerkin method. Parameters of the problem are related to constitutive properties of the fluid or structural problem, or to geometrical parameters related to the domain configuration at the initial time.

A multi-physics reduced order model for the analysis of Lead Fast Reactor single channel

Journal: 

Annals of Nuclear Energy, 87, p. pp. 198–208

Date: 

2016

Authors: 

A. Sartori, A. Cammi, L. Luzzi, and G. Rozza

In this work, a Reduced Basis method, with basis functions sampled by a Proper Orthogonal Decomposition technique, has been employed to develop a reduced order model of a multi-physics parametrized Lead-cooled Fast Reactor single-channel. Being the first time that a reduced order model is developed in this context, the work focused on a methodological approach and the coupling between the neutronics and the heat transfer, where the thermal feedbacks on neutronics are explicitly taken into account, in time-invariant settings.

Reduced basis approximation and a-posteriori error estimation for the coupled Stokes-Darcy system

Journal: 

Advances in Computational Mathematics, 41(5), p. pp. 1131–1157

Date: 

2015

Authors: 

I. Martini, G. Rozza, and B. Haasdonk

The coupling of a free flow with a flow through porous media has many potential applications in several fields related with computational science and engineering, such as blood flows, environmental problems or food technologies. We present a reduced basis method for such coupled problems. The reduced basis method is a model order reduction method applied in the context of parametrized systems. Our approach is based on a heterogeneous domain decomposition formulation, namely the Stokes-Darcy problem. Thanks to an offline/online-decomposition, computational times can be drastically reduced.

Reduced Basis Approximation for the Structural-Acoustic Design based on Energy Finite Element Analysis (RB-EFEA)

Journal: 

CEMRACS 2013 – Modelling and simulation of complex systems: stochastic and deterministic approaches), vol. 48, p. pp. 98–115

Date: 

2015

Authors: 

D. Devaud and G. Rozza

In many engineering applications, the investigation of the vibro-acoustic response of structures is of great interest. Hence, great effort has been dedicated to improve methods in this field in the last twenty years. Classical techniques have the main drawback that they become unaffordable when high frequency impact waves are considered. In that sense, the Energy Finite Element Analysis (EFEA) is a good alternative to those methods. Based on an approximate model, EFEA gives time and locally space-averaged energy densities and has been proven to yield accurate results.

Certified Reduced Basis Methods for Parametrized Partial Differential Equations, 1 ed.

Journal: 

Switzerland: Springer, 2015.

Date: 

2015

Authors: 

J. S. Hesthaven, G. Rozza, and B. Stamm

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.

Reduced basis approximation of parametrized advection-diffusion PDEs with high Péclet number

Journal: 

Lecture Notes in Computational Science and Engineering, 103, p. pp. 419–426

Date: 

2015

Authors: 

P. Pacciarini and G. Rozza

In this work we show some results about the reduced basis approximation of advection dominated parametrized problems, i.e. advection-diffusion problems with high Péclet number. These problems are of great importance in several engineering applications and it is well known that their numerical approximation can be affected by instability phenomena. In this work we compare two possible stabilization strategies in the framework of the reduced basis method, by showing numerical results obtained for a steady advection-diffusion problem.

Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations

Journal: 

International Journal for Numerical Methods in Engineering, 102(5), p. pp. 1136–1161

Date: 

2015

Authors: 

F. Ballarin, A. Manzoni, A. Quarteroni, and G. Rozza

In this work, we present a stable proper orthogonal decomposition–Galerkin approximation for parametrized
steady incompressible Navier–Stokes equations with low Reynolds number.

Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations

Journal: 

Numerische Mathematik, 133(1), p. pp. 67–102

Date: 

2015

Authors: 

P. Chen, A. Quarteroni, and G. Rozza

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.

Fundamentals of Reduced Basis Method for problems governed by parametrized PDEs and applications

Journal: 

Separated representations and PGD-based model reduction: fundamentals and applications), Springer, vol. 554

Date: 

2014

Authors: 

G. Rozza

In this chapter we consider Reduced Basis (RB) approximations of parametrized Partial Differential Equations (PDEs). The the idea behind RB is to decouple the generation and projection stages (Offline/Online computational procedures) of the approximation process in order to solve parametrized PDEs in a fast, inexpensive and reliable way. The RB method, especially applied to 3D problems, allows great computational savings with respect to the classical Galerkin Finite Element (FE) Method.

Reduced Order Methods for Modeling and Computational Reduction, 1 ed.

Journal: 

Springer, 2014, vol. 9.

Date: 

2014

Authors: 

A. Quarteroni and G. Rozza

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.

A weighted empirical interpolation method: A priori convergence analysis and applications

Journal: 

ESAIM: Mathematical Modelling and Numerical Analysis, 48(4), p. pp. 943–953

Date: 

2014

Authors: 

P. Chen, A. Quarteroni, and G. Rozza

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.

Stabilized reduced basis method for parametrized advection-diffusion PDEs

Journal: 

Computer Methods in Applied Mechanics and Engineering, 274, p. pp. 1–18

Date: 

2014

Authors: 

P. Pacciarini and G. Rozza

Comparison between reduced basis and stochastic collocation methods for elliptic problems

Journal: 

Journal of Scientific Computing, 59(1), p. pp. 187–216

Date: 

2014

Authors: 

P. Chen, A. Quarteroni, and G. Rozza

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.

An improvement on geometrical parameterizations by transfinite maps

Journal: 

Comptes Rendus Mathematique, 352(3), p. pp. 263–268

Date: 

2014

Authors: 

C. Jäggli, L. Iapichino, and G. Rozza

We present a method to generate a non-affine transfinite map from a given reference domain to a family of deformed domains. The map is a generalization of the Gordon-Hall transfinite interpolation approach. It is defined globally over the reference domain. Once we have computed some functions over the reference domain, the map can be generated by knowing the parametric expressions of the boundaries of the deformed domain. Being able to define a suitable map from a reference domain to a desired deformation is useful for the management of parameterized geometries.

Model order reduction in fluid dynamics: challenges and perspectives

Journal: 

Reduced Order Methods for Modeling and Computational Reduction, A. Quarteroni and G. Rozza (eds.), Springer MS&A Series, vol. 9, p. pp. 235–274

Date: 

2014

Authors: 

T. Lassila, A. Manzoni, A. Quarteroni, and G. Rozza

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.

Stabilized reduced basis method for parametrized scalar advection-diffusion problems at higher Péclet number: Roles of the boundary layers and inner fronts

Journal: 

11th World Congress on Computational Mechanics, WCCM 2014, 5th European Conference on Computational Mechanics, ECCM 2014 and 6th European Conference on Computational Fluid Dynamics, ECFD 2014, p. pp. 5614–5624.

Date: 

2014

Authors: 

P. Pacciarini and G. Rozza

Advection-dominated problems, which arise in many engineering situations, often require a fast and reliable approximation of the solution given some parameters as inputs. In this work we want to investigate the coupling of the reduced basis method - which guarantees rapidity and reliability - with some classical stabilization techiques to deal with the advection-dominated condition. We provide a numerical extension of the results presented in [1], focusing in particular on problems with curved boundary layers and inner fronts whose direction depends on the parameter.

Shape Optimization by Free-Form Deformation: Existence Results and Numerical Solution for Stokes Flows

Journal: 

Journal of Scientific Computing, 60(3), p. pp. 537–563

Date: 

2014

Authors: 

F. Ballarin, A. Manzoni, G. Rozza, and S. Salsa

Shape optimization problems governed by PDEs result from many applications in computational fluid dynamics. These problems usually entail very large computational costs and require also a suitable approach for representing and deforming efficiently the shape of the underlying geometry, as well as for computing the shape gradient of the cost functional to be minimized. Several approaches based on the displacement of a set of control points have been developed in the last decades, such as the so-called free-form deformations.

A reduced order model for multi-group time-dependent parametrized reactor spatial kinetics

Journal: 

International Conference on Nuclear Engineering, Proceedings, ICONE

Date: 

2014

Authors: 

A. Sartori, D. Baroli, A. Cammi, L. Luzzi, and G. Rozza

In this work, a Reduced Order Model (ROM) for multigroup time-dependent parametrized reactor spatial kinetics is presented. The Reduced Basis method (built upon a high-fidelity ``truth'' finite element approximation) has been applied to model the neutronics behavior of a parametrized system composed by a control rod surrounded by fissile material. The neutron kinetics has been described by means of a parametrized multi-group diffusion equation where the height of the control rod (i.e., how much the rod is inserted) plays the role of the varying parameter.

Comparison of a Modal Method and a Proper Orthogonal Decomposition approach for multi-group time-dependent reactor spatial kinetics

Journal: 

Annals of Nuclear Energy, 71, p. pp. 217–229

Date: 

2014

Authors: 

A. Sartori, D. Baroli, A. Cammi, D. Chiesa, L. Luzzi, R. Ponciroli, E. Previtali, M. E. Ricotti, G. Rozza, and M. Sisti

In 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.

Reduced basis method for the Stokes equations in decomposable domains using greedy optimization

Journal: 

ECMI 2014 proceedings, p. pp. 1–7

Date: 

2014

Authors: 

L. Iapichino, A. Quarteroni, G. Rozza, and S. Volkwein

In this paper we present a reduced order method for the solution of parametrized Stokes equations in domain composed by an arbitrary number of predefined shapes. The novelty of the proposed approach is the possibility to use a small set of precomputed bases to solve Stokes equations in very different computational domains, defined by combining one or more reference geometries. The selection of the basis functions is performed through an optimization greedy algorithm.

A reduced computational and geometrical framework for inverse problems in hemodynamics

Journal: 

International Journal for Numerical Methods in Biomedical Engineering, 29(7), p. pp. 741–776

Date: 

2013

Authors: 

T. Lassila, A. Manzoni, A. Quarteroni, and G. Rozza

The solution of inverse problems in cardiovascular mathematics is computationally expensive. In this paper, we apply a domain parametrization technique to reduce both the geometrical and computational complexities of the forward problem and replace the finite element solution of the incompressible Navier-Stokes equations by a computationally less-expensive reduced-basis approximation. This greatly reduces the cost of simulating the forward problem.

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

Journal: 

ESAIM: Mathematical Modelling and Numerical Analysis, 47(4), p. pp. 1107–1131

Date: 

2013

Authors: 

T. Lassila, A. Manzoni, A. Quarteroni, and G. Rozza

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.

Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: Roles of the inf-sup stability constants

Journal: 

Numerische Mathematik, 125(1), p. pp. 115–152

Date: 

2013

Authors: 

G. Rozza, D. B. P. Huynh, and A. Manzoni

In this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady Stokes flows in affinely parametrized geometries, focusing on the role played by the Brezzi's and Babuška's stability constants. The crucial ingredients of the methodology are a Galerkin projection onto a low-dimensional space of basis functions properly selected, an affine parametric dependence enabling to perform competitive Offline-Online splitting in the computational procedure and a rigorous a posteriori error estimation on field variables.

Reduced basis method for parametrized elliptic optimal control problems

Journal: 

SIAM Journal on Scientific Computing, 35(5), p. pp. A2316–A2340

Date: 

2013

Authors: 

F. Negri, G. Rozza, A. Manzoni, and A. Quarteroni

Free Form Deformation Techniques Applied to 3D Shape Optimization Problems

Journal: 

Communications in Applied and Industrial Mathematics

Date: 

2013

Authors: 

A. Koshakji, A. Quarteroni, and G. Rozza

Generalized reduced basis methods and n-width estimates for the approximation of the solution manifold of parametric PDEs

Journal: 

Analysis and Numerics of Partial Differential Equations, F. Brezzi, P. Colli Franzone, U. Gianazza, and G. Gilardi (eds.), , vol. 4, p. pp. 307–329

Date: 

2013

Authors: 

T. Lassila, A. Manzoni, A. Quarteroni, and G. Rozza

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.

Reduction strategies for shape dependent inverse problems in haemodynamics

Journal: 

IFIP Advances in Information and Communication Technology, 391 AICT, p. pp. 397–406

Date: 

2013

Authors: 

T. Lassila, A. Manzoni, and G. Rozza

This work deals with the development and application of reduction strategies for real-time and many query problems arising in fluid dynamics, such as shape optimization, shape registration (reconstruction), and shape parametrization. The proposed strategy is based on the coupling between reduced basis methods for the reduction of computational complexity and suitable shape parametrizations - such as free-form deformations or radial basis functions - for low-dimensional geometrical description.

A combination between the reduced basis method and the ANOVA expansion: On the computation of sensitivity indices

Journal: 

Comptes Rendus Mathematique, 351(15-16), p. pp. 593–598

Date: 

2013

Authors: 

D. Devaud, A. Manzoni, and G. Rozza

We 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.

Simulation-based uncertainty quantification of human arterial network hemodynamics

Journal: 

International Journal for Numerical Methods in Biomedical Engineering, 29(6), p. pp. 698–721

Date: 

2013

Authors: 

P. Chen, A. Quarteroni, and G. Rozza

This work aims at identifying and quantifying uncertainties from various sources in human cardiovascular system based on stochastic simulation of a one-dimensional arterial network. A general analysis of different uncertainties and probability characterization with log-normal distribution of these uncertainties is introduced.

Reduction strategies for PDE-constrained optimization problems in haemodynamics

Journal: 

ECCOMAS 2012 – European Congress on Computational Methods in Applied Sciences and Engineering, p. pp. 1749–1768

Date: 

2012

Authors: 

G. Rozza, A. Manzoni, and F. Negri

Solving optimal control problems for many different scenarios obtained by varying a set of parameters in the state system is a computationally extensive task. In this paper we present a new reduced framework for the formulation, the analysis and the numerical solution of parametrized PDE-constrained optimization problems.