Publications by Alfio Quarteroni

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For the numerical simulation of the circulatory system, geometrical multiscale models based on the coupling of systems of differential equations with different spatial dimensions are becoming common practice. In this paper we address the mathematical analy ...

Simulating arterial trees in the cardiovascular system can be made by the help of different models, depending on the outputs of interest and the desired degree of accuracy. In particular, one-dimensional fluid-structure interaction models for arteries are ...

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 complexity of the forward problem, and replace the finit ...

We consider a fluid-structure interaction (FSI) problem discretized by finite elements featuring two different grids that do not necessarily agree on the interface separating the computational domain of the fluid from the one of the structure. After identifying a master domain (the structural domain) and a slave domain (the fluid domain), we build up two Radial Basis Function (RBF) inter-grid operators, one Pfs from master to slave, and the other Psf from slave to master. Then, we enforce the kinematic cond

Multiphysics simulations frequently require transferring solution fields between subproblems with nonmatching spatial discretizations, typically using interpolation techniques. Standard methods are usually based on measuring the closeness between points by ...

We consider the numerical approximation of geometric Partial Differential Equations (PDEs) defined on surfaces in the 3D space. In particular, we focus on the geometric PDEs deriving from the minimization of an energy functional by L2-gradient ow. We analyze two energy functionals: the area, which leads to the mean curvature ow, a nonlinear second order PDE, and the Willmore energy, leading to the Willmore ow, a nonlinear fourth order PDE. We consider surfaces represented by single-patch NURBS and discretiz

Mathematics forms bridges between knowledge, tradition, and contemporary life. The continuous development and growth of its many branches, both classical and modern, permeates and fertilizes all aspects of applied science and technology, and so has a vital impact on our modern society. The book will focus on these aspects and will benefit from the contribution of several world-famous scientists from mathematics and related sciences.

A major challenge in the computational fluid dynamics modeling of the heart function is the simulation of isovolumetric phases when the hemodynamics problem is driven by a prescribed boundary displacement. During such phases, both atrioventricular and semi ...

We present some results on coupling Navier-Stokes with shallow water equations for surface flows, and with Darcy's equation for groundwater flows. We discuss suitable interface conditions and show the well-posedness of the coupled problem in the case of a linear Stokes problem. An iterative method is proposed to compute the solution. At each step this method requires the solution of one problem in the fluid part and one in the porous medium. Finally, we introduce Steklov-Poincaré equation associated with th

In the context of SARS-CoV-2 pandemic, mathematical modelling has played a funda-mental role for making forecasts, simulating scenarios and evaluating the impact of pre-ventive political, social and pharmaceutical measures. Optimal control theory represent ...

In this paper we present the continuous and discontinuous Galerkin methods in a unified setting for the numerical approximation of the transport dominated advection-reaction equation. Both methods are stabilized by the interior penalty method, more precise ...

In this paper we propose a saddle point approach to solve boundary control problems for the steady Navier-Stokes equations with mixed Dirichlet-Neumann boundary conditions, both in two and three dimensions. We provide a comprehensive theoretical framework ...

We construct an idealized computational model of the left human heart for the study of the blood flow dynamics in the left atrium and ventricle. We solve the Navier-Stokes equations in the ALE formulation and we prescribe the left heart wall displacement b ...

The ensemble Kalman filter is a computationally efficient technique to solve state and/or parameter estimation problems in the framework of statistical inversion when relying on a Bayesian paradigm. Unfortunately its cost may become moderately large for systems de- scribed by nonlinear time-dependent PDEs, because of the cost entailed by each PDE query. In this paper we propose a reduced basis ensemble Kalman filter technique to address the above problems. The reduction stage yields intrinsic approximation

In the framework of cardiac electrophysiology for the human heart, we apply multipatch NURBS-based Isogeometric Analysis for the space discretization of the Monodomain model. Isogeometric Analysis (IGA) is a technique for the solution of Partial Differenti ...

This paper provides new results of consistence and convergence of the lumped parameters (ODE models) toward one-dimensional (hyperbolic or parabolic) models for blood flow. Indeed, lumped parameter models (exploiting the electric circuit analogy for the circulatory system) are shown to discretize continuous 1D models at first order in space. We derive the complete set of equations useful for the blood flow networks, new schemes for electric circuit analogy, the stability criteria that guarantee the converge

The coupling between cardiac mechanics and electric signaling is addressed in a nonstandard framework in which the electrical potential dictates the active strain (not stress) of the muscle. The physiological and mathematical motivations leading us to this ...

INTERNODES is a general method to deal with non-conforming discretizations of second order partial differential equations on regions partitioned into two or several subdomains. It exploits two intergrid interpolation operators, one for transfering the Dirichlet trace across the interface, the others for the Neumann trace. In this paper we provide several interpretations of the method and we carry out its stability and convergence analysis. In every subdomain the original problem is discretized by the finite

We propose a suitable model reduction paradigm -- the certied reduced basis method (RB) -- for the rapid and reliable solution of parametrized optimal control problems governed by partial dierential equations (PDEs). In particular, we develop the methodolo ...

The Yosida method was introduced in (Quarteroni et al., to appear) for the numerical approximation of the incompressible unsteady Navier-Stokes equations. From the algebraic viewpoint, it can be regarded as an inexact factorization of the matrix arising from the space and time discretization of the problem. However, its differential interpretation resides on an elliptic stabilization of the continuity equation through the Yosida regularization of the Laplacian (see (Brezis, 1983, Ciarlet and Lions, 1991)).

Alfio Quarteroni

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Analysis of a geometrical multiscale blood flow model based on the coupling of ODEs and hyperbolic PDEs

Implicit coupling of one-dimensional and three-dimensional blood flow models with compliant vessels

A reduced computational and geometrical framework for inverse problems in haemodynamics

MATHICSE Technical Report : A fluid-structure interaction algorithm using radial basis function interpolation between non-conforming interfaces

ROBUST RADIAL BASIS FUNCTION INTERPOLATION BASED ON GEODESIC DISTANCE FOR THE NUMERICAL COUPLING OF MULTIPHYSICS PROBLEMS

MATHICSE Technical Report : Isogeometric analysis of geometric partial differential equations

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Modeling isovolumetric phases in cardiac flows by an Augmented Resistive Immersed Implicit Surface method

Mathematical and numerical models for coupling surface and groundwater flows

Optimized numerical solutions of SIRDVW multiage model controlling SARS-CoV-2 vaccine roll out: An application to the Italian scenario

Interior penalty continuous and discontinuous finite element approximations of hyperbolic equations

A saddle point approach to an optimal boundary control problem for steady Navier-Stokes equations

Computational fluid dynamics of blood flow in an idealized left human heart

MATHICSE Technical Report : Efficient state/Parameter estimation in nonlinear unsteady PDEs by reduced basis ensemble Kalman filter

Multipatch Isogeometric Analysis for electrophysiology: Simulation in a human heart

Analysis of lumped parameter models for blood flow simulations and their relation with 1D models

Electromechanical Coupling In Cardiac Dynamics: The Active Strain Approach

MATHICSE Technical Report : Analysis of the internodes method for non-conforming discretizations of elliptic equations

Reduced basis method for parametrized elliptic optimal control problems

Analysis of the Yosida Method for the Incompressible Navier-Stokes Equations