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In this chapter we consider problems of the following form $$ \left\{ {\begin{array}{*{20}{c}} {Lu = - {\text{div}}(\mu \nabla u) + {\mathbf{b}}\cdot\nabla u + \sigma u = f\;{\text{in}}\Omega ,} \\ {u = 0,{\text{on}}\;\partial \Omega ,} \end{array}} \right.$$ (12.1) where µ, σ, f and b are given functions (or constants). In the most general case, we will suppose that µ ∈ L ∞ (Ω) with µ(x) ≥ μ 0 > 0, σ ∈ L2(Ω) with σ(x) ≥ 0 a.e. in Ω, b ∈ [L∞(Ω)]2 with div(b) ∈ L2(Ω), and f ∈ L2(Ω).

Discretization methods such as finite differences or finite elements were usually employed to provide high fidelity solution approximations for reduced order modeling of parameterized partial differential equations. In this paper, a novel discretization technique-Isogeometric Analysis (IGA) is used in combination with proper orthogonal decomposition (POD) for model order reduction of the time parameterized acoustic wave equations. We propose a new fully discrete IGA-Newmark-POD approximation and we analyze the associated numerical error, which features three components due to spatial discretization by IGA, time discretization with the Newmark scheme, and modes truncation by POD. We prove stability and convergence. Numerical examples are presented to show the effectiveness and accuracy of IGA-based POD techniques for the model order reduction of the acoustic wave equation.

Questo capitolo tratta un caposaldo del Calcolo Scientifico, quello della risoluzione numerica di problemi ai limiti, stazionari e evolutivi. Introduciamo le classiche tecniche alle differenze finite, seguite da quelle basate sul metodo agli elementi finiti. Applichiamo questi metodi al caso dei problemi ai limiti ellitrtici, parabolici ed iperbolici. Ricordiamo i principali risultati di consistenza, stabilita, convergenza e indichiamo come affrontare la risoluzione numerica dei problemi algebrici corrispondenti. Vengono proposti svariati esempi, prima di concludere il capitolo con una robusta serie di esercizi.

Approssimare una funzione f significa trovare una funzione ƒ di forma pio semplice che verra usata come surrogato di f. Questa strategia e frequentemente utilizzata nell’;integrazione numerica in cui invece di calcolare \( \int_a^b {f(x)dx} \) si calcola \( \int_a^b {\tilde f(x)dx} \) ove \( \tilde f \) sia una funzione facile da integrare (ad esempio, un polinomio), come mostreremo nel prossimo capitolo. In altri contesti, la funzione f potrebbe essere nota solo parzialmente attraverso i valori che essa assume in determinati punti. In tal caso la determinazione di \( \tilde f \) consentira di approssimare con una funzione continua l’;andamento della “legge f” che ha generato l’insieme di dati. I problemi che seguono danno un’idea di questo approccio.

The finite volume method is a very popular method for the space discretization of partial differential problems in conservation form. For an in-depth presentation of the method, we suggest the monographs [LeV02a], [Wes01] and [Tor09].

In this chapter we will introduce the basic concepts of optimal control for linear elliptic partial differential equations. At first we present the classical theory in functional spaces “a la J.L.Lions”, see [ Lio71, Lio72]; then we will address the methodology based on the use of the Lagrangian functional (see, e.g., [ Mau81, BKR00, Jam88]). Finally, we will show two different numerical approaches for control problems, based on the Galerkin finite element method.

In this chapter we focus more deeply on a number of aspects relating to the translation of the finite-element method into computer code. This implementation process can hide some pitfalls. Beyond the syntactic requirements of a given programming language, the need for a high computational efficiency leads to an implementation that is generally not the immediate translation of what has been seen during the theoretical presentation. Efficiency depends on many factors, including the language used and the architecture on which one works1. Personal experience can play a role as fundamental as learning from a textbook. Moreover, although spending time searching for a bug in the code or for a more efficient data structure can sometimes appear to be a waste of time, it (almost) never is. For this reason, we wish to propose the present chapter as a sort of “guideline” for trials that the reader can perform on his own, rather than a chapter to be studied in the traditional sense.

As we have seen, the finite element method for the solution of partial differential equations requires a “triangulation” of the computational domain, i.e. a partition of the domain in simpler geometric entities (for instance, triangles or quadrangles in two dimensions, tetrahedra, prisms or hexahedra in three dimensions), called the elements, which verify a number of conditions. Similar partitions stand at the base of other approximation methods, such as the finite volume method (see Chapter 9) and the spectral element method (see Chapter 10). The set of all elements is the so-called computational grid (or, simply, grid, or mesh).

Purpose: Morphological and functional remodeling of the left atrium (LA) caused by atrial fibrillation (AF) favors blood stasis and, consequently, stroke risk. Several clinical studies suggest that stroke risk stratification could be improved by using hemodynamic information on the LAand the left atrial appendage (LAA). The goal of this study was to develop a patient-specific computational fluid dynamics model (CFD) of the LA which may help quantify the hemodynamic implications of AF on a patient-specific basis. Methods: Specifically designed algorithms were applied to derive the 3D patient specific LA anatomical model and the LA motion field throughout the cardiac cycle from dynamic CT imaging in two patients with persistent AF. To perform CFD simulation in the LA, the arbitrary Lagrangian Eulerian formulation of the Navier-Stokes equations was applied in both sinus rhythm (SR) and AF conditions. The CFD model was constrained by assigning realistic inflow boundary conditions obtained from real Doppler measurements in AF patients. Results: Simulated velocity profile at the mitral valve showed a “physiological” behavior with amplitude compatible with AF condition. Moreover, in the SR simulation, peak A-wave velocity was 38 cmˑs-1 while the A-wave was absent in AF. LAA blood flow stasis was quantified by counting the number of particles remaining in the LAA throughout the cardiac cycles. After three cardiac cycles, 26% of the particles remained in the LAA in the SR condition, while 45.6% of them remained in the LAA in the AF condition. Conclusion: We presented our initial efforts towards the development of a patient specific CFD model of AF. To our knowledge, this is the first presented in literature. The model returned realistic blood flow patterns both at SR and during AF. In addition, it confirmed that AF episodes result in a reduced washout of the LAA which might lead to thrombi formation.

This chapter serves as a quick and elementary introduction of some of the basic algorithms that are used to solve a system of linear algebraic equations. For a more thorough presentation we advise the reader to refer to, e.g., [ QSS07, Chaps. 3 and 4], [Saa96] and [vdV03].

In this chapter we deal with time-dependent problems of hyperbolic type. For their origin and an in-depth analysis see e.g. [ Sal08, Chap. 4]. We will limit ourselves to considering the numerical approximation using the finite difference method, which was historically the first one to be applied to this type of equations. To introduce in a simple way the basic concepts of the theory, most of our presentation will concern problems depending on a single space variable. Finite element approximations will be addressed in Chapter 14, the extension to nonlinear problems in Chapter 15.

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. 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 illustrat...

Coronary artery disease, caused by the buildup of atherosclerotic plaques in the coronary vessel wall, is one of the leading causes of death in the world. For high-risk patients, coronary artery bypass graft is the preferred treatment. Despite overall excellent patency rates, bypasses may fail due to restenosis. In this context, the purpose of this work was to perform a parametric computational study of the fluid dynamics in patient-specific geometries with the aim of investigating a possible relationship between coronary stenosis degree and risk of graft failure. Firstly, we propose a strategy to prescribe realistic boundary conditions in the absence of measured data, based on an extension of Murray's law to provide the flow division at bifurcations in case of stenotic vessels and non-Newtonian blood rheology. Then, we carry out numerical simulations in three patients affected by severe coronary stenosis and treated with a graft, in which the stenosis degree is virtually varied in order to compare the resulting fluid dynamics in terms of hemodynamic indices potentially involved in restenosis development. Our findings suggest that low degrees of coronary stenosis produce a more disturbed fluid dynamics in the graft, resulting in hemodynamic conditions that may promote a higher risk of graft failure.

Navier-Stokes equations describe the motion of a fluid with constant density ρ in a domain Ω ⊂ ℝd (with d = 2,3). They read as follows $$\left\{ {\begin{array}{*{20}{l}} {\frac{{\partial {\mathbf{u}}}}{{\partial t}} - {\text{div}}[v(\nabla {\mathbf{u}} + \nabla {{\mathbf{u}}^T})] + ({\mathbf{u}}.\nabla ){\mathbf{u}} + \nabla {\mathbf{p}} = {\mathbf{f}},\quad x \in \Omega ,t > 0,} \\ {{\text{div}}{\mathbf{u}} = 0,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad x \in \Omega ,t > 0,} \end{array}} \right.$$ (16.1) u being the fluid’s velocity, p the pressure divided by the density (which will simply be called “pressure”), \(v={\mu \over \rho}\) the kinematic viscosity, µ the dynamic viscosity, and f a forcing term per unit of mass that we suppose belongs in L2(ℝ+; [L2(Ω)] d ) (see Sect. 5.2). The first equation is that of conservation of linear momentum, the second one that of conservation of mass, which is also called the continuity equation. The term (u • ∇)u describes the process of convective transport, while —div [v(∇u + ∇u T )] the process of molecular diffusion. System (16.1) can be derived by the analogous system for compressible flows introduced in Chapter 15 by assuming p constant, using the continuity equation (which, under current assumptions, takes the simplified form div u = 0) to simplify the various terms, and finally dividing the equation by ρ. Note that in the incompressible case (16.2) the energy equation has disappeared. Indeed, even though such an equation can still be written for incompressible flows, its solution can be found independently once the velocity field is obtained from the solution of (16.1).

In this work we present a stable proper orthogonal decomposition (POD)-Galerkin approximation for parametrized steady incompressible Navier-Stokes equations with low Reynolds number. Supremizers solutions are added to the reduced velocity space in order to obtain a stable reduced-order system, considering in particular the fulfillment of an inf-sup condition. The stability analysis is first carried out from a theoretical standpoint, then confirmed by numerical tests performed on a parametrized two-dimensional backward facing step flow.

To properly describe the electrical activity of the left ventricle, it is necessary to model the Purkinje fibers, responsible for the fast and coordinate ventricular activation, and their interaction with the muscular propagation. The aim of this work is to propose a methodology for the generation of a patient-specific Purkinje network driven by clinical measurements of the activation times related to pathological propagations. In this case, one needs to consider a strongly coupled problem between the network and the muscle, where the feedback from the latter to the former cannot be neglected as in a normal propagation. We apply the proposed strategy to data acquired on three subjects, one of them suffering from muscular conduction problems owing to a scar and the other two with a muscular pre-excitation syndrome (Wolff–Parkinson–White). To assess the accuracy of the proposed method, we compare the results obtained by using the patient-specific Purkinje network generated by our strategy with the ones obtained by using a non-patient-specific network. The results show that the mean absolute errors in the activation time is reduced for all the cases, highlighting the importance of including a patient-specific Purkinje network in computational models. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we consider the numerical approximation of high order Partial Differential Equations (PDEs) by means of NURBS-based Isogeometric Analysis (IGA) in the framework of the Galerkin method, for which global smooth basis functions with degree of continuity higher than C0 can be used. We derive a priori error estimates for high order elliptic PDEs under h-refinement, by extending existing results for second order PDEs approximated with IGA and specifically addressing the case of errors in lower order norms. We present some numerical results which both validate the proposed error estimates and highlight the accuracy of IGA. Then, we apply NURBS-based IGA to solve the fourth order stream function formulation of the Navier-Stokes equations for which we derive and numerically validate a priori error estimates under h-refinement. We solve the benchmark lid-driven cavity problem for Reynolds numbers up to 5000, by considering both the classical square cavity and the semi-circular cavity, which is exactly represented by NURBS.

The propagation of the electrical signal in the Purkinje network is the starting point for the activation of the ventricular muscular cells leading to the contraction of the ventricle. In the computational models, describing the electrical activity of the ventricle is therefore important to account for the Purkinje fibers. Until now, the inclusion of such fibers has been obtained either by using surrogates such as space-dependent conduction properties or by generating a network based on an a priori anatomical knowledge. The aim of this work was to propose a new method for the generation of the Purkinje network using clinical measures of the activation times on the endocardium related to a normal electrical propagation, allowing to generate a patient-specific network. The measures were acquired by means of the EnSite NavX system. This system allows to measure for each point of the ventricular endocardium the time at which the activation front, that spreads through the ventricle, has reached the subjacent muscle. We compared the accuracy of the proposed method with the one of other strategies proposed so far in the literature for three subjects with a normal electrical propagation. The results showed that with our method we were able to reduce the absolute errors, intended as the difference between the measured and the computed data, by a factor in the range 9-25 %, with respect to the best of the other strategies. This highlighted the reliability of the proposed method and the importance of including a patient-specific Purkinje network in computational models.

This paper is concerned with the solution of heterogeneous problems by the interface control domain decomposition (ICDD) method, a strategy introduced for the solution of partial differential equations in computational domains partitioned into subdomains that overlap. After reformulating the original boundary value problem by introducing new additional control variables, the unknown traces of the solution at internal subdomain interfaces; the latter are determined by requiring that the (a priori) independent solutions in each subdomain undergo the minimization of a suitable cost functional.We provide an abstract formulation for coupled heterogeneous problems and a general theorem of well-posedness for the associated ICDD problem. Then, we illustrate and validate an efficient algorithm based on the solution of the Schur-complement system restricted solely to the interface control variables by considering two kinds of heterogeneous boundary value problems: the coupling between pure advection and advection-diffusion equations and the coupling between Stokes and Darcy equations. In the latter case, we also compare the ICDD method with a classical approach based on the Beavers-Joseph-Saffman conditions. Copyright (c) 2014 John Wiley & Sons, Ltd.

In this paper, we are interested in the spatio-temporal dynamics of the transmembrane potential in paced isotropic and anisotropic cardiac tissues. In particular, we observe a specific precursor of cardiac arrhythmias that is the presence of alternans in the action potential duration. The underlying mathematical model consists of a reaction–diffusion system describing the propagation of the electric potential and the nonlinear interaction with ionic gating variables. Either conforming piecewise continuous finite elements or a finite volume-element scheme are employed for the spatial discretization of all fields, whereas operator splitting strategies of first and second order are used for the time integration. We also describe an efficient mechanism to compute pseudo-ECG signals, and we analyze restitution curves and alternans patterns for physiological and pathological cardiac rhythms.

We are interested in the numerical solution of the unsteady Navier-Stokes equations on large scale parallel architectures. We consider efficient preconditioners, such as the Pressure Convection-Diffusion (PCD), the Yosida preconditioner, the SIMPLE preconditioner, and the algebraic additive Schwarz preconditioner, for the linear systems arising from finite element discretizations using tetrahedral unstructured meshes and time advancing finite difference schemes. To achieve parallel efficiency, we introduce approximate versions of these preconditioners, based on their factorizations where each factor can be either inverted exactly or using an add-hoc preconditioner. We investigate their strong scalability for both classical benchmark problems and simulations relevant to hemodynamics, using up to 8192 cores. (C) 2013 Elsevier Ltd. All rights reserved.

In this paper we develop and analyze an efficient computational method for solving stochastic optimal control problems constrained by an elliptic partial differential equation (PDE) with random input data. We first prove both existence and uniqueness of the optimal solution. Regularity of the optimal solution in the stochastic space is studied in view of the analysis of stochastic approximation error. For numerical approximation, we employ a finite element method for the discretization of physical variables, and a stochastic collocation method for the discretization of random variables. In order to alleviate the computational effort, we develop a model order reduction strategy based on a weighted reduced basis method. A global error analysis of the numerical approximation is carried out, and several numerical tests are performed to verify our analysis.

Several computational challenges arise when evaluating the failure probability of a given system in the context of risk prediction or reliability analysis. When the dimension of the uncertainties becomes high, well established direct numerical methods can not be employed because of the "curse-of-dimensionality". Many surrogate models have been proposed with the aim of reducing computational effort. However, most of them fail in computing an accurate failure probability when the limit state surface defined by the failure event in the probability space lacks smoothness. In addition, for a stochastic system modeled by partial differential equations (PDEs) with random input, only a limited number of the underlying PDEs (order of a few tens) are affordable to solve in practice due to the considerable computational effort, therefore preventing the application of many numerical methods especially for high dimensional random inputs. In this work we develop hybrid and goal-oriented adaptive reduced basis methods to tackle these challenges by accurately and efficiently computing the failure probability of a stochastic PDE. The curse-of-dimensionality is significantly alleviated by reduced basis approximation whose bases are constructed by goal-oriented adaptation. Moreover, an accurate evaluation of the failure probability for PDE system with solution of low regularity in probability space is guaranteed by the certified a posteriori error bound for the output approximation error. At the end of this paper we suitably extend our proposed method to deal with more general PDE models. Finally we perform several numerical experiments to illustrate its computational accuracy and efficiency. (C) 2013 Elsevier B.V. All rights reserved.

We propose in this paper an Eulerian finite element approximation of a coupled chemical fluid-structure interaction problem arising in the study of mesoscopic cardiac biomechanics. We simulate the active response of a myocardial cell (here considered as an anisotropic, hyperelastic, and incompressible material), the propagation of calcium concentrations inside it, and the presence of a surrounding Newtonian fluid. An active strain approach is employed to account for the mechanical activation, and the deformation of the cell membrane is captured using a level set strategy. We address in detail the main features of the proposed method, and we report several numerical experiments aimed at model validation. Copyright (c) 2013 John Wiley & Sons, Ltd.

We study the Interface Control Domain Decomposition (ICDD) for the Stokes equation. We reformulate this problem introducing auxiliary control variables that represent either the traces of the fluid velocity or the normal stress across subdomain interfaces. Then, we characterize suitable cost functionals whose minimization permits to recover the solution of the original problem. We analyze the well-posedness of the optimal control problems associated to the different choices of the cost functionals, and we propose a discretization of the problem based on hp finite elements. The effectiveness of the proposed methods is illustrated through several numerical tests.

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. We then consider the solution of inverse problems both in the deterministic sense, by solving a least-squares problem, and in the statistical sense, by using a Bayesian framework for quantifying uncertainty. Two inverse problems arising in hemodynamics modeling are considered: (i) a simplified fluid-structure interaction model problem in a portion of a stenosed artery for quantifying the risk of atherosclerosis by identifying the material parameters of the arterial wall on the basis of pressure measurements; (ii) a simplified femoral bypass graft model for robust shape design under uncertain residual flow in the main arterial branch identified from pressure measurements.

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. Deriving from a deterministic one dimensional fluid structure interaction model, we establish the stochastic model as a coupled hyperbolic system incorporated with parametric uncertainties to describe the blood flow and pressure wave propagation in the arterial network. By applying a stochastic collocation method with sparse grid technique, we study systemically the statistics and sensitivity of the solution with respect to many different uncertainties in a relatively complete arterial network with potential physiological and pathological implications for the first time.

This work focuses on the finite element discretization of boundary value problems whose solution features either a discontinuity or a discontinuous conormal derivative across an in- terface inside the computational domain. The interface is characterized via a level set function. The discontinuities are accounted for by using suitable extension operators whose numerical implementa- tion requires a very low computational effort. After carrying out the error analysis, numerical results to validate our approach are presented in one, two, and three dimensions.

In this work we present numerical simulations of continuous flow left ventricle assist device implantation with the aim of comparing difference in flow rates and pressure patterns depending on the location of the anastomosis and the rotational speed of the device. Despite the fact that the descending aorta anastomosis approach is less invasive, since it does not require a sternotomy and a cardiopulmonary bypass, its benefits are still controversial. Moreover, the device rotational speed should be correctly chosen to avoid anomalous flow rates and pressure distribution in specific location of the cardiovascular tree. With the aim of assessing the differences between these two approaches and device rotational speed in terms of flow rate and pressure waveforms, we set up numerical simulations of network of one-dimensional models where we account for the presence of an outflow cannula anastomosed to different locations of the aorta. Then, we use the resulting network to compare the results of the two different cannulations for several stages of heart failure and different rotational speed of the device. The inflow boundary data for the heart and the cannulas are obtained from a lumped parameters model of the entire circulatory system with an assist device, which is validated with clinical data. The results show that ascending and descending aorta cannulations lead to similar waveforms and mean flow rate in all the considered cases. Moreover, regardless of the anastomosis region, the rotational speed of the device has an important impact on wave profiles; this effect is more pronounced at high RPM.

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.

Questa presentazione intende fornire una sintetica rassegna dell’attività svolta nel Dipartimento di Matematica del Politecnico e nell’Istituto Lombardo in un quarto di secolo in ricerca nell’ambito dei modelli matematici e nelle loro applicazioni in seguito alla fondazione del MOX, Laboratorio di modellistica e calcolo scientifico. Inoltre illustro la collaborazione realizzata con diversi colleghi di altri Dipartimenti del Politecnico, con riferimento ad applicazioni in differenti campi della matematica computazionale. Infine accenno ad alcuni progetti sviluppati dal MOX a livello nazionale ed internazionale in questi ultimi 10 anni.

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 hemodynamics 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 patient-specific medical images to fast parameterized 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 parameterization is employed to compare the effect of stenosis and anastomosis variation on the outcome of the surgery in several relevant cases.

In this paper, we consider large eddy simulations (LES) for human stenotic carotids in presence of atheromasic plaque, a pathological condition where transitional effects to turbulence may occur, with relevant clinical implications such as plaque rupture. We provide a reference numerical solution obtained at high resolution without any subgrid scale model, to be used to assess the accuracy of LES simulations. In the context we are considering, ie, hemodynamics, we cannot refer to a statistically homogeneous, isotropic, and stationary turbulent regime; hence, the classical Kolmogorov theory cannot be used. For this reason, a mesh size and a time step are deemed fine enough if they allow to capture all the features of the velocity field in the shear layers developed after the bifurcation. To assess these requirements, we consider a simplified model of the evolution of a 2D shear layer, a relevant process in the formation of transitional effects in our case. Then, we compare the results of LES σ model (both static and dynamic) and mixed LES models (where also a similarity contribution is considered). In particular, we consider a realistic scenario of a human carotid, and we use the reference solution as gold standard. The results highlight the accuracy of the LES σ models, especially for the static model.

We present a model for the electrophysiology in the heart to handle the electrical propagation through the Purkinje system and in the myocardium, with two-way coupling at the Purkinje-muscle junctions. In both the subproblems the monodomain model is considered, whereas at the junctions a resistor element is included that induces an orthodromic propagation delay from the Purkinje network towards the heart muscle. We prove a sufficient condition for convergence of a fixed-point iterative algorithm to the numerical solution of the coupled problem. Numerical comparison of activation patterns is made with two different combinations of models for the coupled Purkinje network/myocardium system, the eikonal/eikonal and the monodomain/monodomain models. Test cases are investigated for both physiological and pathological activation of a model left ventricle. Finally, we prove the reliability of the monodomain/monodomain coupling on a realistic scenario. Our results underlie the importance of using physiologically realistic Purkinje-trees with propagation solved using the monodomain model for simulating cardiac activation. (C) 2015 Elsevier Inc. All rights reserved.

Reduced basis methods are introduced for elliptic linear parametrized PDEs. Any reduced basis (RB) approximation is, in a nutshell, a (Petrov-)Galerkin projection onto an N-dimensional space V N (the RB space) that approximates the high-fidelity (say, finite element) solution of the given PDE, for any choice of the parameter within a prescribed parameter set. We illustrate the main steps needed to set up such methods efficiently. We discuss in detail projection methods, which represent the main feature of these techniques, and highlight the difference between Galerkin and least-squares RB methods. We show how to obtain a suitable offline/online decomposition meant to lower the computational complexity and then derive a posteriori error estimates for bounding the error of the RB solution with respect to the underlying high-fidelity solution. We consider the rather general case of inf-sup stable operators, of which coercive operators can be regarded as a particular — yet very relevant — instance. Proper orthogonal decomposition (POD) and greedy algorithms, two major techniques employed to build reduced spaces, are described thoroughly in Chaps. 6 and 7.

Partial differential equations (PDEs) represent the foundation upon which many mathematical models for real-life applications are erected. In order to solve these equations one almost invariably has to resort to efficient approximation techniques (such as the finite element method, for example). These are also called high-fidelity approximations, and represent the building blocks of any kind of reduced-order model, such as the reduced basis (RB) method for parametrized PDEs presented in this book. We review the formulation, analysis and approximation of three important classes of variational problems, namely strongly coercive, weakly coercive (also called noncoercive) and saddle-point (also called mixed variational) problems. Each case is accompanied by some examples of interest.

RB methods are revisited from both an algebraic and a geometric standpoint. A number of relationships between the Galerkin RB approximation (as well as least-squares RB approximation) and the Galerkin high-fidelity approximation (3.11) are highlighted, for the purpose of illustrating, in a more fitting way and from a different perspective, the mathematical structure underpinning RB methods. The key role played by the transformation matrix in defining orthogonal and oblique projections is emphasized.

This review paper addresses the so called geometric multiscale approach for the numerical simulation of blood flow problems, from its origin (that we can collocate in the second half of '90s) to our days. By this approach the blood fluid-dynamics in the whole circulatory system is described mathematically by means of heterogeneous problems featuring different degree of detail and different geometric dimension that interact together through appropriate interface coupling conditions. Our review starts with the introduction of the stand-alone problems, namely the 3D fluid-structure interaction problem, its reduced representation by means of 1D models, and the so-called lumped parameters (aka 0D) models, where only the dependence on time survives. We then address specific methods for stand-alone 3D models when the available boundary data are not enough to ensure the mathematical well posedness. These so-called "defective problems" naturally arise in practical applications of clinical relevance but also because of the interface coupling of heterogeneous problems that are generated by the geometric multiscale process. We also describe specific issues related to the boundary treatment of reduced models, particularly relevant to the geometric multiscale coupling. Next, we detail the most popular numerical algorithms for the solution of the coupled problems. Finally, we review some of the most representative works-from different research groups-which addressed the geometric multiscale approach in the past years. A proper treatment of the different scales relevant to the hemodynamics and their interplay is essential for the accuracy of numerical simulations and eventually for their clinical impact. This paper aims at providing a state-of-the-art picture of these topics, where the gap between theory and practice demands rigorous mathematical models to be reliably filled. (C) 2016 Elsevier B.V. All rights reserved.

I metodi a basi ridotte (reduced basis, RB in inglese) si applicano a equazioni a derivate parziali (EDP) dipendenti da parametri. Tali meto|di sono un esempio di tecniche di riduzione computazionale che consentono di ottenere in modo rapido e accurato la soluzione di tali equazioni, o la valutazione di un output espresso da una specifica funzione della soluzione.

International audience; In this work we propose a new numerical procedure for the simulation of time-dependent problems based on the coupling between the finite element method and the lattice Boltzmann method. The procedure is based on the Parareal paradigm and allows to couple efficiently the two numerical methods, each one working with its own grid size and time-step size. The motivations behind this approach are manifold. Among others, we have that one technique may be more efficient, or physically more appropriate or less memory consuming than the other depending on the target of the simulation and/or on the sub-region of the computational domain. Furthermore, the coupling with finite element method may circumvent some difficulties inherent to lattice Boltzmann discretization, for some domains with complex boundaries, or for some boundary conditions. The theoretical and numerical framework is presented for parabolic equations, in order to describe and validate numerically the methodology in a simple situation.

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. (C) 2015 Elsevier Ltd. All rights reserved.

We propose and analyse a new discontinuous reduced basis element method for the approximation of parametrized elliptic PDEs in partitioned domains. The method is built upon an offline stage (parameter independent) and an online (parameter dependent) one. In the offline stage we build a non-conforming (discontinuous) global reduced space as a direct sum of local basis functions generated independently on each subdomain. In the online stage, for any given value of the parameter, the approximate solution is obtained by ensuring the weak continuity of the fluxes and of the solution itself thanks to a discontinuous Galerkin approach. The new method extends and generalizes the methods introduced in [L. Iapichino, Ph.D. thesis, EPF Lausanne (2012); L. Iapichino, A. Quarteroni and G. Rozza, Comput. Methods Appl. Mech. Eng. 221–222 (2012) 63–82]. We prove its stability and convergence properties, as well as the spectral properties of the associated online algebraic system. We also propose a two-level preconditioner for the online problem which exploits the pre-existing decomposition of the domain and is based upon the introduction of a global coarse finite element space. Numerical tests are performed to verify our theoretical results.

We present a selection of numerical results dealing with the RB approximation of the parametrized problems formulated in the previous chapter. For each problem we highlight the RB method’s computational performance, assess its accuracy by means of a posteriori error bounds, and show various options for the construction of the RB space (either via POD or the greedy algorithm) and different projection criteria (G-RB versus LS-RB methods). Here we focus on linear affine PDEs, and defer nonaffine and nonlinear problems to Chaps. 10 and 11 respectively.

We discuss relevant theoretical features of the RB methods seen in the previous chapters. We specifically highlight those properties of the solution manifold that are directly inherited from the parametrized differential operators. We define the Kolmogorov n-width to measure how well suited n-dimensional subspaces are to approximate the solution manifold. At the end we show that a wise selection of snapshots yields exponential convergence when approximating the solution manifold. This key property, that warrants the computational efficiency of RB methods, is provided by greedy algorithms and POD techniques used to construct RB spaces that are introduced in the subsequent chapters.

The RB method is extended to the case of parametrized nonlinear PDEs. Examples are discussed concerning the Navier-Stokes equations and an elliptic semilinear equation. Both high-fidelity and RB approximations, as well as their interplay with Newton linearization, are analyzed, before considering in detail the case of Navier-Stokes equations. The underlying RB construction is essentially the same as for the previous linear PDEs, hence the focus is put on the characteristic aspects of the efficient treatment of nonlinear terms.