Keywords
 A6.1.1. Continuous Modeling (PDE, ODE)
 A6.1.4. Multiscale modeling
 A6.1.5. Multiphysics modeling
 A6.2.1. Numerical analysis of PDE and ODE
 A6.3.1. Inverse problems
 A6.3.2. Data assimilation
 A6.3.4. Model reduction
 B2.2.1. Cardiovascular and respiratory diseases
 B2.4.1. Pharmaco kinetics and dynamics
1 Team members, visitors, external collaborators
Research Scientists
 Miguel Angel Fernandez Varela [Team leader, Inria, Senior Researcher, HDR]
 Céline Grandmont [Inria, Senior Researcher, HDR]
 Damiano Lombardi [Inria, Researcher, HDR]
 Marina Vidrascu [Inria, Emeritus]
Faculty Members
 Muriel Boulakia [Sorbonne Université, Associate Professor, HDR]
 Olga Mula [Université de Dauphine, Associate Professor]
PostDoctoral Fellow
 Mihai Nechita [Inria, from Nov 2020]
PhD Students
 Mocia Agbalessi [Cardiac Simulation Imaging Software, CIFRE]
 Marguerite Champion [Inria, from Nov 2020]
 Daniele Carlo Corti [Inria, from Oct 2020]
 Sara Costa Faya [Inria, from Sep 2020]
 Maria Fuente Ruiz [Inria, from Mar 2020]
 Felipe Galarce Marin [Inria]
 Fannie Gerosa [Inria]
 Fabien Lespagnol [Politecnico di Milano, from Nov 2020]
 Haibo Liu [Notocord Systems, from Sep 2020]
 Fabien Raphel [Notocord Systems, CIFRE]
Technical Staff
 Daniele Carlo Corti [Inria, Engineer, until Sep 2020]
 Vicente Mataix Ferrandiz [Inria, Engineer, from Oct 2020]
Interns and Apprentices
 Fabien Lespagnol [Inria, from Apr 2020 until Jul 2020]
 Sacha Marmouget [Inria, from Apr 2020 until Jul 2020]
Administrative Assistants
 Laurence Bourcier [Inria]
 Julien Guieu [Inria]
2 Overall objectives
COMMEDIA is a joint projectteam of the Inria Research Center of Paris and the JacquesLouis Lions Laboratory (LJLL) of Sorbonne Université and CNRS (UMR7598). The research activity of COMMEDIA focuses on the numerical simulation of biofluid flows in the human body, more specifically, blood flows in the cardiovascular system and air flows in the respiratory system. These simulations are intended to complement available clinical data with the following purpose: help clinicians or bioengineers to enhance the understanding of physiological phenomena, to improve diagnosis and therapy planning or to optimize medical devices. The main main objectives of COMMEDIA are:
 the development of appropriate mathematical models and efficient numerical methods for the simulations and for the interaction of simulations with measured data;
 the mathematical analysis of these models and numerical techniques;
 the development and validation of scientific computing software which implements these numerical techniques.
A distinctive feature of the mathematical models considered in COMMEDIA is that they often couple different types of partial differential equations (PDEs). This heterogeneous character in the models is a mathematical manifestation of the multiphysics nature of the considered problems.
3 Research program
3.1 Multiphysics modeling and simulation
The research activity in terms of modeling and simulation (i.e., the socalled forward problem) is driven by two application domains related to the cardiovascular and the respiratory systems.
3.1.1 Cardiovascular hemodynamics
We distinguish between cardiac hemodynamics (blood flow inside the four chambers of the heart) and vascular hemodynamics (blood flow in the vessels of the body).
Cardiac hemodynamics. The numerical simulation of cardiac hemodynamics presents many difficulties. We can mention, for instance, the large deformation of the cardiac chambers and the complex fluidstructure interaction (FSI) phenomena between blood, the valves and the myocardium. Blood flow can be described by the incompressible NavierStokes equations which have to be coupled with a biophysical model of the myocardium electromechanics and a mechanical model of the valves. The coupling between the fluid and the solid media is enforced by kinematic and dynamic coupling conditions, which guarantee the continuity of velocity and stresses across the interface. In spite of the significant advances achieved since the beginning of this century (see, e.g., 72, 80, 71, 74, 64), the simulation of all the fluidstructure interaction phenomena involved in the heart hemodynamics remains a complex and challenging problem.
Heart valves are definitely a bottleneck of the problem, particularly due to their fast dynamics and the contact phenomena at high pressuredrops. Computational cost is recognized as one of the key difficulties, related to the efficiency of the FSI coupling method and the robustness of the contact algorithm. Furthermore, the numerical discretization of these coupled systems requires to deal with unfitted fluid and solid meshes, which are known to complicate the accuracy and/or the robustness of the numerical approximations (see Section 3.3.2 below).
The ultimate goal of the proposed research activity is the simulation of the complete fluidstructurecontact interaction phenomena involved within the heart. Most of this work will be carried out in close collaboration with the M3DISIM projectteam, which has a wide expertise on the modeling, simulation and estimation of myocardium electromechanics. We will also consider simplified approaches for cardiac hemodynamics (see, e.g., 44, 59, 62). The objective is to develop mathematically sound models of reduced valve dynamics with the purpose of enhancing the description of the pressure dynamics right after the opening/closing of the valve (traditional models yield spurious pressure oscillations).
Vascular hemodynamics. The modeling and simulation of vascular hemodynamics in large vessels has been one of the core research topics of some members of COMMEDIA, notably as regards the fluidstructure interaction phenomena. Here we propose to investigate the modeling of pathological scenarios, such as the hemorrhage phenomena in smaller vessels. Modeling of hemorrhage is motivated by the medical constatation that, after a primary vessel wall rupture, secondary vessel wall ruptures are observed. Biologists postulate that the mechanical explanation of this phenomena might be in the change of applied stress due to blood bleeding. We propose to model and simulate the underlying coupled system, blood vessel flow through the external tissue, to estimate the effect of the subsequent stress variation.
3.1.2 Respiratory flows
The motivation of the proposed research activities is to develop a hierarchy of easily parametrizable models allowing to describe and efficiently simulate the physical, mechanical and biological phenomena related to human respiration, namely,
ventilation, particle deposition, gas diffusion and coupling with the circulatory system.
Ventilation. The current modeling approaches (either 3D–0D coupled models where the 3D NavierStokes equations are solved in truncated geometries of the bronchial tree with appropriate lumped boundary conditions, or 0D–3D coupled models where the lung parenchyma is described by a 3D elastic media irrigated by a simplified bronchial tree) provide satisfactory results in the case of mechanical ventilation or normal breathing. Realistic volumeflow phase portraits can also be simulated in the case of forced expiration (see 46, 56, 77), but the magnitude of the corresponding pressure is not physiological. The current models must be enriched since they do not yet correctly describe all the physiological phenomena at play. We hence propose to extend the 0D–3D (bronchial tree–parenchyma) model developed in the team, by considering a nonlinear, viscoelastic and possibly poroelastic description of the parenchyma with appropriate boundary conditions that describe ribs and adjacent organs and taking into account an appropriate resistive model.
So far, the motion of the trachea and proximal bronchi has been neglected in the ventilation models (see, e.g., 78). These features can be critical for the modeling of pathologic phenomena such as sleep apnea and occlusion of the airways. This would be a longterm goal where fluidstructure interaction and the possible contact phenomena will be taken into account, as in the simulation of cardiac hemodynamics (see Section 3.1.1).
Aerosol and gas diffusion. The dynamics of aerosols in the lung have been widely studied from the mathematical modeling standpoint. They can be described by models at different scales: the microscopic one for which each particle is described individually, the mesoscopic (or kinetic) one for which a density of probability is considered, or the macroscopic one where reactiondiffusion equations describing the behavior of the constituant concentration are considered. The objective of COMMEDIA will mainly be to develop the kinetic approach that allows a precise description of the deposition area at controlled computational costs. Part of this study could be done in collaboration with colleagues from the Research Center for Respiratory Diseases at Inserm Tours (UMR1100).
The macroscopic description is also appropriate for the diffusion of gases (oxygen and carbon dioxide) in the bronchial tree (see 73). Regarding the influence of the carrier gas, if the patient inhales a different mixture of air such as a HeliumOxygen mixture, the diffusion mechanisms could be modified. In this context, the goal is to evaluate if the crossdiffusion (and thus the carrier gas) modifies the quantities of oxygen diffused. Part of this work will be carried out in collaboration with members of the LJLL and of the MAP5.
As a long term goal, we propose to investigate the coupling of these models to models of diffusion in the blood or to perfusion models of the parenchyma, and thus, have access thanks to numerical simulations to new indices of ventilation efficiency (such as dissolved oxygen levels), depending on the pathology considered or the resting or exercise condition of the patient.
3.2 Simulation with data interaction
The second research axis of COMMEDIA is devoted to the interaction of numerical simulations with measured data. Several research directions related to two specific applications are described below: blood flows and cardiac electrophysiology, for which the mathematical models have been validated against experimental data. This list is not exhaustive and additional problems (related to cardiac and respiratory flows) shall be considered depending on the degree of maturity of the developed models.
3.2.1 Fluid flow reconstruction from medical imaging
A first problem which is currently under study at COMMEDIA is the reconstruction of the flow state from Doppler ultrasound measurements. This is a cheap and largely available imaging modality where the measure can be interpreted as the average on a voxel of the velocity along the direction of the ultrasound beam. The goal is to perform a fullstate estimation in a time compatible with a realistic application.
A second problem which is relevant is the flow and wall dynamics reconstruction using 4Dflow MRI. This imaging modality is richer than Doppler ultrasound and provides directly a measure of the 3D velocity field in the voxels. This enables the use of direct estimation methods at a reduced computational cost with respect to the traditional variational data assimilation approaches. Yet, the sensitivity of the results to subsampling and noise is still not well understood.
We also propose to address the issues related to uncertainty quantification. Indeed, measurements are corrupted by noise and the parameters as well as the available data of the system are either hidden or not known exactly (see 70). This uncertainty makes the estimation difficult and has a large impact on the precision of the reconstruction, to be quantified in order to provide a reliable tool.
3.2.2 Inverse problem in electrocardiography
The objective of the inverse problem in electrocardiography is to recover information about the cardiac electrical activity from electrical measurements on the body surface (for instance from electrocardiograms). We propose to investigate approaches based on recent methods for the Cauchy problem reported in 54. Basically, the idea consists in regularizing the discrete inverse problem using stabilized finite element methods, without the need of integrating a priori knowledge of the solution, only regularity on the exact solution is required.
3.2.3 Safety pharmacology
One of the the most important problems in pharmacology is cardiotoxicity (see 69). The objective is to predict whether or not a molecule alters in a significant way the normal functioning of the cardiac cells. This problem can be formulated as inferring the impact of a drug on the ionic currents of each cell based on the measured electrical signal (e.g., electrograms from MicroElectrodes Arrays). The proposed approach in collaboration with two industrial partners (NOTOCORD and Ncardia) consists in combining available realistic data with virtual ones obtained by numerical simulations. These two datasets can be used to construct efficient classifiers and regressors using machine learning tools (see 51) and hence providing a rapid way to estimate the impact of a molecule on the electrical activity. The methodological aspects of this work are addressed in Section 3.3.3.
3.3 Methodological core
The work described in this section is aimed at investigating fundamental mathematical and numerical problems which arise in the first two research axes.
3.3.1 Mathematical analysis of PDEs
The mathematical analysis of the multiscale and multiphysics models are a fundamental tool of the simulation chain. Indeed, wellposedness results provide precious insights on the properties of solutions of the systems which can, for instance, guide the design of the numerical methods or help to discriminate between different modeling options.
Fluidstructure interaction. Most of the existing results concern the existence of solutions locally in time or away from contacts. One fundamental problem, related to the modeling and simulation of valve dynamics (see Sections 3.1.1 and 3.3.2), is the question of whether or not the model allows for contact (see 68, 66). The proposed research activity is aimed at investigating the case of both immersed rigid or elastic structures and explore if the considered model allows for contact and if existence can be proved beyond contact. The question of the choice of the model is crucial and considering different types of fluid (newtonian or non newtonian), structure (smooth or rough, elastic, viscoelastic, poroelastic), or various interface conditions has an influence on whether the model allows contact or not.
Fluid–structure mixture. The main motivation to study fluidsolid mixtures (i.e., porous media consisting of a skeleton and connecting pores filled with fluid) comes from the modeling of the lung parenchyma and cerebral hemorrhages (see Sections 3.1.1–3.1.2). The Biot model is the most widely used in the literature for the modeling of poroelastic effects in the arterial wall. Here, we propose to investigate the recent model proposed by the M3DISIM projectteam in 58, which allows for nonlinear constitutive behaviors and viscous effects, both in the fluid and the solid. Among the questions which will be addressed, some of them in collaboration with M3DISIM, we mention the justification of the model (or its linearized version) by means of homogenization techniques and its wellposedness.
Fluid–particle interaction. Mathematical analysis studies on the NavierStokesVlasov system for fluidparticle interaction in aerosols can be found in 48, 50. We propose to extend these studies to more realistic models which take into account, for instance, changes in the volume of the particles due to humidity.
3.3.2 Numerical methods for multiphysics problems
In this section we describe the main research directions that we propose to explore as regards the numerical approximation of multiphysics problems.
Fluidstructure interaction. The spatial discretization of fluidstructure interaction (FSI) problems generally depends on the amount of solid displacement within the fluid. Problems featuring moderate interface displacements can be successfully simulated using (moving) fitted meshes with an arbitrary LagrangianEulerian (ALE) description of the fluid. This facilitates, in particular, the accurate discretization of the interface conditions. Nevertheless, for problems involving large structural deflections, with solids that might come into contact or that might break up, the ALE formalism becomes cumbersome. A preferred approach in this case is to combine an Eulerian formalism in the fluid with an unfitted mesh discretization, in which the fluidstructure interface deforms independently of a background fluid mesh. In general, traditional unfitted mesh approaches (such as the immersed boundary and the fictitious domain methods 76, 47, 65, 45) are known to be inaccurate in space. These difficulties have been recently circumvented by a Nitschebased cutFEM methodolgy (see 42, 52). The superior accuracy properties of cutFEM approaches comes at a price: these methods demand a much more involved computer implementation and require a specific evaluation of the interface intersections.
As regards the time discretization, significant advances have been achieved over the last decade in the development and the analysis of timesplitting schemes that avoid strong coupling (fully implicit treatment of the interface coupling), without compromising stability and accuracy. In the vast majority these studies, the spatial discretization is based on body fitted fluid meshes and the problem of accuracy remains practically open for the coupling with thickwalled structures (see, e.g., 63). Within the unfitted mesh framework, splitting schemes which avoid strong coupling are much more rare in the literature.
Computational efficiency is a major bottleneck in the numerical simulation of fluidstructure interaction problems with unfitted meshes. The proposed research activity is aimed at addressing these issues. Another fundamental problem that we propose to face is the case of topology changes in the fluid, due to contact or fracture of immersed solids. This challenging problem (fluidstructurecontactfracture interaction) has major role in many applications (e.g., heart valves repair or replacement, breakup of drugloaded microcapsules) but most of the available studies are still merely illustrative. Indeed, besides the numerical issues discussed above, the stability and the accuracy properties of the numerical approximations in such a singular setting are not known.
Fluid–particle interaction and gas diffusion.
Aerosols can be described through mesoscopic equations of kinetic type, which provide a tradeoff between model complexity and accuracy. The strongly coupled fluidparticle system involves the incompressible NavierStokes equations and the Vlasov equation. The proposed research activity is aimed at investigating the theoretical stability of timesplitting schemes for this system. We also propose to extend these studies to more complex models that take into account the radius growth of the particles due to humidity, and for which stable, accurate and mass conservative schemes have to be developed.
As regards gas diffusion, the mathematical models are generally highly nonlinear (see, e.g., 73, 75, 49). Numerical difficulties arise from these strong non linearities and we propose to develop numerical schemes able to deal with the stiff geometrical terms and that guarantee mass conservation. Moreover, numerical diffusion must be limited in order to correctly capture the time scales and the crossdiffusion effects.
3.3.3 Statistical learning and mathematical modeling interactions
Machine learning and in general statistical learning methods (currently intensively developed and used, see 43) build a relationship between the system observations and the predictions of the QoI based on the a posteriori knowledge of a large amount of data. When dealing with biomedical applications, the available observations are signals (think for instance to images or electrocardiograms, pressure and Doppler measurements). These data are high dimensional and the number of available individuals to set up precise classification/regression tools could be prohibitively large. To overcome this major problem and still try to exploit the advantages of statistical learning approaches, we try to add, to the a posteriori knowledge of the available data an a priori knowledge, based on the mathematical modeling of the system. A large number of numerical simulations is performed in order to explore a set of meaningful scenarios, potentially missing in the dataset. This in silico database of virtual experiments is added to the real dataset: the number of individuals is increased and, moreover, this larger dataset can be used to compute semiempirical functions to reduce the dimension of the observed signals.
Several investigations have to be carried out to systematically set up this framework. First, often there is not a single mathematical model describing a physiological phenomenon, but hierarchies of model of different complexity. Every model is characterized by a model error. How can this be accounted for? Moreover, several statistical estimators can be set up and eventually combined together in order to improve the estimations (see 81). Other issues have an actual impact and has to be investigated: what is the optimal number of in silico experiments to be added? What are the most relevant scenarios to be simulated in relation to the statistical learning approach considered in order to obtain reliable results? In order to answer to these questions, discussions and collaborations with statistics and machine learning groups have to be developed.
3.3.4 Tensor approximation and HPC
Tensor methods have a recent significant development because of their pertinence in providing a compact representation of large, highdimensional data. Their applications range from applied mathematics and numerical analysis to machine learning and computational physics. Several tensor decompositions and methods are currently available (see 67). Contrary to matrices, for tensors of order higher or equal to three, there does not exist, in general, a best low rank approximation, the problem being ill posed (see 79). Two main points will be addressed: (i) The tensor construction and the multilinear algebra operations involved when solving highdimensional problems are still sequential in most of the cases. The objective is to design efficient parallel methods for tensor construction and computations; (ii) When solving highdimensional problems, the tensor is not assigned; instead, it is specified through a set of equations and tensor data. Our goal is to devise numerical methods able to (dynamically) adapt the rank and the discretization (possibly even the tensor format) to respect the chosen error criterion. This could, in turn, improve the efficiency and reduce the computational burden.
These sought improvements could make the definition of parsimonious discretizations for kinetic theory and uncertainty quantification problems (see Section 3.2.1) more efficient and suitable for a HPC paradigm. This work will be carried out in collaboration with Olga Mula (Université ParisDauphine) and the ALPINES and MATHERIALS projectteams.
4 Application domains
4.1 Cardiovascular hemodynamics
The heart is a double pump whose purpose is to deliver blood to the tissue and organs of the body. This function is made possible through the opening and closing of the heart valves. Cardiac diseases generally manifest by affecting the pumping function of the heart. Numerical simulations of cardiac hemodynamics, in normal and pathological conditions, are recognized as a tool of paramount importance for improving the understanding, diagnosis and treatment of cardiac pathologies, and also for the development of implantable devices (see, e.g., 74, 57). As an example, we can mention the case of cardiac mitral valve regurgitation, one of the most common heart valve diseases. For this pathology, clinical data are known to be insufficient for determining the optimal timing for surgery, the best surgical strategy and the longterm outcome of a surgical repair. Contrary to imaging techniques, numerical simulations provide local information, such as pressure and stresses, which are of fundamental importance for the prediction of the mechanical behavior of native valves and of implantable devices.
4.2 Respiratory flows
Respiration involves the transport of air through the airways from the mouth to the alveoli of the lungs. These units where diffusion of oxygen and carbon dioxide take place, are surrounded by a viscoelastic medium (the parenchyma) consisting of blood vessels and collagen fibers. Air flows due to the displacement of the diaphragm, which drives the pulmonary parenchyma. Accidental inhalations of foreign bodies or pathologies such as asthma, emphysema and fibrosis might prevent the lung of fulfilling its function. Therapies mostly use aerosols (set of small particles, solid or liquid), which must reach the specific areas of the lung targeted for treatment. Understanding the airflow mechanisms within the respiratory network is a fundamental ingredient for predicting the particles motion and their deposition (see, e.g., 55). Moreover, understanding of the gas diffusion in the lung is also of major importance since the main fonction of this organ is to deliver oxygen to the blood.
4.3 Safety pharmacology
The problem of safety pharmacology can be summarized as follows: given a molecule which is a candidate to become a drug, is its use dangerous due to side effects? Among all the different problems to be addressed, one of the most relevant questions in pharmacology is cardiotoxicity (see 69). More precisely, the objective is to determine whether or not a molecule alters in a significant way the normal functioning of the cardiac cells. To answer these questions, the CiPA initiative promotes the introduction of novel techniques and their standardisation (see 61). One of the proposed tests of the CiPA panel is to measure the the electrical activity using MicroElectrodes Array: these are microchips that record the electrical activity of an ensemble of cells. The task is to infer the impact of a drug on the ionic currents of each cell based on the electrical signal measured (electrograms) and, in perspective, to be able to assess whether a molecule can induce arrhythmia (see 60).

5 Highlights of the year

Céline Grandmont participated to the Inria covid mission project Prelifa.
6 New software and platforms
6.1 New software
6.1.1 FELiScE
 Name: Finite Elements for Life SCiences and Engineering problems
 Keywords: Finite element modelling, Cardiac Electrophysiology, Cardiovascular and respiratory systems
 Functional Description: FELiScE is a finite element code which the M3DISIM and REO projectteams initially jointly develop in order to build up on their respective experiences concerning finite element simulations. One specific objective of this code is to provide in a unified software environment all the stateoftheart tools needed to perform simulations of the complex respiratory and cardiovascular models considered in the two teams – namely involving fluid and solid mechanics, electrophysiology, and the various associated coupling phenomena. FELISCE is written in C++, and may be later released as an opensource library. FELiScE was registered in July 2014 at the Agence pour la Protection des Programmes under the Inter Deposit Digital Number IDDN.FR.001.350015.000.S.P.2014.000.10000.

URL:
https://
team. inria. fr/ commedia/ software/ felisce/  Authors: JeanFrédéric Gerbeau, Miguel Ángel Fernández, Dominique Chapelle, Marina Vidrascu, Philippe Moireau
 Contact: Miguel Ángel Fernández
 Participants: Matteo Aletti, Daniele Carlo Corti, Dominique Chapelle, Miguel Ángel Fernández, Benoit Fabreges, Axel Fourmont, JeanFrédéric Gerbeau, Fannie Gerosa, Sébastien Gilles, Mikel Landajuela Larma, Damiano Lombardi, Vicente Mataix Ferrandiz, Philippe Moireau, Irène VignonClementel, Marina Vidrascu
6.1.2 FELiScENS
 Keywords: Incompressible flows, Thinwalled solids
 Functional Description: FELiScENS is a set finite elements solvers for incompressible fluids (fractionalstep schemes) and nonlinear thinwalled structures (3D shells, and 2D curved beams) developed in the framework of the FELiScE library. FELiSCeNS was registered in 2018 at the Agence pour la Protection des Programmes Inter Deposit Digital Number IDDN.FR.001.270015.000.S.A.2018.000.31200.
 Authors: Benoit Fabreges, Axel Fourmont, Miguel Ángel Fernández, JeanFrédéric Gerbeau, Marina Vidrascu
 Contact: Miguel Ángel Fernández
 Participants: Benoit Fabreges, Miguel Ángel Fernández, Axel Fourmont, JeanFrédéric Gerbeau, Marina Vidrascu
6.1.3 DCIMaL
 Keyword: Cardiac Electrophysiology
 Functional Description: DCIMaL is a Python and C++ software for safety pharmacology studies and particularly field potentials signals measured with microelectrode array (MEA). The software includes a solver for field potential simulations and a dictionary of entries corresponding to features which can be extracted from real or simulated potential signals. It also includes an algorithm for drug classification (channel blockade or torsadogenic risk) and a tool for estimating ion channel activity (based on the CMAES library). DCIMaL was registered in 2018 at the Agence pour la Protection des Programmes Inter Deposit Digital Number IDDN.FR.001.270003.000.S.P.2018.000.31230
 Authors: JeanFrédéric Gerbeau, Damiano Lombardi, Fabien Raphel
 Contact: Damiano Lombardi
 Participants: Fabien Raphel, JeanFrédéric Gerbeau, Damiano Lombardi
7 New results
7.1 Fluid flow reconstruction from medical imaging
Participants: Felipe Galarce Marin, Damiano Lombardi, Olga Mula.
In 22 we develop a reducedorder approach to perform a fast estimation of hemodynamics quantities of interest based on measurements which can be modelled as linear forms applied to the system state, corrupted by some noise. A prototypical example of application is the estimation of the pressure (or of the wall shear stress) by using data coming from Doppler Ultrasound imaging or 4dflow MRI.
7.2 Safety pharmacology
Participants: Damiano Lombardi, Fabien Raphel.
In 26 we propose to use a double greedy algorithm to approximate the observabletoparameters map in an electrophysiology model. This approximation is used as a nonlinear preconditioner in a parameter estimation problem solved by means of an Unscented Kalman filter. The results shown that the nonlinear preconditioning strategy produced a significant speedup of the filter convergence and reduced the error mean and standard deviation.7.3 Mathematical analysis of PDEs
Participants: Muriel Boulakia, Céline Grandmont.
In 41 we consider a quasistatic fluidstructure interaction problem where the fluid is modeled by the Stokes equations and the structure is an active and elastic medium. More precisely, the displacement of the structure verifies the equations of elasticity with an active stress, which models the presence of internal biological motors in the structure. Under smallness assumptions on the data, we prove the existence of a unique solution for this strongly coupled system. These kind of models describe selfpropelled structures such as cilia and flagella, that are examples of such soft materials that deform themselves using internal biological motors and thus, induce a flow within the surrounding fluid.
7.4 Numerical methods for multiphysics problems
Participants: Miguel Ángel Fernández Varela, Fannie Gerosa.
In 36, robust a priori error estimates are derived for the unfitted meshe semiimplicit coupling scheme recently introduced in 20, for the simulation of incompresible fluidstructure interaction involving thinwalled solids. The analysis shows that, under a hyperbolicCFL condition, the leading term in the energy error scales as $\mathcal{O}\left({h}^{r\frac{1}{2}}\right)$, where $r=1,2$ stands for the extrapolation order of the solid velocity in the viscous luid substep. The theoretical findings are illustrated via a numerical experiments which show, in particular, that the considered method avoids the spatial nonuniformity issues of standard loosely coupled schemes and that it delivers practically the same accuracy as the fully implicit scheme.
In 35, we consider a fully discrete loosely coupled scheme for incompressible fluidstructure interaction based on the time semidiscrete splitting method introduced in 53. The splittling method uses a RobinRobin type coupling that allows for a segregated solution of the solid and the fluid systems, without inner iterations. For the discretisation in space we consider piecewise affine continuous finite elements for all the fields and ensure the infsup condition by using a BrezziPitkäranta type pressure stabilization. The interfacial fluidstresses are evaluated in a variationally consistent fashion, that is shown to admit an equivalent Lagrange multiplier formulation. We prove that the method is unconditionally stable and robust with respect to the amount of addedmass in the system. Furthermore, we provide an error estimate that shows the error in the natural energy norm for the system is $\mathcal{O}\left(\sqrt{T(\Delta t+h)}\right)$ where $T$ is the final time, $\Delta t$ the timestep length and $h$ the space discretization parameter.
The numerical approximation of incompressible fluidstructure interaction problems with Lagrange multiplier is generally based on strongly coupled schemes. This delivers unconditional stability but at the expense of solving a computationally demanding coupled system at each timestep. For the case of the coupling with immersed thinwalled solids, in 31 we introduce a class of semiimplicit coupling schemes which avoids strongly coupling without compromising stability and accuracy. A priori energy and error estimates are derived. The theoretical results are illustrated through numerical experiments in an academic benchmark.
7.5 Statistical learning and mathematical modeling interactions
Participants: Damiano Lombardi, Olga Mula, Fabien Raphel.
In 30 the aim is to develop training algorithms that deliver local minima of better quality than the ones obtained with usual training approaches such as stochastic gradient descent. We also attempt to bring new quantitative results on the generalization properties of the constructed networks. For this, we have adopted a recent point of view which connects deep learning with optimal control as a way to define a notion of a continuous underlying learning problem. In this view, neural networks can be interpreted as a discretization of a parametric ordinary differential equation which, in the limit, defines a continuousdepth neural network. The learning task then consists in finding the best ODE parameters for the problem under consideration, and their number increases with the accuracy of the time discretization. Although important steps have been taken to realize the advantages of such continuous formulations, most current learning techniques fix a discretization, which implies that the number of layers is fixed. In this work, we introduce an iterative adaptive algorithm where we progressively refine the time discretization. This, in turn, means that we increase the number of layers and the depth of the network across the iterations. Provided that certain tolerances are met across the iterations, we have proved that the strategy converges to the underlying continuous problem. One salient advantage of such a shallowtodeep approach is that it helps to benefit in practice from the high approximation properties of deep networks by mitigating overparametrization issues in the training.
7.6 Tensor approximation and HPC
Participants: Damiano Lombardi.
In 40 we develop a method to compress a given tensor into a Canonical Polyadic format. This is known to be in general an illposed problem. By suitably modifying the TTSVD method (used in general to construct a Tensor Train format approximation) we propose a method which can produce a stable CP approximation. The tests show that the proposed approach has encouraging performances, especially in highdimensional settings, when compared to other methods proposed in the literature, such as ALS or ASVD.
7.7 Miscellaneous
Participants: Damiano Lombardi, Olga Mula.
In 23 we propose a reducedorder method for the fast reconstruction of facial muscles from CTscan images. In this short note we investigate how the information available in the form of points lying on the muscle surface could be exploited in order to obtain a full 3D reconstruction of the muscle.
In 25 we propose a simplified fluidstructure interaction model for arterioles in order to provide a mechanical insight of experimental observations and validate an hypothesis on the biological processes leading to microhaemorrhages. The simulations performed confirmed the medical doctor hypothesis on the most critical configurations leading to microhaemorrhages in the nervous system microcirculation.
In 11 we develop a general forecasting method to predict series of hospitalized and dead people using a model reduction method involving SIR compartmental models. The obtained results seem satisfactory not only to us: eminent French epidemiologists, physicians and virologists with whom we have discussed, recognise that our approach has predictive qualities worthy of interest. They have invited us to deploy our approach by making it available as a platform for predicting the evolution of the disease. Also, as a further step, we are currently starting to enlarge the methodology in order to include interregional population mobillity and information on the viral level concentration which can be extracted from the analysis of wasted waters.
The paper 15 is a contribution on state estimation problems using reduced model algorithms. Here we study the notion of optimality of state estimation algorithms: we define a certain criterion to describe the reconstruction quality and we study what is the optimal reconstruction algorithm that provides it. In general, this optimal algorithm cannot be computed in practice. However, we show that if we restrict ourselves to lineal algorithms, the optimal linear algorithm is computable and we provide a numerical illustration of it.
In 24, we have made a contribution to the topic of domain decomposition of the time domain. We consider the parareal algorithm which is a predictorcorrector algorithm involving propagations in parallel of an accurate fine solver (which is computationally expensive), and a coarse solver. This algorithm is very popular due to its simplicity of implementation but it suffers from poor parallel efficiency (which is unfortunately a common burden in time parallel algorithms). The main obstacle for better efficiency in parareal is the cost of its fine solver so, in order to improve it, we have developed an adaptive parareal strategy in which the accuracy of the fine solver is increased across the iterations. We prove for an idealized setting that the algorithm would provide full parallel efficiency. In practice, although we show that the fine solver is better handled across iterations with our strategy, the cost of the coarse solver starts to enter into the picture, and this prevented us from obtaining full efficiency. Despite this, our results improved by about a factor 2 the efficiency of the traditional algorithm.
In 16 we develop a fully adaptive strategy to solve the radiative transfer equation, which is a lineal Boltzmann equation. The main importance of the approach is that it comes with certified a posteriori error bounds. For this, we formulate a fixedpoint iteration in a suitable, infinite dimensional function space that is guaranteed to converge with a fixed error reduction per step. The numerical scheme is then based on approximately realizing this outer iteration within dynamically updated accuracy tolerances that still ensure convergence to the exact solution. To guarantee that these error tolerances are met, we employ rigorous a posteriori error bounds based on a Discontinuous Petrov–Galerkin (DPG) scheme. These a posteriori bounds are also used to generate adapted angular dependent spatial meshes to signifiicantly reduce overall computational complexity. The scheme also requires the evaluation of the global scattering operator at increasing accuracy at every iteration and its computation is accelerated through lowrank approximation and matrix compression techniques. We illustrate the theoretical findings with numerical experiments involving nontrivial scattering kernels.
In 39 we develop a piecewise affine strategy to build a state estimation algorithm for which we prove that we can asymptotically provide the optimal reconstruction performance.
8 Bilateral contracts and grants with industry
8.1 Bilateral contracts with industry
Notocord Systems
Participants: Damiano Lombardi, Fabien Raphel.
This work is devoted to the investigation on new approaches and efficient algorithms in the context of safety pharmacology and the analysis of biological signals.
Casis
Participants: Mocia Agbalessi, Miguel Ángel Fernández Varela, Damiano Lombardi.
This work is devoted to the combination of 4DMRI data and fluidstructure interaction models of blood flow to asses indicators of aneurysm rupture.
9 Partnerships and cooperations
9.1 International initiatives
9.1.1 Inria associate team not involved in an IIL
IMFIBIO: Innovative Methods for Forward and Inverse problems in BIOmedical applications
Participants: Muriel Boulakia, Daniele Corti, Miguel Ángel Fernández Varela, Fannie Gerosa, Céline Grandmont.
 Duration: 20202022
 Coordinator: Muriel Boulakia
 Partner: Department of Mathematics, University College London (UK)
 Summary: The purpose of the IMFIBIO Associate Team is to exploit the complementary expertise of both partners in mathematical analysis, numerical analysis, scientific computing and data assimilation in order to develop innovative methods for the study of forward and inverse problems in the context of biomedical applications.

Web site:
https://
team. inria. fr/ imfibio/
9.1.2 Visits to international teams
Research stays abroad
 Fannie Gerosa
 One week visit at the Mathematics Department of UCL, February 2020
 Céline Grandmont
 Member of the research group "Analyse et équations aux dérivées partielles" at ULB Belgium
9.2 European initiatives
9.2.1 FP7 & H2020 Projects
INSPIRE: INnovation in Safety Pharmacology for Integrated cardiovascular safety assessment to REduce adverse events and late stage drug attrition
Participants: Muriel Boulakia, Sara Costa Faya, Miguel Ángel Fernández Varela, Céline Grandmont, Haibo Liu, Damiano Lombardi.
 Funding: Horizon 2020  MSCAITN
 Duration: 20202023
 Coordinator: University of Antwerp
 Local coordinator: Damiano Lombardi
 Partners: see the link
 Summary: INSPIRE is an European Training Network (ETN) projet funding 15 Early Stage Researchers (ESRs) aimed to exploit innovative techniques for better assessment and prediction of cardiovascular safety liabilities.

Web site:
https://
www. uantwerpen. be/ en/ projects/ inspiresafetypharmacology/
9.3 National initiatives
9.3.1 ANR
ADAPT: Adaptive Dynamical Approximations by Parallel Tensor methods
Participants: Maria FuenteRuiz, Damiano Lombardi, Olga Mula.
 Funding: ANR JCJC
 Duration: 20182022
 Coordinator: Damiano Lombardi
 Summary: The main goal of the ANR is to investigate the numerical approximation of the solution of highdimensional problems. In particular, the applications that motivate this study are the Uncertainty Quantification and the Kinetic theory. The main objective is to construct in an adaptive way parsimonious discretisations starting from arbitrarily chosen separated discretisations.

Web site:
https://
project. inria. fr/ adapt/
SIMR: Simulation and Imaging for Mitral Regurgitation
Participants: Daniele Carlo Corti, Miguel Ángel Fernández Varela, Fannie Gerosa, Céline Grandmont, Marina Vidrascu.
 Funding: ANR PRC
 Duration: 20202023
 Coordinator: Miguel Ángel Fernández Varela
 Partners: CREATIS, HCL, LGEF, M3DISIM, TIMC
 Summary: The SIMR project aims at evaluating the physical consequences of mitral repair using efficient numerical simulations, advanced imaging techniques and an innovative measurement tools in a clinical study.

Web site:
https://
project. inria. fr/ simr/
10 Dissemination
10.1 Promoting scientific activities
10.1.1 Scientific events: organisation
 Damiano Lombardi
 Coorganizer of the CEMRACS 2021 summer school.
 Coorganizer of InriaLJLL meeting in scientific computing
 Olga Mula
 Coorganizer of the CEMRACS 2021 summer school.
 Minisymposium coorganizer (with B. Desprès, M. Campos Pinto and O. Laffite.) Numerical Aspects of Transport, Boltzmann and Kinetic Equations,WCCMECCOMAS 2020 conference, Paris
10.1.2 Journal
Member of the editorial boards
 Céline Grandmont
 Member of the editorial board of Mathematical Modelling of Natural Phenomena
 Member of the editorial board of Journal of Mathematical Fluid Mechanics
 Olga Mula
 Member of the editorial board of Calcolo
10.1.3 Scientific expertise
 Olga Mula
 Member of "Facing the virus", a scientific initiative on Covid19 launched by researchers from PSL Univ.
 Member of the "Liaison Committee of SIGMA", the activity group on SignalImageGeometryModellingApproximation.
10.1.4 Research administration
 Miguel Ángel Fernández Varela
 Head of Science, Inria Paris
 Member of the Inria Evaluation Committee
 Céline Grandmont
 Member of the Inria Evaluation Committee
 Member of the Inria Parity Committee
 Member of the scientific board of the EDMH, Paris Saclay
 Damiano Lombardi
 Member of LJLL Conseil du Laboratoire
10.1.5 Conferences
 Muriel Boulakia
 Seminar, Workshop on Collective behavior of particles in fluids , December 2020, IHP Paris
 Seminar MEDISIMPOEMSDEFI, June 2020, Inria Saclay
 Daniele Corti
 Contributed talk, WCCMECCOMAS 2020 conference, Paris
 Miguel Ángel Fernández Varela
 Contributed talk, WCCMECCOMAS 2020 conference, Paris
 Fannie Gerosa
 Contributed talk, WCCMECCOMAS 2020 conference, Paris
 Céline Grandmont
 Invited Speaker, Mathematics of Complex Systems in Biology and Medicin conference, Feb 2020, Cirm Marseille
 Damiano Lombardi
 Seminar, Eindhoven Univ of Technology, September 2020
 Olga Mula
 Seminar Rencontres INRIALJLL, December 2020
 Seminar of the JeanLeray Institute, November 2020
 Seminar of the MaNu activity group, June 2020
 Seminar MEDISIMPOEMSDEFI (Inria Saclay), June 2020
 Working Group at Dauphine on COVID19, June 2020
 Seminar ANEDP at Univ de Lille, March 2020
 Journée des 50 ans du CEREMADE (Univ ParisDauphine), January 2020
 Fabien Raphel
 Mathbio GDT Marseille
 Marina Vidrascu
 Contributed talk at WCCMECCOMAS 2020 conference, Paris
10.2 Teaching  Supervision  Juries
10.2.1 Teaching
 Licence:
 Muriel Boulakia
 Projects on differential equations, 12h, L3, Polytech Sorbonne, Sorbonne Univ
 Nonlinear systems and optimization, 36h, L3, Polytech Sorbonne, Sorbonne Univ
 Mathematics for scientific studies, 50h, L1, Sorbonne Univ
 Felipe Galarce Marin
 TD Numerical methods for differential equations, 24h, L3, Sorbonne Univ
 Fannie Gerosa
 TD Numerical methods for differential equations, 26h, L3, Sorbonne Univ
 Damiano Lombardi
 PGD methods , 3 h, ENPC.
 TP, Numerical Analysis, L3, 24h, Sorbonne Univ
 Fabien Raphel
 TP Numerical Analysis in Python 12h, L3, Sorbonne Univ
 Muriel Boulakia
 Master:
 Muriel Boulakia
 Tutorials on Basis of Functional Analysis (distance learning), 30h, M1, Sorbonne Univ
 Preparatory course for teaching admission examination “Agrégation”, 60h, M2, Sorbonne Univ
 Miguel Ángel Fernández Varela
 Modeling and numerical methods for hemodynamics, 30h, M2, Sorbonne Univ
 Damiano Lombardi
 Minicourse Modeling the electrophysiology of heart, 4.5h, M2, Ecole des Mines Paristech
 TD, Numerical Methods, M1, 15 h, Sorbonne Univ
 Muriel Boulakia
10.2.2 Supervision
 PhD in progress: Mocia Agbalessi, Modeling and patient specific fluidstructure interaction simulations of aortic pathological configurations. Since April, 2019, Supervisors: M.A. Fernández Varela & D. Lombardi
 PhD in progress: Marguerite Champion, Modeling and numerical simulation of implantable aortic blood pumps. Since November 2020. Supervisors: M.A. Fernández Varela, C. Grandmont & M. Vidrascu
 PhD in progress: Daniele Corti, Modeling and numerical simulation of the mitral apparatus. Since October 2020. Supervisors: M.A. Fernández Varela, F. Alauzet and G. Delay & M. Vidrascu
 PhD in progress: Sara Costa Faya, An in silico approach to monitor and predict haemodynamics during safety pharmacology studies., since September 2020. M.A. Fernández Varela, C. Grandmont & D.Lombardi
 PhD in progress: Maria Fuente Ruiz , Adaptive tensor methods for scientific computing. Since March 2020. Supervisors: D. Lombardi & V. Ehrlacher
 PhD in progress: Felipe Galarce, Enhancing hemodynamics measurements with mathematical modeling, since December 2017. Supervisors: J.F. Gerbeau, D. Lombardi & O. Mula
 PhD in progress: Fannie Gerosa, Immersed boundary methods for fluidstructure interaction with topological changes, since January 2018. Supervisor: M.A. Fernández Varela
 PhD in progress: Fabien Lespagnol, A new computational approach for fluidstructure interaction of slender bodies immersed in threedimen sional flow. Since September 2020. Supervisors: M. Boulakia, M.A. Fernández Varela, C. Grandmont & Paolo Zunino (MOX, Politechnico de Milano)
 PhD in progress: Haibo Liu, Data assimilation for highthroughput screening in safety pharmacology. Since September 2020. Supervisors: D. Lombardi & M. Boulakia
 PhD in progress: Fabien Raphel, Mathematical modeling and learning of biomedical signals for safety pharmacology. Since April 2019. Supervisors: J.F. Gerbeau & D. Lombardi
10.2.3 Juries
 Muriel Boulakia
 PHD committee: Nicolas Molina, Univ ParisDauphine (reviewer), Imene Djebour, IECL Nancy
 Miguel Ángel Fernández Varela
 Hiring committees: Inria CRCN Paris and CRCN National
 PHD committee: Daniel Grinberg MD, INSA Lyon (reviewer)
 Céline Grandmont
 Hiring committee Assistant Professor Univ Orléans
 Hiring committees: Inria CRCN Lille, Inria DR2
 PHD committee: Nicola Molinas, Univ Dauphine
 Blaise Pascal SMAI Prize
 Damiano Lombardi
 PHD committee: Roman Weinhadl, Otto von Guericke Univ and Max Planck Magdebourg (reviewer) and Sebastien Riffaud, Univ. Bordeaux
 Olga Mula
 Hiring committee Assistant Professor UTC Compiègne, Univ Stasbourg
 PHD committee: Antonio Galia, Univ ParisSaclay
10.3 Popularization
10.3.1 Interventions
 Céline Grandmont
 Conference Filles et Maths, Villetaneuse, February 2020
11 Scientific production
11.1 Major publications
 1 articleA loosely coupled scheme for fictitious domain approximations of fluidstructure interaction problems with immersed thinwalled structuresSIAM Journal on Scientific Computing412February 2019, 351374
 2 articleWellposedness for the coupling between a viscous incompressible fluid and an elastic structureNonlinearity322019, 35483592
 3 article Quantification of the unique continuation property for the nonstationary Stokes problem Mathematical Control and Related Fields March 2016
 4 article Existence of global strong solutions to a beamfluid interaction system Archive for Rational Mechanics and Analysis 2016
 5 articleCoupling schemes for the FSI forward prediction challenge: comparative study and validationInternational Journal for Numerical Methods in Biomedical Engineering3342017, e02813
 6 article A nonparametric knearest neighbor entropy estimator Physical Reviev E January 2016
 7 article Predicted airway obstruction distribution based on dynamical lung ventilation data: a coupled modelingmachine learning methodology International Journal for Numerical Methods in Biomedical Engineering 34 9 May 2018
 8 article Augmented Resistive Immersed Surfaces valve model for the simulation of cardiac hemodynamics with isovolumetric phases International Journal for Numerical Methods in Biomedical Engineering May 2019
 9 articleComposite biomarkers derived from MicroElectrode Array measurements and computer simulations improve the classification of druginduced channel blockFrontiers in Physiology810962018, 130
11.2 Publications of the year
International journals
 10 article Approximation of Optimal Transport problems with marginal moments constraints Mathematics of Computation 2020
 11 articleEpidemiological Forecasting with Model Reduction of Compartmental Models. Application to the COVID19 PandemicBiology101December 2020, 22
 12 articleFluidkinetic modelling for respiratory aerosols with variable size and temperatureESAIM: Proceedings and Surveys672020, 100119
 13 article Data assimilation finite element method for the linearized NavierStokes equations in the low Reynolds regime Inverse Problems 36 8 May 2020
 14 article Numerical reconstruction based on Carleman estimates of a source term in a reactiondiffusion equation. ESAIM: Control, Optimisation and Calculus of Variations 2021
 15 article Optimal reduced model algorithms for databased state estimation SIAM Journal on Numerical Analysis June 2020
 16 article An Adaptive Nested Source Term Iteration for Radiative Transfer Equations Mathematics of Computation January 2020
 17 article Adaptive hierarchical subtensor partitioning for tensor compression SIAM Journal on Scientific Computing 2021
 18 article Nonlinear model reduction on metric spaces. Application to onedimensional conservative PDEs in Wasserstein spaces ESAIM: Mathematical Modelling and Numerical Analysis 2020
 19 article An unfitted mesh semiimplicit coupling scheme for fluidstructure interaction with immersed solids International Journal for Numerical Methods in Engineering May 2020
 20 articleSplitting schemes and unfitted mesh methods for the coupling of an incompressible fluid with a thinwalled structureIMA Journal of Numerical Analysis402April 2020, 1407–1453
 21 article Fast reconstruction of 3D blood flows from Doppler ultrasound images and reduced models Computer Methods in Applied Mechanics and Engineering March 2021
 22 article Reconstructing Haemodynamics Quantities of Interest from Doppler Ultrasound Imaging International Journal for Numerical Methods in Biomedical Engineering 2020
 23 article Fast semiautomatic segmentation based on reduced basis Comptes Rendus Mathématique 2020
 24 article An Adaptive Parareal Algorithm Journal of Computational and Applied Mathematics May 2020
 25 articleReducing Hypermuscularization of the Transitional Segment between Arterioles and Capillaries Protects Against Spontaneous Intracerebral HemorrhageCirculation14125March 2020, 20782094
 26 article Characterization of SpatioTemporal Cardiac Action Potential Variability at Baseline and under betaAdrenergic Stimulation by Combined Unscented Kalman Filter and Double Greedy Dimension Reduction IEEE Journal of Biomedical and Health Informatics April 2020
 27 articleAugmented Resistive Immersed Surfaces valve model for the simulation of cardiac hemodynamics with isovolumetric phasesInternational Journal for Numerical Methods in Biomedical Engineering363February 2020, e3223
 28 articleA pipeline for image based intracardiac CFD modeling and application to the evaluation of the PISA methodComputer Methods in Applied Mechanics and Engineering358January 2020, 112627
Doctoral dissertations and habilitation theses
 29 thesis Contributions to scientific computing for datasimulation interaction in biomedical applications Sorbonne Université July 2020
Reports & preprints
 30 misc DepthAdaptive Neural Networks from the Optimal Control viewpoint July 2020
 31 misc Splitting schemes for a Lagrange multiplier formulation of FSI with immersed thinwalled structure: stability and convergence analysis July 2020
 32 report Initiative face au virus. Regards croisés sur l'épidemie de Covid19 apportés par les données sanitaires et de géolocalisation (mars à octobre 2020) Université PSL; Inria; CNRS December 2020
 33 report Initiative face au virus Observations sur la mobilité pendant l'épidémie de Covid19 Université PSL May 2020
 34 report Indicateurs de suivi de l'activité scientifique de l'Inria Inria December 2020
 35 misc Fully discrete loosely coupled RobinRobin scheme for incompressible fluidstructure interaction: stability and error analysis July 2020
 36 misc Error analysis of an unfitted mesh semiimplicit coupling scheme for fluidstructure interaction December 2020
 37 report Évaluation des Logiciels Inria January 2021
 38 report Software Evaluation Inria January 2021
 39 misc Nonlinear reduced models for state and parameter estimation September 2020
 40 misc CPTT: using TTSVD to greedily construct a Canonical Polyadic tensor approximation November 2020
 41 misc Existence and uniqueness for a quasistatic interaction problem between a viscous fluid and an active structure February 2020
11.3 Cited publications
 42 articleNitscheXFEM for the coupling of an incompressible fluid with immersed thinwalled structuresComput. Methods Appl. Mech. Engrg.3012016, 300335
 43 book Introduction to machine learning MIT press 2009
 44 articleA robust and efficient valve model based on resistive immersed surfacesInt. J. Numer. Meth. Biomed. Engng.2892012, 937959
 45 articleA fictitious domain/mortar element method for fluidstructure interactionInt. Jour. Num. Meth. Fluids352001, 743761
 46 articleMultiscale modeling of the respiratory tractMath. Models Methods Appl. Sci.2012010, 5993
 47 articleFinite element approach to immersed boundary method with different fluid and solid densitiesMath. Models Methods Appl. Sci.21122011, 25232550
 48 articleGlobal existence of solutions for the coupled Vlasov and NavierStokes equationsDifferential Integral Equations2211122009, 12471271
 49 incollectionDiffusion models of multicomponent mixtures in the lungCEMRACS 2009: Mathematical modelling in medicine30ESAIM Proc.EDP Sci., Les Ulis2010, 90103
 50 articleGlobal existence of solutions to the incompressible NavierStokesVlasov equations in a timedependent domainJ. Differential Equations26232017, 13171340
 51 book Classification and regression trees Routledge 2017
 52 articleAn unfitted Nitsche method for incompressible fluidstructure interaction using overlapping meshesComput. Methods Appl. Mech. Engrg.2792014, 497514
 53 misc Stability and error analysis of a splitting method using RobinRobin coupling applied to a fluidstructure interaction problem 2020
 54 articleStabilized finite element methods for nonsymmetric, noncoercive, and illposed problems. Part I: Elliptic equationsSIAM J. Sci. Comput.3562013, 27522780
 55 articleNumerical simulation of respiratory flow patterns within human lungRespir. Physiol. Neurobiol.13022002, 201221
 56 articleHomogenization of a multiscale viscoelastic model with nonlocal damping, application to the human lungsMath. Models Methods Appl. Sci.2562015, 11251177
 57 articleRole of Computational Simulations in Heart Valve Dynamics and Design of Valvular ProsthesesCardiovasc. Eng. Technol.112010, 1838
 58 articleGeneral coupling of porous flows and hyperelastic formulations—From thermodynamics principles to energy balance and compatible time schemesEur. J. Mech. B Fluids.462014, 8296
 59 articleImageBased Simulations Show Important Flow Fluctuations in a Normal Left Ventricle: What Could be the Implications?Ann. Biomed. Eng.44112016, 33463358
 60 articleThe comprehensive in vitro proarrhythmia assay (CiPA) initiative—update on progressJ. Pharmacol. Toxicol. Methods812016, 1520
 61 articleAn evaluation of 30 clinical drugs against the comprehensive in vitro proarrhythmia assay (CiPA) proposed ion channel panelJ. Pharmacol. Toxicol. Methods812016, 251262
 62 articleA patientspecific aortic valve model based on moving resistive immersed implicit surfacesBiomech. Model. Mechanobiol.1652017, 17791803
 63 articleConvergence and error analysis for a class of splitting schemes in incompressible fluidstructure interactionIMA J. Numer. Anal.3642016, 17481782
 64 articleA coupled mitral valveleft ventricle model with fluidstructure interactionMed. Eng. Phys.4709 2017, 128136
 65 articleA distributed Lagrange mutiplier/fictitious domain method for particulate flowsInt. J. of Multiphase Flow251999, 755794
 66 articleExistence of global strong solutions to a beamfluid interaction systemArch. Ration. Mech. Anal.22032016, 12831333
 67 articleA literature survey of lowrank tensor approximation techniquesGAMMMitt.3612013, 5378
 68 articleLack of collision between solid bodies in a 2D incompressible viscous flowComm. Partial Differential Equations32792007, 13451371
 69 articleDruginduced functional cardiotoxicity screening in stem cellderived human and mouse cardiomyocytes: effects of reference compoundsJ. Pharmacol. Toxicol. Methods6812013, 97111
 70 book Statistical and computational inverse problems 160 Applied Mathematical Sciences SpringerVerlag, New York 2005
 71 articleImmersogeometric cardiovascular fluidstructure interaction analysis with divergenceconforming BsplinesComput. Methods Appl. Mech. Engrg.3142017, 408472
 72 articleAn immersed boundary method with formal secondorder accuracy and reduced numerical viscosityJ. Comp. Phys.16022000, 705719
 73 articleModeling of the oxygen transfer in the respiratory processESAIM Math. Model. Numer. Anal.4742013, 935960
 74 articleComputational modeling of cardiac hemodynamics: current status and future outlookJ. Comput. Phys.3052016, 10651082
 75 article Aerosol transport throughout inspiration and expiration in the pulmonary airways Int. J. Numer. Methods Biomed. Eng. 33 9 2017
 76 articleThe immersed boundary methodActa Numer.112002, 479517
 77 article A comprehensive computational human lung model incorporating interacinar dependencies: Application to spontaneous breathing and mechanical ventilation Int. J. Numer. Method. Biomed. Eng. 33 1 e02787 2016
 78 article Fluidstructure interaction including volumetric coupling with homogenised subdomains for modeling respiratory mechanics Int. J. Numer. Method. Biomed. Eng. 33 4 e2812 2016
 79 articleTensor rank and the illposedness of the best lowrank approximation problemSIAM J. Matrix Anal. Appl.3032008, 10841127
 80 articleA combined fictitious domain/adaptive meshing method for fluid–structure interaction in heart valvesInternational Journal for Numerical Methods in Fluids4652004, 533544
 81 book Targeted learning Springer Series in Statistics Springer, New York 2011