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30
Numerical mathematics
, 2000
"... Abstract. In this paper we introduce some basic differential models for the description of blood flow in the circulatory system. We comment on their mathematical properties, their meaningfulness and their limitation to yield realistic and accurate numerical simulations, and their contribution for a ..."
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Abstract. In this paper we introduce some basic differential models for the description of blood flow in the circulatory system. We comment on their mathematical properties, their meaningfulness and their limitation to yield realistic and accurate numerical simulations, and their contribution for a better understanding of cardiovascular physiopathology. Mathematics Subject Classification (2000). 92C50,96C10,76Z05,74F10,65N30,65M60. Keywords. Cardiovascular mathematics; mathematical modeling; fluid dynamics; Navier– Stokes equations; numerical approximation; finite element method; differential equations. 1.
Multilevel Schwarz and Multigrid Preconditioners for the Bidomain System
"... Summary. Two parallel and scalable multilevel preconditioners for the Bidomain system in computational electrocardiology are introduced and studied. The Bidomain system, consisting of two degenerate parabolic reactiondiffusion equations coupled with a stiff system of several ordinary differential e ..."
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Summary. Two parallel and scalable multilevel preconditioners for the Bidomain system in computational electrocardiology are introduced and studied. The Bidomain system, consisting of two degenerate parabolic reactiondiffusion equations coupled with a stiff system of several ordinary differential equations, generates very illconditioned discrete systems when discretized with semiimplicit methods in time and finite elements in space. The multilevel preconditioners presented in this paper attain the best performance to date, both in terms of convergence rate and solution time and outperform the simpler onelevel preconditioners previously introduced. Parallel numerical results, using the PETSc library and run on Linux Clusters, show the scalability of the proposed preconditioners and their efficiency on largescale simulations of a complete cardiac cycle. 1
Towards the numerical simulation of electrocardiograms
"... Abstract. We present preliminary results of the numerical simulation of electrocardiograms (ECG). We consider the bidomain equations to model the electrical activity of the heart and a Laplace equation for the torso. The ionic activity is modeled with a MitchellSchaeffer dynamics. We use adaptive s ..."
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Cited by 7 (2 self)
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Abstract. We present preliminary results of the numerical simulation of electrocardiograms (ECG). We consider the bidomain equations to model the electrical activity of the heart and a Laplace equation for the torso. The ionic activity is modeled with a MitchellSchaeffer dynamics. We use adaptive semiimplicit BDF schemes for the time discretization and a NeumannRobin domain decomposition algorithm for the space discretization. The obtained ECGs, although not completely satisfactory, are promising. They allow to discuss various modelling assumptions, for example the relevance of cells heterogeneity, the fiber orientation and the coupling conditions with the torso. To cite this article: In Proceedings of the
A multiresolution spacetime adaptive scheme for the bidomain model in electrocardiology, in "Numerical Methods for Partial Differential Equations
"... The bidomain model of electrical activity of myocardial tissue consists of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a timedependent ODE modeling the evolution ..."
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The bidomain model of electrical activity of myocardial tissue consists of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a timedependent ODE modeling the evolution of the gating variable. In the simpler subcase of the monodomain model, the elliptic PDE reduces to an algebraic equation. Since typical solutions of the bidomain and monodomain models exhibit wavefronts with steep gradients, we propose a finite volume scheme enriched by a fully adaptive multiresolution method, whose basic purpose is to concentrate computational effort on zones of strong variation of the solution. Time adaptivity is achieved by two alternative devices, namely locally varying time stepping and a RungeKuttaFehlbergtype adaptive time integration. A series of numerical examples demonstrates that these methods are efficient and sufficiently accurate to simulate the electrical activity in myocardial tissue with affordable effort. In addition, the optimal choice of the threshold for discarding nonsignificant information in the multiresolution representation of the solution is addressed, and the numerical efficiency and accuracy of the method is measured in
An efficient generalization of the RushLarsen method for solving electrophysiology membrane equations
 Electronic Transactions on Numerical Analysis
, 2009
"... Abstract. In this paper we address a secondorder class of methods for solving ordinary differential systems coming from some problems in electrophysiology. The set of methods generalizes to the second order a previous proposal by Rush and Larsen (1978). We prove that the methods are secondorder ..."
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Abstract. In this paper we address a secondorder class of methods for solving ordinary differential systems coming from some problems in electrophysiology. The set of methods generalizes to the second order a previous proposal by Rush and Larsen (1978). We prove that the methods are secondorder convergent and are in general more stable than the corresponding multistep methods. Moreover, they feature better positivity properties. We present their timeadaptive formulation, which is well suited for our electrophysiology problems. In particular, numerical results are presented on the Monodomain model coupled to LuoRudy I ionic models for the propagation of the cardiac potential.
Parallel solution of cardiac reactiondiffusion models
 Procedings of the 15th International Conference on Domain Decomposition Methods, Lecture Notes in Computational Science and Engineering
, 2004
"... Summary. We present and study a parallel iterative solver for reactiondiffusion systems in three dimensions arising in computational electrocardiology, such as the Bidomain and Monodomain models. The models include intramural fiber rotation and anisotropic conductivity coefficients that can be full ..."
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Summary. We present and study a parallel iterative solver for reactiondiffusion systems in three dimensions arising in computational electrocardiology, such as the Bidomain and Monodomain models. The models include intramural fiber rotation and anisotropic conductivity coefficients that can be fully orthotropic or axially symmetric around the fiber direction. These cardiac models are coupled with a membrane model for the ionic currents, consisting of a system of ordinary differential equations. The solver employs structured isoparametric Q1 finite elements in space and a semiimplicit adaptive method in time. Parallelization and portability are based on the PETSc parallel library and largescale computations with up to O(10 7) unknowns have been run on parallel computers. These simulation of the full Bidomain model (without operator or variable splitting) for a full cardiac cycle are, to our knowledge, among the most complete in the available literature. 1 The cardiac Bidomain and Monodomain models Cardiac tissue is traditionally modeled as an arrangement of cardiac fibers that rotate counterclockwise from the epicardium to the endocardium, (see
Null controllability of a system of viscoelasticity with a moving control
 J. Math. Pures Appl
"... Abstract. In this paper, we consider the wave equation with both a viscous KelvinVoigt and frictional damping as a model of viscoelasticity in which we incorporate an internal control with a moving support. We prove the null controllability when the control region, driven by the flow of an ODE, cov ..."
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Abstract. In this paper, we consider the wave equation with both a viscous KelvinVoigt and frictional damping as a model of viscoelasticity in which we incorporate an internal control with a moving support. We prove the null controllability when the control region, driven by the flow of an ODE, covers all the domain. The proof is based upon the interpretation of the system as, roughly, the coupling of a heat equation with an ordinary differential equation (ODE). The presence of the ODE for which there is no propagation along the space variable makes the controllability of the system impossible when the control is confined into a subset in space that does not move. The null controllability of the system with a moving control is established in using the observability of the adjoint system and some Carleman estimates for a coupled system of a parabolic equation and an ODE with the same singular weight, adapted to the geometry of the moving support of the control. This extends to the multidimensional case the results by P. Martin et al. on the onedimensional case, employing 1 − d Fourier analysis techniques. 1.
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"... a posteriori error estimator for model adaptivity in electrocardiology by ..."
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a posteriori error estimator for model adaptivity in electrocardiology by
problem in electrocardiology
"... model preconditioner for the Bidomain problem in electrocardiology by ..."
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