1,721,052 research outputs found
A multilevel hybrid Newton-Krylov-Schwarz method for the Bidomain model of electrocardiology
No full textA multilevel hybrid Newton-Krylov-Schwarz (NKS) method is constructed and studied numerically for implicit time discretizations of the Bidomain reaction-diffusion system in three dimensions. This model describes the bioelectrical activity of the heart by coupling two degenerate parabolic equations with a stiff system of ordinary differential equations. The NKS Bidomain solver employs an outer inexact Newton iteration to solve the nonlinear finite element system originating at each time step of the implicit discretization. The Jacobian update during the Newton iteration is solved by a Krylov method employing a multilevel hybrid overlapping Schwarz preconditioner, additive within the levels and multiplicative among the levels. Several parallel tests on Linux clusters are performed, showing that the convergence of the method is independent of the number of subdomains (scalability), the discretization parameters and the number of levels (optimality)
A hybrid multilevel Schwarz method for the bidomain model
No full textA hybrid multilevel Schwarz method is studied numerically for the anisotropic Bidomain model in both two and three dimensions. This multiscale system models the electrical activity of the heart and it consists of two degenerate parabolic non-linear reaction–diffusion equations, coupled with a stiff system of ordinary differential equations. The numerical discretization of the whole system by finite elements in space and semi-implicit methods in time generates ill-conditioned linear systems that must be solved at each time step. The multilevel algorithm studied employs a hierarchy of nested meshes with overlapping Schwarz preconditioners on each level and is additive within the levels and multiplicative among the levels. We perform several parallel tests on two Linux clusters, showing that the convergence of the method is independent of the number of subdomains (scalability), the discretization parameters and the number of levels (optimality). Moreover the comparison with the traditional Block Jacobi ILU parallel preconditioner and the V-cycle Multigrid parallel preconditioner shows that, on a whole heart cycle simulation, the proposed method attains the best performances in terms of CPU times
BPX preconditioners for the Bidomain model of electrocardiology
Full text linkThe aim of this work is to develop a BPX preconditioner for the Bidomain model of electrocardiology. This model describes the bioelectrical activity of the cardiac tissue and consists of a system of a non-linear parabolic reaction–diffusion partial differential equation (PDE) and an elliptic linear PDE, modeling at macroscopic level the evolution of the transmembrane and extracellular electric potentials of the anisotropic cardiac tissue. The evolution equation is coupled through the non-linear reaction term with a stiff system of ordinary differential equations, the so-called membrane model, describing the ionic currents through the cellular membrane. The discretization of the coupled system by finite elements in space and semi-implicit finite differences in time yields at each time step the solution of an ill-conditioned linear system. The goal of the present study is to construct, analyze and numerically test a BPX preconditioner for the linear system arising from the discretization of the Bidomain model. Optimal convergence rate estimates are established and verified by two- and three-dimensional numerical tests on both structured and unstructured meshes. Moreover, in a full heartbeat simulation on a three-dimensional wedge of ventricular tissue, the BPX preconditioner is about 35% faster in terms of CPU times than ILU(0) and an Algebraic Multigrid preconditioner
Multilevel additive Schwarz preconditioners for the Bidomain reaction-diffusion system
No full textMultilevel additive Schwarz methods are analyzed and studied numerically for the anisotropic cardiac Bidomain model in three dimensions. This is the most complete model to date of the bioelectrical activity of the heart tissue, consisting of a degenerate parabolic system of nonlinear reaction-diffusion equations coupled with a stiff system of several ordinary differential equations describing the ionic currents through the cellular membrane. Due to the presence of very different scales in both space and time, the numerical discretization of this system by finite elements in space and semi-implicit methods in time produces very ill-conditioned linear systems that must be solved at each time step. The proposed multilevel algorithm employs a hierarchy of nested meshes with overlapping Schwarz preconditioners on each level and is fully additive, hence parallel, within and among levels. Convergence estimates are proved for the resulting multilevel algorithm, showing that its convergence rate is independent of the number of subdomains (scalability), of the mesh sizes of each level and of the number of levels (optimality). Several parallel tests on a Linux cluster confirm the scalability and optimality of the method, as well as its parallel efficiency on both Cartesian and deformed domains in three dimensions
Parallel multilevel Schwarz and Block preconditioners for the bidomain parabolic-parabolic and parabolic-elliptic formulations
No full textThe aim of this work is to develop parallel multilevel and block preconditioners for the Bidomain model of electrocardiology. The Bidomain model describes the electrical activity of the heart tissue and consists of a system of two parabolic nonlinear partial differential equations (PDEs) of reaction-diffusion type (PP formulation) or alternatively of a system of a parabolic nonlinear PDE and an elliptic linear PDE (PE formulation). In both formulations, the PDEs are coupled with a system of ordinary differential equations, modeling the cellular membrane ionic currents. The first goal of the present study is to construct, analyze, and numerically test a multilevel additive Schwarz preconditioner for the PE formulation of the Bidomain model, extending previous results obtained for the PP formulation. Optimal convergence rate estimates are established and confirmed by 3D numerical test on Linux clusters. The second goal of the present study is to analyze the scalability of multilevel Schwarz block-diagonal and block-factorized preconditioners for both PP and PE formulations of the Bidomain model and to compare them with multilevel Schwarz coupled preconditioners. The 3D parallel numerical tests show that block preconditioners for the PP formulation are not scalable, while they are scalable for the PE formulation, but less efficient than the coupled preconditioners
Characteristics of FC-induced changes of transmembrane potential in maize roots
No full textCharacteristics of FC-induced changes of transmembrane potential in maize roots FC(fusicoccin) a glucoside very active in promoting cell enlargement and proton secretion also induces an increase of the negative transmembrane potential in pea internode segments, squash cotyledons, maize coleoptiles and maize roots segments. In maize roots segments maximum hyperpolarization values ore obtained with concentrations of FC higher than 5.10-5M, with 5.10-5M FC the period of time required for reaching maximum value is of ca.6 min. The hyperpolarizing effect of FC is not influenced by the osmolarity of the medium and by the rate of cell enlargement. Low temperature(6 C} induces a rapid decrease in PD both in the controls and in the FC-treated samples, FC-induced PD is more temperature sensitive than basal PD, thus suggesting that effect of FC is metabolism dependent. FCCP and DNP induce a rapid and marked decrease of PD, larger in the FC-treated samples than in the controls. This can be interpreted as a consequence eiter of the incoupling of oxidative phosphorylation and thus of the decrease of ATP for ATPase involved in electrogenesis, or of a direct effect of FCCP and DNP on permeability of plasmalemma to proton. Partial anaerobiosis such markedly inhibit PD in the controls, does not prevent the hyperpolarization effect of FC. The effect of FC on PD is dependent on the ionic composition of the medium: K+ and ,at lesser extent, Rb+ at concentration higher than 5.10-4 M induce a rapid depolarization(new level is reached ca. 5 min. after change of
the external cation concentration}, larger in presence than in absence of FC. In absence of K or Rb PO showed a constant tendency to increase, and in the medium containing Na, K + free,
the maximal PD FC-induced is observed. In this condition Na+ is ineffective when present at concentration up to 10 mM; 30mM Na+ induces a clear depolarization in the PD probably due
to the high driving force of the negative PD; 100 mM Na+ induces a rapid depolarization larger in the presence of FC than in the control. Li+ and Cs+ did not influence PD when present at con-
centration up to 10 mM; at 30 mM depolarizing activity becomes apparent for Cs and at 100 mM for Cs and Li , also for these cations the depolarizing activity was much larger in the presence
of FC. FC markedly increased the monovalent cation uptake rates at low salt concentration this effect is much larger for K+ and Rb+ than for Na+ and Cs+ ;the rate of cation uptake is only qualitatively related to depolarizing activity. The selectivity of the transport system for K+ and Rb+ as compared to the other cation is markedly increased by FC. These results are in agreement with the hypothesis that FC-induced hyperpolarization of PD is a consequence of the activation of a proton/ cation exchange mechanism having a high affinity for K+
A scalable Newton-Krylov-Schwarz method for the Bidomain reaction-diffusion system
No full textA novel two-level Newton-Krylov-Schwarz (NKS) solver is constructed and analyzed for implicit time discretizations of the bidomain reaction-diffusion system in three dimensions. This multiscale system describes the bioelectrical activity of the heart by coupling two degenerate parabolic equations with several ordinary differential equations at each point in space. Together with a finite element discretization in space, the proposed NKS Bidomain solver employs an outer inexact Newton iteration to solve the nonlinear finite element system originating at each time step of the decoupled implicit discretization. The Jacobian update during the Newton iteration is solved by a Krylov method employing a two-level overlapping Schwarz preconditioner. A convergence rate estimate is proved for the resulting preconditioned operator, showing that its condition number is independent of the number of subdomains (scalability) and bounded by the ratio of the subdomains characteristic size and the overlap size. This theoretical result is confirmed by several parallel simulations employing up to more than 2000 processors for scaled and standard speedup tests in three dimensions. The results show the scalability of the proposed NKS Bidomain solver in terms of both nonlinear and linear iterations, in both Cartesian slabs and ellipsoidal cardiac domains
Exploring anodal and cathodal make and break cardiac excitation mechanisms in a 3D anisotropic bidomain model
No full textPublished studies have investigated the relevance of cardiac virtual electrode responses to unipolar cathodal and anodal stimulations for explaining the make and break excitation mechanisms. Most of these studies have considered 2D bidomain models or cylindrical domains that by symmetry reduce to the 2D case, so the triggering mechanisms and onset of excitation have not yet been fully elucidated in 3D anisotropic models. The goal of this work is to revisit these excitation mechanisms with 3D bidomain simulations considering two tissue types with unequal anisotropy ratio, including transmural fiber rotation and augmenting the Luo–Rudy I membrane model with the so-called funny and the electroporation currents. In addition to usual snapshots of transmembrane potential patterns, we compute from the action potential waveforms the activation time and associated isochrone sequences, yielding a detailed 3D description of the instant and location of excitation origin, shape and propagation of activation wavefronts. A specific aim of this work is to detect the location of the excitation onset and whether its trigger mechanism is (a) electrotonic, i.e. originating from discharge diffusion of currents flowing between virtual cathodes and anods and/or (b) membrane-based, i.e. arising only from intrinsic depolarizing membrane currents. Our results show that the electrotonic mechanism is observed independently of the degree of unequal anisotropy in diastolic anode make and systolic cathode break. The membrane-based mechanism is observed in diastolic cathode make, diastolic anode break, only for a relative weak anisotropy, and systolic anode break. The excitation trigger mechanism, the location of the excitation origin and the pattern of the isochrone sequence are independent of the degree of anisotropy for diastolic cathode make, systolic cathode and anode break, while they might depend on the degree of anisotropy for diastolic anode make and break. Moreover, the tissue anisotropy has a strong influence on the threshold amplitude of the stimulation pulse triggering these mechanisms
A reliability analysis of cardiac repolarization time markers
No full textOnly a limited number of studies have addressed the reliability of extracellular markers of cardiac repolarization time, such as the classical marker RTeg defined as the time of maximum upslope of the electrogram T wave. This work presents an extensive three-dimensional simulation study of cardiac repolarization time, extending the previous one-dimensional simulation study of a myocardial strand by Steinhaus [B.M. Steinhaus, Estimating cardiac transmembrane activation and recovery times from unipolar and bipolar extracellular electrograms: a simulation study, Circ. Res. 64 (3) (1989) 449]. The simulations are based on the bidomain - Luo-Rudy phase I system with rotational fiber anisotropy and homogeneous or heterogeneous transmural intrinsic membrane properties. The classical extracellular marker RTeg is compared with the gold standard of fastest repolarization time RTtap, defined as the time of minimum derivative during the downstroke of the transmembrane action potential (TAP). Additionally, a new extracellular marker RT 90eg is compared with the gold standard of late repolarization time RT 90tap, defined as the time when the TAP reaches 90% of its resting value. The results show a good global match between the extracellular and transmembrane repolarization markers, with small relative mean discrepancy (≤ 1.6 %) and high correlation coefficients (≥ 0.92), ensuring a reasonably good global match between the associated repolarization sequences. However, large local discrepancies of the extracellular versus transmembrane markers may ensue in regions where the curvature of the repolarization front changes abruptly (e.g. near front collisions) or is negligible (e.g. where repolarization proceeds almost uniformly across fiber). As a consequence, the spatial distribution of activation-recovery intervals (ARI) may provide an inaccurate estimate of (and weakly correlated with) the spatial distribution of action potential durations (APD)
Overlapping schwarz methods for isogeometric analysis
Full text linkWe construct and analyze an overlapping Schwarz preconditioner for elliptic problems discretized with isogeometric analysis. The preconditioner is based on partitioning the domain of the problem into overlapping subdomains, solving local isogeometric problems on these subdomains, and solving an additional coarse isogeometric problem associated with the subdomain mesh. We develop an -analysis of the preconditioner, showing in particular that the resulting algorithm is scalable and its convergence rate depends linearly on the ratio between subdomain and „overlap sizes” for fixed polynomial degree and regularity of the basis functions. Numerical results in two- and three-dimensional tests show the good convergence properties of the preconditioner with respect to the isogeometric discretization parameters , number of subdomains , overlap size, and also jumps in the coefficients of the elliptic operator
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