Simone Scacchi

Multilevel Additive Schwarz Preconditioners for the Bidomain Reaction-Diffusion System

Multilevel 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.

Isogeometric BDDC Preconditioners with Deluxe Scaling

A balancing domain decomposition by constraints (BDDC) preconditioner with a novel scaling, introduced by Dohrmann for problems with more than one variable coefficient and here denoted as deluxe scaling, is extended to isogeometric analysis of scalar elliptic problems. This new scaling turns out to be more powerful than the standard

A Scalable Newton-Krylov-Schwarz Method for the Bidomain Reaction-Diffusion System

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Modeling ventricular repolarization : Effects of transmural and apex-to-base heterogeneities in action potential durations

- and stiffness scalings considered in a previous isogeometric BDDC study. Our

A reliability analysis of cardiac repolarization time markers.

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Parallel Multilevel Schwarz and Block Preconditioners for the Bidomain Parabolic-Parabolic and Parabolic-Elliptic Formulations

-analysis shows that the condition number of the resulting deluxe BDDC preconditioner is scalable with a quasi-optimal polylogarithmic bound which is also independent of coefficient discontinuities across subdomain interfaces. Extensive numerical experiments support the theory and show that the deluxe scaling yields a remarkable improvement over the older scalings, in particular for large isogeometric polynomial degree and high regularity.

Determining recovery times from transmembrane action potentials and unipolar electrograms in normal heart tissue

A 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.

DYNAMICAL EFFECTS OF MYOCARDIAL ISCHEMIA IN ANISOTROPIC CARDIAC MODELS IN THREE DIMENSIONS

Heterogeneities in the densities of membrane ionic currents of myocytes cause regional variations in action potential duration (APD) at various intramural depths and along the apico-basal and circumferential directions in the left ventricle. This work extends our previous study of cartesian slabs to ventricular walls shaped as an ellipsoidal volume and including both transmural and apex-to-base APD heterogeneities. Our 3D simulation study investigates the combined effect on repolarization sequences and APD distributions of: (a) the intrinsic APD heterogeneity across the wall and along the apex-to-base direction, and (b) the electrotonic currents that modulate the APDs when myocytes are embedded in a ventricular wall with fiber rotation and orthotropic anisotropy. Our findings show that: (i) the transmural and apex-to-base heterogeneities have only a weak influence on the repolarization patterns on myocardial layers parallel to the epicardium; (ii) the patterns of APD distribution on the epicardial surface are mostly affected by the apex-to-base heterogeneities and do not reveal the APD transmural heterogeneity; (iii) the transmural heterogeneity is clearly discernible in both repolarization and APD patterns only on transmural sections; (iv) the apex-to-base heterogeneity is clearly discernible only in APD patterns on layers parallel to the epicardium. Thus, in our orthotropic ellipsoidal wall, the complex 3D electrotonic modulation of APDs does not fully mix the effects of the transmural and apex-to-base heterogeneity. The intrinsic spatial heterogeneity of the APDs is unmasked in the modulated APD patterns only in the appropriate transmural or intramural sections. These findings are independent of the stimulus location (epicardial, endocardial) and of Purkinje involvement.

Cardiac excitation mechanisms, wavefront dynamics and strength - interval curves predicted by 3D orthotropic bidomain simulations

Only a limited number of studies have addressed the reliability of extracellular markers of cardiac repolarization time, such as the classical marker RT(eg) 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 RT(eg) is compared with the gold standard of fastest repolarization time RT(tap), defined as the time of minimum derivative during the downstroke of the transmembrane action potential (TAP). Additionally, a new extracellular marker RT90(eg) is compared with the gold standard of late repolarization time RT90(tap), 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 (<or=1.6%) and high correlation coefficients (>or=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).

Exploring anodal and cathodal make and break cardiac excitation mechanisms in a 3D anisotropic bidomain model

The 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.