BPX preconditioners for the Bidomain model of electrocardiology

The 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\u2013diffusion 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

The LifeV library: engineering mathematics beyond the proof of concept

LifeV is a library for the finite element (FE) solution of partial differential equations in one, two, and three dimensions. It is written in C++ and designed to run on diverse parallel architectures, including cloud and high performance computing facilities. In spite of its academic research nature, meaning a library for the development and testing of new methods, one distinguishing feature of LifeV is its use on real world problems and it is intended to provide a tool for many engineering applications. It has been actually used in computational hemodynamics, including cardiac mechanics and fluid-structure interaction problems, in porous media, ice sheets dynamics for both forward and inverse problems. In this paper we give a short overview of the features of LifeV and its coding paradigms on simple problems. The main focus is on the parallel environment which is mainly driven by domain decomposition methods and based on external libraries such as MPI, the Trilinos project, HDF5 and ParMetis. Dedicated to the memory of Fausto Saleri.Comment: Review of the LifeV Finite Element librar

Accelerating Cardiac Bidomain Simulations Using Graphics Processing Units

Anatomically realistic and biophysically detailed multiscale computer models of the heart are playing an increasingly important role in advancing our understanding of integrated cardiac function in health and disease. Such detailed simulations, however, are computationally vastly demanding, which is a limiting factor for a wider adoption of in-silico modeling. While current trends in high-performance computing (HPC) hardware promise to alleviate this problem, exploiting the potential of such architectures remains challenging since strongly scalable algorithms are necessitated to reduce execution times. Alternatively, acceleration technologies such as graphics processing units (GPUs) are being considered. While the potential of GPUs has been demonstrated in various applications, benefits in the context of bidomain simulations where large sparse linear systems have to be solved in parallel with advanced numerical techniques are less clear. In this study, the feasibility of multi-GPU bidomain simulations is demonstrated by running strong scalability benchmarks using a state-of-the-art model of rabbit ventricles. The model is spatially discretized using the finite element methods (FEM) on fully unstructured grids. The GPU code is directly derived from a large pre-existing code, the Cardiac Arrhythmia Research Package (CARP), with very minor perturbation of the code base. Overall, bidomain simulations were sped up by a factor of 11.8 to 16.3 in benchmarks running on 6-20 GPUs compared to the same number of CPU cores. To match the fastest GPU simulation which engaged 20 GPUs, 476 CPU cores were required on a national supercomputing facility

Parallel multilevel solvers for the cardiac electro-mechanical coupling

We develop a parallel solver for the cardiac electro-mechanical coupling. The electric model consists of two non-linear parabolic partial differential equations (PDEs), the so-called Bidomain model, which describes the spread of the electric impulse in the heart muscle. The two PDEs are coupled with a non-linear elastic model, where the myocardium is considered as a nearly-incompressible transversely isotropic hyperelastic material. The discretization of the whole electro-mechanical model is performed by Q1 finite elements in space and a semi-implicit finite difference scheme in time. This approximation strategy yields at each time step the solution of a large scale ill-conditioned linear system deriving from the discretization of the Bidomain model and a non-linear system deriving from the discretization of the finite elasticity model. The parallel solver developed consists of solving the linear system with the Conjugate Gradient method, preconditioned by a Multilevel Schwarz preconditioner, and the non-linear system with a Newton\u2013Krylov-Algebraic Multigrid solver. Three-dimensional parallel numerical tests on a Linux cluster show that the parallel solver proposed is scalable and robust with respect to the domain deformations induced by the cardiac contraction

MULTIGRID METHODS FOR THE BIDOMAIN EQUATIONS

The study of cardiac electrophysiology has many applications in medical practice. One important model is the bidomain equations. In the thesis, the bidomain equations for the muscle and for the muscle and the bath are considered. By implementing multigrid algorithms as the preconditioner, we explore the block factorization approach for solving the bidomain equations. The dissertation consists two parts, aiming to present the biological background and dis- cretization for the bidomain equations, as well as the multigrid algorithms. In the first part, we present the derivation of the formula of bidomain equations, the finite difference and finite element discretization for the bidomain system, and semi-implicit time stepping. In the second part, we study the key facts of both geometric multigrid and algebraic multigrid method. We consider the with and without fibrosis cases. We implement the two multigrid methods as both the solver for the bidomain system and the preconditioner for the block factorization approach, and conclude that block factorization works efficiently, especially compared with the performance of the algebraic multigrid solver. We also test the block factorization with algebraic multigrid preconditioner on a realistic three-dimensional geometry, and obtain only a small increase in solver iterations as the mesh becomes finer. We discuss useful extensions of this block factorization approach on solving the bidomain system. Since algebraic multigrid works best for Poisson-like problems, we can factorize the original matrix into blocks with poisson like form, and apply algebraic multigrid as preconditioner to each block to achieve good convergence.Doctor of Philosoph

Parallel Newton-Krylov-BDDC and FETI-DP deluxe solvers for implicit time discretizations of the cardiac Bidomain equations

Two novel parallel Newton-Krylov Balancing Domain Decomposition by Constraints (BDDC) and Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) solvers are here constructed, analyzed and tested numerically for implicit time discretizations of the three-dimensional Bidomain system of equations. This model represents the most advanced mathematical description of the cardiac bioelectrical activity and it consists of a degenerate system of two non-linear reaction-diffusion partial differential equations (PDEs), coupled with a stiff system of ordinary differential equations (ODEs). A finite element discretization in space and a segregated implicit discretization in time, based on decoupling the PDEs from the ODEs, yields at each time step the solution of a non-linear algebraic system. The Jacobian linear system at each Newton iteration is solved by a Krylov method, accelerated by BDDC or FETI-DP preconditioners, both augmented with the recently introduced {\em deluxe} scaling of the dual variables. A polylogarithmic convergence rate bound is proven for the resulting parallel Bidomain solvers. Extensive numerical experiments on linux clusters up to two thousands processors confirm the theoretical estimates, showing that the proposed parallel solvers are scalable and quasi-optimal

Optimized schwarz methods for the bidomain system in electrocardiology

The propagation of the action potential in the heart chambers is accurately described by the Bidomain model, which is commonly accepted and used in the specialistic literature. However, its mathematical structure of a degenerate parabolic system entails high computational costs in the numerical solution of the associated linear system. Domain decomposition methods are a natural way to reduce computational costs, and Optimized Schwarz Methods have proven in the recent years their effectiveness in accelerating the convergence of such algorithms. The latter are based on interface matching conditions more efficient than the classical Dirichlet or Neumann ones. In this paper we analyze an Optimized Schwarz approach for the numerical solution of the Bidomain problem. We assess the convergence of the iterative method by means of Fourier analysis, and we investigate the parameter optimization in the interface conditions. Numerical results in 2D and 3D are given to show the effectiveness of the method