An agglomeration-based multigrid solver for the discontinuous Galerkin discretization of cardiac electrophysiology
📝 Abstract
This work presents a novel agglomeration-based multilevel preconditioner designed to accelerate the convergence of iterative solvers for linear systems arising from the discontinuous Galerkin discretization of the monodomain model in cardiac electrophysiology. The proposed approach exploits general polytopic grids at coarser levels, obtained through the agglomeration of elements from an initial, potentially fine, mesh. By leveraging a robust and efficient agglomeration strategy, we construct a nested hierarchy of grids suitable for multilevel solver frameworks. The effectiveness and performance of the methodology are assessed through a series of numerical experiments on two- and three-dimensional domains, involving different ionic models and realistic unstructured geometries. The results demonstrate strong solver effectiveness and favorable scalability with respect to both the polynomial degree of the discretization and the number of levels selected in the multigrid preconditioner.
💡 Analysis
This work presents a novel agglomeration-based multilevel preconditioner designed to accelerate the convergence of iterative solvers for linear systems arising from the discontinuous Galerkin discretization of the monodomain model in cardiac electrophysiology. The proposed approach exploits general polytopic grids at coarser levels, obtained through the agglomeration of elements from an initial, potentially fine, mesh. By leveraging a robust and efficient agglomeration strategy, we construct a nested hierarchy of grids suitable for multilevel solver frameworks. The effectiveness and performance of the methodology are assessed through a series of numerical experiments on two- and three-dimensional domains, involving different ionic models and realistic unstructured geometries. The results demonstrate strong solver effectiveness and favorable scalability with respect to both the polynomial degree of the discretization and the number of levels selected in the multigrid preconditioner.
📄 Content
Computational modeling of the heart has been actively pursued as a tool for accelerating cardiovascular research. However, the clinical applicability of such models is constrained by the high complexity and computational cost, for instance when moving towards whole heart modeling and coupling different physics and scales. Hence, the development of numerical methods able to reduce the computing time while keeping accuracy becomes essential for speeding-up fundamental research and, ultimately, for translation of modeling into clinical practice [49,71].
In this work, we focus specifically on cardiac electrophysiology models describing the electrical activation of the myocardium. The electrical activity of the heart is usually modeled through the bidomain or monodomain equations, which describe the propagation of the transmembrane potential across the heart tissue. Both models consist of an ellipticparabolic system of nonlinear reaction-diffusion partial differential equations (PDEs). The mathematical analysis of the system has already been addressed in [36], and a wide strand of discretization approaches have been proposed during the years, for instance in [10,21] and references therein. In order to take into account the electrochemical reactions that occur at a cellular level, these systems are coupled, through a non-linear reaction term, to a system of Ordinary Differential Equations (ODEs), modelling the inward and outward flow of ionic currents across the cell membrane [49]. During the years, a large variety of models have been developed, ranging from reduced models with only one or few unknowns such as the Rogers-McCulloch ionic model [73], or the FitzHugh and Nagumo model [48], up to the Bueno-Orovio and ten Tusscher-Panfilov ionic models [30,78,79]. Due to the quick upstroke of the action potential, which is caused by voltage-dependent sodium channels, a numerically robust calculation of the propagation of the wave across the tissue is well known to be computationally challenging; indeed, the rapid increase of the transmembrane potential over a few milliseconds results in a steep wave front in space, requiring high resolution in both temporal and spatial discretizations.
Overall, the resulting large-scale linear systems are prohibitive for direct solvers. In addition, the system matrix often turns out to be very ill-conditioned, leading to a deterioration in the convergence rate of standard iterative methods. Hence, the main challenge to be addressed is the reduction of the overall time-to-solution of the iterative method through the design of effective preconditioners [25].
The development of suitable preconditioners for such models is a vast and active area of research, and many approaches have been proposed over the years. Among them, we mention domain-decomposition techniques such as Balancing Domain Decomposition with Constraints (BDDC) [42] and Finite Element Tearing and Interconnecting (FETI) methods [59]. Extensions of these methodologies, along with their application to electrophysiology problems, have been extensively studied [1,56,68,81].
Traditional strategies exploit linear continuous elements on very fine computational grids, in order to capture the physical features of the solution. To cope with such complexity, higher order continuous elements or discontinuous Galerkin (DG) approaches have recently received more attention [4,28,55], thanks to their ability to capture sharp gradients. For these reasons, in this work we will consider a DG discretization of the monodomain model. Our main focus is the development of a preconditioner for accelerating the convergence of the large-scale linear system of equations.
For linear systems stemming from the discretization of elliptic operators, multigrid methods are widely recognized as one of the most efficient approaches that scale to large parallel computers. Standard geometric multigrid techniques require the construction of a hierarchy of meshes that is most often obtained by (either uniform or adaptive) refinement of a coarse initial mesh. However, in the case of very complex geometries, such as the ones stemming from realistic models, building such hierarchies is far from trivial, requiring to fallback to non-nested methods [26,35,47] or algebraic multigrid (AMG) solvers [80].
The use of polytopal shapes is attractive in this respect, as coarse grids can be simply generated by agglomerating mesh elements together and various multilevel approaches have been proposed in the literature [22,29,34,39,64,66]. However, providing auto-mated and flexible agglomeration strategies for polyhedral elements remains a challenging task which is subject of intense research [8,9,45]. Motivated by the good results observed in [45], we adopt the agglomeration algorithm developed therein to produce nested sequences of agglomerated grids by exploiting R-trees [53], a spatial indexing data structure which excels at organizing geometric data through bounding box
This content is AI-processed based on ArXiv data.