Comparison of Parallel Solution Techniques for the Eikonal Equation

The Eikonal equation is well established for modelling wave propagation in the field of cardiac electrophysiology[?Franzone]. It offers highly tempting computation speeds in contrast to the bidomain or monodomain models. Consequently, developing effective algorithmic solution techniques to the Eikonal equation is an immediate and valuable task. In this paper, we build on the methods existing in the Chaste (Cancer, Heart and Soft Tissue Environment)[?JPF] simulation environment, working on both conventional PDE Finite Element Method solvers and developing a graph-based algorithm based on the Fast Marching Method[?Sethian_1997]. We found that the FMM techniques were computationally preferential to both the existing Dijkstra-based methods and FEM techniques, although conventional PDE methods provided greater control over errors, especially in regions of high wavefront curvature. Both methods can be parallelised readily, making them suitable for application over large clusters. 1. Solving the Eikonal Equation with FEM Solvers Along with attempts to extend Dijkstra style algorithms to more accurate solutions over the mesh, it is naturally of interest to consider the usual numerical approaches to solving nonlinear PDEs and seeing how these cope with anisotropic meshes and reduced discrete boundary conditions. The favoured approach here, due to the complex nature of the mesh and boundary conditions as well the need for time efficency is the (Galerkin)Finite Element Method (FEM). The formulation used for this equation is a Navier Stokes form [?Tucker_2003]: