Algorithms for the numerical solution of the Eikonal equation discretized with tetrahedra are discussed. Several massively parallel algorithms for GPU computing are developed. This includes domain decomposition concepts for tracking the moving wave fronts in sub-domains and over the sub-domain boundaries. Furthermore a low memory footprint implementation of the solver is introduced which reduces the number of arithmetic operations and enables improved memory access schemes. The numerical tests for different meshes originating from the geometry of a human heart document the decreased runtime of the new algorithms.

Simulation of one heart beat which faithfully account for biophysical details involved in cardiac electrophysiology and mechanics are still far away from real time performance, even when employing several thousands of compute nodes. Therefore, a simpler model based on the Eikonal equation will be considered. This model could be of great utility as a tool for generating activation and repolarisation sequences and its concomitant electrocardiogram by replacing the PDE part of the bi-domain equations with the Eikonal equations, while retaining the ODE parts to account for the full mechanistic detail relevant to ECG computation. Further, the approach can be extended to use Eikonal-based activation sequences as a driver for mechanical contraction models. We will address the implementation of an Eikonal solver for Shared Memory (OpenMP) with a low memory footprint. This solver will be transferred for a coarse model onto a tablet computer and other handheld devices for clinical use. Due to the splitting of the wave front (described by the Eikonal equations), the parallel version results in a slightly different convergence history and in minor differences in the solution. A future CUDA implementation of the parallel algorithm will reduce the run time further such that also interactive simulations will be possible.

The numerical solution of the Eikonal equation follows the fast iterative method [1] with its application for tetrahedral meshes [2]. Therein the main operations in each discretization element τ contain various inner products in the Mmetric as �� e k,s , � e s,� �Mτ ≡ � e T k,s • M τ • � e s,� with � e s,� as connecting edge between vertices s and � in element τ. While the authors of [2] pass all coordinates of the tetrahedron together with the 6 entries of M τ we precompute these inner products and use only them in the wave front computation. This first change requires less memory transfers for each tetrahedron. The second change is caused by the fact that �� e k,s , � e s,� �Mτ (k � = �) represents an angle of a surface triangle whereas �� e k,s , � e k,s �Mτ represents the length of an edge in the Mmetric. Basic geometry as well as vector arithmetics yield to the conclusion that the angle information can be expressed by the combination of three edge lengths. Therefore we only have to precompute the 6 edge lengths of a tetrahedron and compute the remaining 12 angle data on-the-fly which reduces the memory footprint per tetrahedron to 6 numbers. The efficient implementation of the two changes requires a local Gray-code numbering of edges in the tetrahedron and a bunch of bit shifts to assign the appropriate data. First numerical experiments on CPUs show that the reduced memory footprint approach is faster than the original implementation. Detailed investigations as well as a CUDA implementation are ongoing work.

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