Fast many-core solvers for the Eikonal equations in cardiovascular simulations

Abstract

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.

A massively parallel Eikonal solver on unstructured meshes

“…see entry (1,2) in Table 1. For all the other angles of the tetrahedron the calculation is done in the same way and the sign problem falls into one of the previous three cases.…”

“…These are the products marked in blue in Table 4. The other 3 needed products for the mixed edge numbers, respectively (1, 2), (1,5) and (2,5) which reference exactly the off-diagonal products in blue in Table 1, are calculated based on formula (11). The memory footprint reduction is obvious.…”

“…But also in the general case of parallel architectures with limited high-bandwidth memory, the memory footprint of the local solver becomes a bottleneck to performance. We addressed the implementation of an Eikonal solver for Shared Memory (OpenMP) and for streaming architectures in CUDA with a low memory footprint in our previous work [2] wherein we achieved already a very short execution time of the algorithm and good quality results. We use the fast iterative method [1,4,5] for solving the Eikonal equation, especially the version by Fu, Kirby and Whitaker [1] wherein the memory footprint has been reduced by storing temporarily the inner products needed to compute the 3D solution of one vertex.…”

“…Recent work in [1][2][3] has shown that building an efficient 3D tetrahedral Eikonal solver for multi-core and SIMD (single instruction multiple data) architectures poses many challenges. It is important to keep the memory footprint low in order to reduce the costly memory accesses and achieve a good computational density on GPUs and other SIMD architectures with limited memory and register capabilities.…”

Reducing the Memory Footprint of an Eikonal Solver

“…This arrival time ϕ(x) of the wavefront at some point x ∈ Ω can be approximated by the Eikonal equation (3) with given heterogeneous, anisotropic velocity information.…”