Potential benefits of a block-space GPU approach for discrete tetrahedral domains

The study of data-parallel domain re-organization and thread-mapping techniques are relevant topics as they can increase the efficiency of GPU computations when working on spatial discrete domains with non-box-shaped geometry. In this work we study the potential benefits of applying a succint data re-organization of a tetrahedral data-parallel domain of size $\mathcal{O}(n^3)$ combined with an efficient block-space GPU map of the form $g:\mathbb{N} \rightarrow \mathbb{N}^3$. Results from the analysis suggest that in theory the combination of these two optimizations produce significant performance improvement as block-based data re-organization allows a coalesced one-to-one correspondence at local thread-space while $g(\lambda)$ produces an efficient block-space spatial correspondence between groups of data and groups of threads, reducing the number of unnecessary threads from $O(n^3)$ to $O(n^2\rho^3)$ where $\rho$ is the linear block-size and typically $\rho^3 \ll n$. From the analysis, we obtained that a block based succint data re-organization can provide up to $2\times$ improved performance over a linear data organization while the map can be up to $6\times$ more efficient than a bounding box approach. The results from this work can serve as a useful guide for a more efficient GPU computation on tetrahedral domains found in spin lattice, finite element and special n-body problems, among others.