Acceleration and Parallelization of ZENO/Walk-on-Spheres

AbstractThis paper describes our on-going work to accelerate ZENO, a software tool based on Monte Carlo methods (MCMs), used for computing material properties at nanoscale. ZENO employs three main algorithms: (1) Walk on Spheres (WoS), (2) interior sampling, and (3) surface sampling. We have accelerated the first two algorithms. For the sake of brevity, the paper will discuss our work on the first one only as it is the most commonly used and the acceleration techniques were similar in both cases.WoS is a Brownian motion MCM for solving a class of partial differential equations (PDEs). It provides a stochastic solution to a PDE by estimating the probability that a random walk, which started at infinity, will hit the surface of the material under consideration. WoS is highly effective when the problem's geometry is additive, as this greatly reduces the number of walk steps needed to achieve accurate results. The walks start on the surface of an enclosing sphere and can make much larger jumps than in a direct simulation of Brownian motion. Our current implementation represents the molecular structure of nanomaterials as a union of possibly overlapping spheres. The core processing bottleneck in WoS is a Computational Geometry one, as the algorithm repeatedly determines the distance from query point to the material surface in each step of the random walk.In this paper, we present results from benchmarking spatial data structures, including several open-source implementations of k-D trees, for accelerating WoS algorithmically. The paper also presents results from our multicore and cluster parallel implementation to show that it exhibits linear strong scaling with the number of cores and compute nodes; this implementation delivers up to 4 orders of magnitude speedup compared to the original FORTRAN code when run on 8 nodes (each with dual 6-core Intel Xeon CPUs) with 24 threads per node

Space-Efficient Graph Coarsening with Applications to Succinct Planar Encodings

We present a novel space-efficient graph coarsening technique for n-vertex planar graphs G, called cloud partition, which partitions the vertices V(G) into disjoint sets C of size O(log n) such that each C induces a connected subgraph of G. Using this partition ? we construct a so-called structure-maintaining minor F of G via specific contractions within the disjoint sets such that F has O(n/log n) vertices. The combination of (F, ?) is referred to as a cloud decomposition. For planar graphs we show that a cloud decomposition can be constructed in O(n) time and using O(n) bits. Given a cloud decomposition (F, ?) constructed for a planar graph G we are able to find a balanced separator of G in O(n/log n) time. Contrary to related publications, we do not make use of an embedding of the planar input graph. We generalize our cloud decomposition from planar graphs to H-minor-free graphs for any fixed graph H. This allows us to construct the succinct encoding scheme for H-minor-free graphs due to Blelloch and Farzan (CPM 2010) in O(n) time and O(n) bits improving both runtime and space by a factor of ?(log n). As an additional application of our cloud decomposition we show that, for H-minor-free graphs, a tree decomposition of width O(n^{1/2 + ?}) for any ? > 0 can be constructed in O(n) bits and a time linear in the size of the tree decomposition. A similar result by Izumi and Otachi (ICALP 2020) constructs a tree decomposition of width O(k ?n log n) for graphs of treewidth k ? ?n in sublinear space and polynomial time

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum