Johannes Pahlke | Searcharxiv

arXiv2025

On the Computational Power of Particle Methods

We investigate the computational power of particle methods, a well-established class of algorit hms with applications in scientific computing and computer simulation. The computational power of a compute model determines the class of problems it can solve. Automata theory allows describing the computational power of abstract machines (automata) and the problems they can solve. At the top of the Chomsky hierarchy of formal languages and grammars are Turing machines, which resemble the concept on which most modern computers are built. Although particle methods can be interpreted as automata based on their formal definition, their computational power has so far not been studied. We address this by analyzing Turing completeness of particle methods. In particular, we prove two sets of restrictions under which a particle method is still Turing powerful, and we show when it loses Turing powerfulness. This contributes to understanding the theoretical foundations of particle methods and provides insight into the powerfulness of computer simulations.

cs.FL↗

arXiv2024

Proven Distributed Memory Parallelization of Particle Methods

We provide a mathematically proven parallelization scheme for particle methods on distributed-memory computer systems. Particle methods are a versatile and widely used class of algorithms for computer simulations and numerical predictions in various applications, ranging from continuum fluid dynamics and granular flows, using methods such as Smoothed Particle Hydrodynamics (SPH) and Discrete Element Methods (DEM) to Molecular Dynamics (MD) simulations in molecular modeling. Particle methods naturally lend themselves to implementation on parallel-computing hardware. So far, however, a mathematical proof of correctness and equivalence to sequential implementations was only available for shared-memory parallelism. Here, we leverage a formal definition of the algorithmic class of particle methods to provide a proven parallelization scheme for distributed-memory computers. We prove that these parallelized particle methods on distributed memory computers are formally equivalent to their sequential counterpart for a well-defined class of particle methods. Notably, the here analyzed parallelization scheme is well-known and commonly used. Our analysis is, therefore, of immediate practical relevance to existing and new parallel software implementations of particle methods and places them on solid theoretical grounds.

cs.DC↗