arXiv subjects

Wayne Mitchell

Publications and source records attributed to Wayne Mitchell.

5 recordsLinked to original sources

arXiv2026

Skywing: A Platform for Decentralized Mathematical Computing in Unreliable Environments

Emerging edge, autonomous, and cyber-physical systems increasingly require mathematical computation across heterogeneous devices connected by unreliable communication networks. Traditional high-performance computing and distributed data-processing frameworks provide powerful abstractions for managed environments but are less suited to decentralized settings where centralized coordination, reliable communication, and global synchronization cannot be assumed. This paper presents Skywing, an open-source platform for decentralized mathematical computing in unreliable environments. Its programming model consists of three abstractions: agents represent participants in a decentralized computation, processors encapsulate algorithm-specific update rules, and iterations manage distributed execution. Skywing supports asynchronous operation, publish-subscribe communication, managed message handling, and the composition of independent algorithms into complex decentralized workflows. We demonstrate Skywing using representative algorithms from consensus, optimization, and numerical linear algebra. Experiments on the native Skywing runtime include Push Sum and Max Consensus, a composed monitoring and control workflow, resilient Push Sum under delayed communication, and resilient asynchronous Jacobi under malevolent data corruption. These demonstrations show that Skywing supports diverse decentralized algorithms while separating mathematical logic from communication and execution infrastructure. Skywing serves as both a deployment framework for decentralized applications and a research platform for developing resilient mathematical algorithms.

cs.DC↗

arXiv2026

Automated Grammar-based Algebraic Multigrid Design With Evolutionary Algorithms

Although multigrid is asymptotically optimal for solving many important partial differential equations, its efficiency relies heavily on the careful selection of the individual algorithmic components. In contrast to recent approaches that can optimize certain multigrid components using deep learning techniques, we adopt a complementary strategy, employing evolutionary algorithms to construct efficient multigrid cycles from proven algorithmic building blocks. Here, we will present its application to generate efficient algebraic multigrid methods with so-called \emph{flexible cycling}, that is, level-specific smoothing sequences and non-recursive cycling patterns. The search space with such non-standard cycles is intractable to navigate manually, and is generated using genetic programming (GP) guided by context-free grammars. Numerical experiments with the linear algebra library, \emph{hypre}, demonstrate the potential of these non-standard GP cycles to improve multigrid performance both as a solver and a preconditioner.

cs.CE↗

arXiv2020

Tight two-level convergence of Linear Parareal and MGRIT: Extensions and implications in practice

Two of the most popular parallel-in-time methods are Parareal and multigrid-reduction-in-time (MGRIT). Recently, a general convergence theory was developed in Southworth (2019) for linear two-level MGRIT/Parareal that provides necessary and sufficient conditions for convergence, with tight bounds on worst-case convergence factors. This paper starts by providing a new and simplified analysis of linear error and residual propagation of Parareal, wherein the norm of error or residual propagation is given by one over the minimum singular value of a certain block bidiagonal operator. New discussion is then provided on the resulting necessary and sufficient conditions for convergence that arise by appealing to block Toeplitz theory as in Southworth (2019). Practical applications of the theory are discussed, and the convergence bounds demonstrated to predict convergence in practice to high accuracy on two standard linear hyperbolic PDEs: the advection(-diffusion) equation, and the wave equation in first-order form.

math.NA↗

arXiv2019

Advances in Implementation, Theoretical Motivation, and Numerical Results for the Nested Iteration with Range Decomposition Algorithm

This paper studies a low-communication algorithm for solving elliptic partial differential equations (PDE's) on high-performance machines, the nested iteration with range decomposition algorithm (NIRD). Previous work has shown that NIRD converges to a high level of accuracy within a small, fixed number of iterations (usually one or two) when applied to simple elliptic problems. This paper makes some improvements to the NIRD algorithm (including the addition of adaptivity during preprocessing, wider choice of partitioning functions, and modified error measurement) that enhance the method's accuracy and scalability, especially on more difficult problems. In addition, an updated convergence proof is presented based on heuristic assumptions that are supported by numerical evidence. Furthermore, a new performance model is developed that shows increased performance benefits for NIRD when problems are more expensive to solve using traditional methods. Finally, extensive testing on a variety of elliptic problems provides additional insight into the behavior of NIRD and additional evidence that NIRD achieves excellent convergence on a wide class of elliptic PDE's and, as such, should be a very competitive method for solving PDE's on large parallel computers.

math.NA↗

arXiv2012

Development and application of a particle-particle particle-mesh Ewald method for dispersion interactions

For inhomogeneous systems with interfaces, the inclusion of long-range dispersion interactions is necessary to achieve consistency between molecular simulation calculations and experimental results. For accurate and efficient incorporation of these contributions, we have implemented a particle-particle particle-mesh (PPPM) Ewald solver for dispersion ($r^{-6}$) interactions into the LAMMPS molecular dynamics package. We demonstrate that the solver's $\mathcal{O}(N\log N)$ scaling behavior allows its application to large-scale simulations. We carefully determine a set of parameters for the solver that provides accurate results and efficient computation. We perform a series of simulations with Lennard-Jones particles, SPC/E water, and hexane to show that with our choice of parameters the dependence of physical results on the chosen cutoff radius is removed. Physical results and computation time of these simulations are compared to results obtained using either a plain cutoff or a traditional Ewald sum for dispersion.

cond-mat.mtrl-sci↗