arXiv subjects

Cory Hauck

Publications and source records attributed to Cory Hauck.

At least 19 recordsLinked to original sources

arXiv2026

Sweep-based, implicit solutions of the multidimensional BGK equation on unstructured grids

We present a nodal discontinuous Galerkin method for solving the Bhatnagar-Gross-Krook (BGK) kinetic equation on multi-dimensional, unstructured grids. The method uses implicit, sweep-based solvers and a moment-preserving projection of the Maxwellian source to enable high-order accuracy in time while avoiding restrictive time steps imposed by boundary layers and other geometry-induced features. We verify that the method is correct in the continuum limit by comparing to closed-form and high-order solutions of the Sod shock problem on 2 and 3D unstructured grids. Linear L2 stability is demonstrated for a B-stable diagonally implicit Runge-Kutta method of third order. The solver uses a hybrid parallel scheme based on spatial domain decomposition with local sweeps performed on CPU and GPU hardware. Platform-portability is demonstrated through the development of new GPU-friendly, graph-based sweep algorithms that are implemented using the Kokkos performance portability library and achieve greater than 20 times speedup on NVIDIA H100 GPUs compared to 64-core AMD EPYC 9654 CPUs. Finally, we show results on the Frontier supercomputer at the Oak Ridge Leadership Computing Facility for a boundary value problem with 2.77 trillion phase space degrees of freedom that executed on 1536 nodes utilizing 6144 AMD MI250X GPUs.

cs.CE↗

arXiv2026

MPEX AI Digital Twins Milestone Report

This is the six month progress report to Fusion Energy Science (FES) and the American Science Cloud (AmSC) on the MPEX AI Digtial Twins project that was started in October 2025. There are two milestones to demonstrate the Artificial Intelligence (AI) advantage for MPEX operations and scientific discovery, that will be completed by June 2026. The first is a Helicon AI Hot-Spot Controller (Sec. 3.1), which is the helicon heating component of the more comprehensive planned MPEX AI Hot Spot Digital Twin (Sec. 3). The second is an E-beam Damage Assessment Digital Twin (Sec. 4.1), which is a reduced electron beam damage modality prototype for the MPEX AI Damage Assessment Digital Twin (Sec. 4). These two phase I milestones are on track for the June demonstration. In addition to these two milestones, progress on configuring the Galaxy software interface for automation, validation and data analysis is reported (Sec. 5). This interface now connects a subset of the main physics simulation codes to DOE HPC resources and will connect to the MPEX data acquisition system so that analysis of data, validation and execution of simulations can be performed by the scientist or by AI-Agents. When AmSC is ready to accept connections and data, Galaxy will be the MPEX interface to AmSC

physics.plasm-ph↗

arXiv2026

MPEX AI Digital Twins

Our vision for the MPEX AI Digital Twins project is to supply experimental and physics model simulation data to train Artificial Intelligence (AI) models for data processing, analysis, operational control, PMI and materials simulation to maximize the scientific output of the MPEX device. Ultimately, an AI digital twin of MPEX material assessment metrics for tested and synthetic material types with simulated PMI will be trained by the AI Modeling Teams on the experimental and physics simulation data submitted to the American Science Cloud by this project

physics.plasm-ph↗

arXiv2025

Data-Driven Filtering of the Spherical Harmonics Method

We investigate a data-driven approach for tuning the filtered spherical harmonics method (\fpn) when solving the radiation transport equation (RTE). The \fpn method extends the classical spherical harmonics approach (\pn) by introducing regularization through a filter operator, which mitigates spurious oscillations caused by Gibbs' phenomenon. This filter includes a tunable parameter, the {filter strength}, that controls the degree of smoothing applied to the solution. However, selecting an optimal filter strength is nontrivial, often requiring inaccessible information such as the true or a high-order reference solution. To overcome this limitation, we model the filter strength as a neural network whose inputs include local state variables and material cross-sections. The optimal filter strength is formulated as the solution to a PDE-constrained optimization problem, and the neural network is trained using a discretize-then-optimize formulation in PyTorch. We evaluate the learned filter strength across a suite of test problems and compare the results to those from a simple, but tunable constant filter strength. In all cases, the neural-network-driven filter substantially improves the accuracy of the \pn approximation. For 1-D test cases, the constant filter outperforms the neural-network filter in some cases, but in 2-D problems, the neural-network filter generally performs better.

math.NA↗

arXiv2025

Exploring the Capabilities of the Frontier Large Language Models for Nuclear Energy Research

The AI for Nuclear Energy workshop at Oak Ridge National Laboratory evaluated the potential of Large Language Models (LLMs) to accelerate fusion and fission research. Fourteen interdisciplinary teams explored diverse nuclear science challenges using ChatGPT, Gemini, Claude, and other AI models over a single day. Applications ranged from developing foundation models for fusion reactor control to automating Monte Carlo simulations, predicting material degradation, and designing experimental programs for advanced reactors. Teams employed structured workflows combining prompt engineering, deep research capabilities, and iterative refinement to generate hypotheses, prototype code, and research strategies. Key findings demonstrate that LLMs excel at early-stage exploration, literature synthesis, and workflow design, successfully identifying research gaps and generating plausible experimental frameworks. However, significant limitations emerged, including difficulties with novel materials designs, advanced code generation for modeling and simulation, and domain-specific details requiring expert validation. The successful outcomes resulted from expert-driven prompt engineering and treating AI as a complementary tool rather than a replacement for physics-based methods. The workshop validated AI's potential to accelerate nuclear energy research through rapid iteration and cross-disciplinary synthesis while highlighting the need for curated nuclear-specific datasets, workflow automation, and specialized model development. These results provide a roadmap for integrating AI tools into nuclear science workflows, potentially reducing development cycles for safer, more efficient nuclear energy systems while maintaining rigorous scientific standards.

physics.comp-ph↗

arXiv2025

Reference solutions for linear radiation transport: the Hohlraum and Lattice benchmarks

Two benchmark problems for linear radiation transport and derived from the literature are presented in detail and several quantities of interest are defined. High-resolution simulations are computed using standard, robust numerical methods and implemented using HPC resources. The goal of these simulations is to provide reference solutions for new discretization approaches, to aid in the development of multi-fidelity uncertainty quantification and optimization algorithms, to provide data for training surrogates models. The code used to perform the simulations and post-process the data is publicly available, as is the raw data that used to generate the results presented in the paper.

physics.comp-ph↗

arXiv2024

Learning to Control the Smoothness of Graph Convolutional Network Features

The pioneering work of Oono and Suzuki [ICLR, 2020] and Cai and Wang [arXiv:2006.13318] initializes the analysis of the smoothness of graph convolutional network (GCN) features. Their results reveal an intricate empirical correlation between node classification accuracy and the ratio of smooth to non-smooth feature components. However, the optimal ratio that favors node classification is unknown, and the non-smooth features of deep GCN with ReLU or leaky ReLU activation function diminish. In this paper, we propose a new strategy to let GCN learn node features with a desired smoothness -- adapting to data and tasks -- to enhance node classification. Our approach has three key steps: (1) We establish a geometric relationship between the input and output of ReLU or leaky ReLU. (2) Building on our geometric insights, we augment the message-passing process of graph convolutional layers (GCLs) with a learnable term to modulate the smoothness of node features with computational efficiency. (3) We investigate the achievable ratio between smooth and non-smooth feature components for GCNs with the augmented message-passing scheme. Our extensive numerical results show that the augmented message-passing schemes significantly improve node classification for GCN and some related models.

cs.LG↗

arXiv2024

On high-order/low-order and micro-macro methods for implicit time-stepping of the BGK model

In this paper, a high-order/low-order (HOLO) method is combined with a micro-macro (MM) decomposition to accelerate iterative solvers in fully implicit time-stepping of the BGK equation for gas dynamics. The MM formulation represents a kinetic distribution as the sum of a local Maxwellian and a perturbation. In highly collisional regimes, the perturbation away from initial and boundary layers is small and can be compressed to reduce the overall storage cost of the distribution. The convergence behavior of the MM methods, the usual HOLO method, and the standard source iteration method is analyzed on a linear BGK model. Both the HOLO and MM methods are implemented using a discontinuous Galerkin (DG) discretization in phase space, which naturally preserves the consistency between high- and low-order models required by the HOLO approach. The accuracy and performance of these methods are compared on the Sod shock tube problem and a sudden wall heating boundary layer problem. Overall, the results demonstrate the robustness of the MM and HOLO approaches and illustrate the compression benefits enabled by the MM formulation when the kinetic distribution is near equilibrium.

math.NA↗

arXiv2023

Numerical analysis of a spherical harmonic discontinuous Galerkin method for scaled radiative transfer equations with isotropic scattering

In highly diffusion regimes when the mean free path $\varepsilon$ tends to zero, the radiative transfer equation has an asymptotic behavior which is governed by a diffusion equation and the corresponding boundary condition. Generally, a numerical scheme for solving this problem has the truncation error containing an $\varepsilon^{-1}$ contribution, that leads to a nonuniform convergence for small $\varepsilon$. Such phenomenons require high resolutions of discretizations, which degrades the performance of the numerical scheme in the diffusion limit. In this paper, we first provide a--priori estimates for the scaled spherical harmonic ($P_N$) radiative transfer equation. Then we present an error analysis for the spherical harmonic discontinuous Galerkin (DG) method of the scaled radiative transfer equation showing that, under some mild assumptions, its solutions converge uniformly in $\varepsilon$ to the solution of the scaled radiative transfer equation. We further present an optimal convergence result for the DG method with the upwind flux on Cartesian grids. Error estimates of $\left(1+\mathcal{O}(\varepsilon)\right)h^{k+1}$ (where $h$ is the maximum element length) are obtained when tensor product polynomials of degree at most $k$ are used.

math.NA↗

arXiv2023

Modeling Information Flow with a Multi-Stage Queuing Mode

In this paper, we introduce a nonlinear stochastic model to describe the propagation of information inside a computer processor. In this model, a computational task is divided into stages, and information can flow from one stage to another. The model is formulated as a spatially-extended, continuous-time Markov chain where space represents different stages. This model is equivalent to a spatially-extended version of the M/M/s queue. The main modeling feature is the throttling function which describes the processor slowdown when the amount of information falls below a certain threshold. We derive the stationary distribution for this stochastic model and develop a closure for a deterministic ODE system that approximates the evolution of the mean and variance of the stochastic model. We demonstrate the validity of the closure with numerical simulations.

math.PR↗

arXiv2023

Asymptotic Preserving Discontinuous Galerkin Methods for a Linear Boltzmann Semiconductor Model

A key property of the linear Boltzmann semiconductor model is that as the collision frequency tends to infinity, the phase space density $f = f(x,v,t)$ converges to an isotropic function $M(v)ρ(x,t)$, called the drift-diffusion limit, where $M$ is a Maxwellian and the physical density $ρ$ satisfies a second-order parabolic PDE known as the drift-diffusion equation. Numerical approximations that mirror this property are said to be asymptotic preserving. In this paper we build two discontinuous Galerkin methods to the semiconductor model: one with the standard upwinding flux and the other with a $\varepsilon$-scaled Lax-Friedrichs flux, where 1/$\varepsilon$ is the scale of the collision frequency. We show that these schemes are uniformly stable in $\varepsilon$ and are asymptotic preserving. In particular, we discuss what properties the discrete Maxwellian must satisfy in order for the schemes to converge in $\varepsilon$ to an accurate $h$-approximation of the drift diffusion limit. Discrete versions of the drift-diffusion equation and error estimates in several norms with respect to $\varepsilon$ and the spacial resolution are also included.

math.NA↗

arXiv2022

Numerical schemes for a multi-species BGK model with velocity-dependent collision frequency

We consider a kinetic description of a multi-species gas mixture modeled with Bhatnagar-Gross-Krook (BGK) collision operators, in which the collision frequency varies not only in time and space but also with the microscopic velocity. In this model, the Maxwellians typically used in standard BGK operators are replaced by a generalization of such target functions, which are defined by a variational procedure \cite{arXiv:2101.09047}. In this paper we present a numerical method for simulating this model, which uses an Implicit-Explicit (IMEX) scheme to minimize a certain potential function, mimicking the Lagrange functional that appears in the theoretical derivation. We show that theoretical properties such as conservation of mass, total momentum and total energy as well as positivity of the distribution functions are preserved by the numerical method, and illustrate its usefulness and effectiveness with numerical examples.

math.NA↗

arXiv2022

A Predictor-Corrector Strategy for Adaptivity in Dynamical Low-Rank Approximations

In this paper, we present a predictor-corrector strategy for constructing rank-adaptive dynamical low-rank approximations (DLRAs) of matrix-valued ODE systems. The strategy is a compromise between (i) low-rank step-truncation approaches that alternately evolve and compress solutions and (ii) strict DLRA approaches that augment the low-rank manifold using subspaces generated locally in time by the DLRA integrator. The strategy is based on an analysis of the error between a forward temporal update into the ambient full-rank space, which is typically computed in a step-truncation approach before re-compressing, and the standard DLRA update, which is forced to live in a low-rank manifold. We use this error, without requiring its full-rank representation, to correct the DLRA solution. A key ingredient for maintaining a low-rank representation of the error is a randomized singular value decomposition (SVD), which introduces some degree of stochastic variability into the implementation. The strategy is formulated and implemented in the context of discontinuous Galerkin spatial discretizations of partial differential equations and applied to several versions of DLRA methods found in the literature, as well as a new variant. Numerical experiments comparing the predictor-corrector strategy to other methods demonstrate robustness to overcome short-comings of step truncation or strict DLRA approaches: the former may require more memory than is strictly needed while the latter may miss transients solution features that cannot be recovered. The effect of randomization, tolerances, and other implementation parameters is also explored.

math.NA↗

arXiv2021

A consistent BGK model with velocity-dependent collision frequency for gas mixtures

We derive a multi-species BGK model with velocity-dependent collision frequency for a non-reactive, multi-component gas mixture. The model is derived by minimizing a weighted entropy under the constraint that the number of particles of each species, total momentum, and total energy are conserved. We prove that this minimization problem admits a unique solution for very general collision frequencies. Moreover, we prove that the model satisfies an H-Theorem and characterize the form of equilibrium.

math.AP↗

arXiv2019

A mathematical model of asynchronous data flow in parallel computers

We present a simplified model of data flow on processors in a high performance computing framework involving computations necessitating inter-processor communications. From this ordinary differential model, we take its asymptotic limit, resulting in a model which treats the computer as a continuum of processors and data flow as an Eulerian fluid governed by a conservation law. We derive a Hamilton-Jacobi equation associated with this conservation law for which the existence and uniqueness of solutions can be proven. We then present the results of numerical experiments for both discrete and continuum models; these show a qualitative agreement between the two and the effect of variations in the computing environment's processing capabilities on the progress of the modeled computation.

cs.DC↗

arXiv2018

Realizability-Preserving DG-IMEX Method for the Two-Moment Model of Fermion Transport

Building on the framework of Zhang \& Shu \cite{zhangShu_2010a,zhangShu_2010b}, we develop a realizability-preserving method to simulate the transport of particles (fermions) through a background material using a two-moment model that evolves the angular moments of a phase space distribution function $f$. The two-moment model is closed using algebraic moment closures; e.g., as proposed by Cernohorsky \& Bludman \cite{cernohorskyBludman_1994} and Banach \& Larecki \cite{banachLarecki_2017a}. Variations of this model have recently been used to simulate neutrino transport in nuclear astrophysics applications, including core-collapse supernovae and compact binary mergers. We employ the discontinuous Galerkin (DG) method for spatial discretization (in part to capture the asymptotic diffusion limit of the model) combined with implicit-explicit (IMEX) time integration to stably bypass short timescales induced by frequent interactions between particles and the background. Appropriate care is taken to ensure the method preserves strict algebraic bounds on the evolved moments (particle density and flux) as dictated by Pauli's exclusion principle, which demands a bounded distribution function (i.e., $f\in[0,1]$). This realizability-preserving scheme combines a suitable CFL condition, a realizability-enforcing limiter, a closure procedure based on Fermi-Dirac statistics, and an IMEX scheme whose stages can be written as a convex combination of forward Euler steps combined with a backward Euler step. Numerical results demonstrate the realizability-preserving properties of the scheme. We also demonstrate that the use of algebraic moment closures not based on Fermi-Dirac statistics can lead to unphysical moments in the context of fermion transport.

physics.comp-ph↗

arXiv2017

Positivity preserving DG schemes for a Boltzmann - Poisson model of electrons in semiconductors in curvilinear momentum coordinates

The work presented in this paper is related to the development of positivity preserving Discontinuous Galerkin (DG) methods for Boltzmann - Poisson (BP) computational models of electronic transport in semiconductors. We pose the Boltzmann Equation for electron transport in curvilinear coordinates for the momentum. We consider the 1D diode problem with azimuthal symmetry, which is a 3D plus time problem. We choose for this problem the spherical coordinate system $\vec{p}(|\vec{p}|,\mu=cos\theta,\varphi)$, slightly different to the choice in previous DG solvers for BP, because its DG formulation gives simpler integrals involving just piecewise polynomial functions for both transport and collision terms. Applying the strategy of Zhang \& Shu, \cite{ZhangShu1}, \cite{ZhangShu2}, Cheng, Gamba, Proft, \cite{CGP}, and Endeve et al. \cite{EECHXM-JCP}, we treat the collision operator as a source term, and find convex combinations of the transport and collision terms which guarantee the positivity of the cell average of our numerical probability density function at the next time step. The positivity of the numerical solution to the pdf in the whole domain is guaranteed by applying the limiters in \cite{ZhangShu1}, \cite{ZhangShu2} that preserve the cell average but modify the slope of the piecewise linear solutions in order to make the function non - negative. In addition of the proofs of positivity preservation in the DG scheme, we prove the stability of the semi-discrete DG scheme under an entropy norm, using the dissipative properties of our collisional operator given by its entropy inequalities. The entropy inequality we use depends on an exponential of the Hamiltonian rather than the Maxwellian associated just to the kinetic energy.

math.NA↗

arXiv2016

A new class of high order semi-Lagrangian schemes for rarefied gas dynamics

In this paper we genealize the fast semi-Lagrangian scheme developed in [J. Comput. Phys., Vol. 255, 2013, pp 680-698] to the case of high order reconstructions of the distribution function. The original first order accurate semi-Lagrangian scheme is supplemented with polynomial reconstructions of the distribution function and of the collisional operator leading to an effective high order accurate numerical scheme for all regimes, from extremely rarefied gas to highly collisional siuation. The main idea relies on updating at each time step the extreme points of the distribution function for each velocity of the lattice instead of updating the solution in the cell centers, these extremes points being located at different positions for any fixed velocity of the lattice. The result is a class of scheme which permits to preserve the structure of the solution over very long times compared to existing schemes from the literature. We propose a proof of concept of this new approach along with numerical tests and comparisons with classical numerical methods.

math.NA↗