arXiv subjects

Ryan G. McClarren

Publications and source records attributed to Ryan G. McClarren.

At least 19 recordsLinked to original sources

arXiv2026

Weak-DMD: A Galerkin approach to the problem of noise in the Dynamic Mode Decomposition algorithm

Dynamic Mode Decomposition (DMD) is a data-driven method for approximating the spatiotemporal modes of a system. The eigenvectors and eigenvalues of the system are approximated from a series of time-snapshots of the state variables. The standard formulation of DMD is subject to strict assumptions concerning the time-spacing of the snapshots and is biased by measurement noise. Variations on the method have been developed to address these shortcomings, but the problem is still open. Motivated by the effectiveness of Galerkin methods in the field of model discovery, a weak formulation of DMD is presented, weak-DMD. Weak-DMD precludes timestep considerations and also filters noise. Results for two nuclear engineering applications and the flow of fluid past a cylinder are given and compared with a state of the art DMD algorithm.

cs.CE↗

arXiv2025

Generative Monte Carlo Sampling for Constant-Cost Particle Transport

We present Generative Monte Carlo (GMC), a novel paradigm for particle transport simulation that integrates generative artificial intelligence directly into the stochastic solution of the linear Boltzmann equation. By reformulating the cell-transmission problem as a conditional generation task, we train neural networks using conditional flow matching to sample particle exit states, including position, direction, and path length, without simulating scattering histories. The method employs optical coordinate scaling, enabling a single trained model to generalize across any material. We validate GMC on two canonical benchmarks, namely a heterogeneous lattice problem characteristic of nuclear reactor cores and a linearized hohlraum geometry representative of high-energy density radiative transfer. Results demonstrate that GMC preserves the statistical fidelity of standard Monte Carlo, exhibiting the expected $1/\sqrt{N}$ convergence rate while maintaining accurate scalar flux profiles. While standard Monte Carlo computational cost scales linearly with optical thickness in the diffusive limit, GMC achieves constant $O(1)$ cost per cell transmission, yielding order-of-magnitude speedups in optically thick regimes. This framework strategically aligns particle transport with modern computing architectures optimized for neural network inference, positioning transport codes to leverage ongoing advances in AI hardware and algorithms.

physics.comp-ph↗

arXiv2025

Scatter-Limited Hybrid Monte Carlo, Deterministic Transport with Quasi-Monte Carlo Sampling

We present a hybrid method for time-dependent particle transport that combines Monte Carlo (MC) estimation with a deterministic discrete ordinates ($S_N$) solve, augmented by quasi-Monte Carlo (QMC) sampling. For spatial discretizations, the MC component computes a piecewise-constant (cell-averaged) solution, while the $S_N$ stage employs bilinear discontinuous finite elements. By hybridizing the formulation, the MC subproblem after a prescribed scatter limit becomes scattering-free, yielding a simple and efficient streaming/attenuation procedure. Between time steps, a simple scatter-free MC step is run to relabel the $S_N$ solution as an MC solution. A key feature of the approach is a tunable parameter $N_{s}$ that controls how many material collisions are handled in the (Q)MC leg before handing off to the deterministic $S_N$ solve; $N_s=0$ recovers a purely uncollided MC leg, while $N_s>0$ produces multi-scatter hybrids. QMC replaces pseudorandom draws with low-discrepancy points in the existing MC sampling maps, enabling a plug-in adoption within the standard MC code with modest, localized changes. We observe significant accuracy and convergence rate improvements through the use of QMC and practically no additional computational cost, which are generally not seen in comparable non-hybrid solves. We believe the multi-scatter approach provides additional flexibility in terms of parallelization and the choice of deterministic solver.

math.NA↗

arXiv2025

Curvilinear coordinates and curvature in radiative transport

We derive a general expression for the streaming term in radiative transport equa- tions and other transport problems when formulated in curvilinear coordinates, emphasizing coordinate systems adapted to the geometry of the domain and the directional dependence of particle transport. By parametrizing the angular vari- able using a local orthonormal frame, we express directional derivatives in terms of curvature-related quantities that reflect the geometry of underlying spatial man- ifolds. Our formulation highlights how the interaction between coordinate choices and curvature influences the streaming operator, offering geometric interpretations of its components. The resulting framework offers intuitive insight into when and how angular dependence can be simplified and may guide the selection of coordinate systems that balance analytical tractability and computational efficiency.

math.NA↗

arXiv2025

A review of low-rank methods for time-dependent kinetic simulations

Time-dependent kinetic models are ubiquitous in computational science and engineering. The underlying integro-differential equations in these models are high-dimensional, comprised of a six--dimensional phase space, making simulations of such phenomena extremely expensive. In this article we demonstrate that in many situations, the solution to kinetics problems lives on a low dimensional manifold that can be described by a low-rank matrix or tensor approximation. We then review the recent development of so-called low-rank methods that evolve the solution on this manifold. The two classes of methods we review are the dynamical low-rank (DLR) method, which derives differential equations for the low-rank factors, and a Step-and-Truncate (SAT) approach, which projects the solution onto the low-rank representation after each time step. Thorough discussions of time integrators, tensor decompositions, and method properties such as structure preservation and computational efficiency are included. We further show examples of low-rank methods as applied to particle transport and plasma dynamics.

math.NA↗

arXiv2025

Using Deep Operators to Create Spatio-temporal Surrogates for Dynamical Systems under Uncertainty

Spatio-temporal data, which consists of responses or measurements gathered at different times and positions, is ubiquitous across diverse applications of civil infrastructure. While SciML methods have made significant progress in tackling the issue of response prediction for individual time histories, creating a full spatial-temporal surrogate remains a challenge. This study proposes a novel variant of deep operator networks (DeepONets), namely the full-field Extended DeepONet (FExD), to serve as a spatial-temporal surrogate that provides multi-output response predictions for dynamical systems. The proposed FExD surrogate model effectively learns the full solution operator across multiple degrees of freedom by enhancing the expressiveness of the branch network and expanding the predictive capabilities of the trunk network. The proposed FExD surrogate is deployed to simultaneously capture the dynamics at several sensing locations along a testbed model of a cable-stayed bridge subjected to stochastic ground motions. The ensuing response predictions from the FExD are comprehensively compared against both a vanilla DeepONet and a modified spatio-temporal Extended DeepONet. The results demonstrate the proposed FExD can achieve both superior accuracy and computational efficiency, representing a significant advancement in operator learning for structural dynamics applications.

stat.ML↗

arXiv2025

Fast, Accurate Numerical Evaluation of Incomplete Planck Integrals

Methods for computing the integral of the Planck blackbody function over a finite spectral range, the so-called incomplete Planck integral, are necessary to perform multigroup radiative transfer calculations. We present a comparison, in terms of speed and accuracy, of a wide array of approaches to numerically evaluating these integrals. Our results indicate that a direct rational polynomial approximation to these integrals has the best combination of accuracy and efficiency. We also present for the first time a derivation of the polylogarithm form of these integrals and show that modern approaches to polylogarithm evaluation are suitable for numerically evaluating incomplete Planck integrals. This article is dedicated to Prof. B.D. Ganapol, the Transport Cowboy, on the occasion of his retirement.

physics.comp-ph↗

arXiv2025

Collision-Based Hybrid Method for Two-Dimensional Neutron Transport Problems

A collision-based hybrid method for the discrete ordinates approximation of the multigroup neutron transport equation is developed for two-dimensional time-dependent problems. At each time step, this algorithm splits the neutron transport equation into two equations, where the external source is part of the uncollided equation and the fission and scattering sources are part of the collided equation. Low fidelity energy and angular grids are used with the collided transport solution to decrease convergence time while high fidelity grids are used with the uncollided transport solution to limit discretization error. The hybrid method is shown to be a better solution in terms of both convergence time and accuracy to traditional monolithic coarsening schemes. This advantage is demonstrated for two-dimensional time-dependent problems with different materials using a second order temporal discretization scheme.

physics.comp-ph↗

arXiv2025

Consistent Solutions of the Radiation Diffusion Equation in Spherical and Cylindrical Geometries

We have extended the radiation diffusion model of Hammer and Rosen to diverging spherical and cylindrical geometries. The effect of curvilinear geometry on the supersonic, expanding wavefront increases as the internal radius of a spherical or cylindrical shell approaches zero. Small spherical geometries are important for modeling systems at the size scale of ICF capsules, at these scales existing quasi-analytic models for planar geometry significantly disagree with the results of simulation. With this method, the benefits of rapid iteration can be applied to common spherical systems at much smaller length scales. We present comparisons between numerical diffusion solutions and the analytic model to give ranges of applicability for the model.

physics.comp-ph↗

arXiv2025

Implicit Collision Multiplicity Adjustment for Efficient Monte Carlo Transport Simulation of Reactivity Excursion

We present an implicit collision method with on-the-fly multiplicity adjustment based on the forward weight window methodology for efficient Dynamic Monte Carlo (MC) simulation of reactivity excursion transport problems. Test problems based on the Dragon experiment of 1945 by Otto Frisch are devised to verify and assess the efficiency of the method. The test problems exhibit the key features of the Dragon experiment, namely nine orders of magnitude neutron flux bursts followed by significant post-burst delayed neutron effects. Such an extreme reactivity excursion is particularly challenging and has never been solved with Dynamic MC. The proposed implicit collision multiplicity adjustment, in conjunction with a simple forced delayed neutron precursor decay technique, profitably trades simulation precision for reduced runtime, leading to an improved figure of merit, enabling efficient Dynamic MC simulation of extreme reactivity excursions.

physics.comp-ph↗

arXiv2025

Derivative Source Method for Monte Carlo Transport Calculation of Sensitivities to Material Densities and Dimensions

The Derivative Source Method (DSM) takes derivatives of a particle transport equation with respect to selected parameters and solves them via the standard Monte Carlo random walk simulation along with the original transport problem. The Monte Carlo solutions of the derivative equations make the sensitivities of quantities of interest to the selected parameters. In this paper, we show that DSM can be embedded in Monte Carlo simulation to simultaneously calculate sensitivities of transport phase-space solution to multiple object dimensions and material densities. We verify and assess the efficiency of DSM by solving a multigroup neutronic system of a source-driven fuel-moderator-absorber slab lattice and calculating the fast and slow flux sensitivity coefficient distributions to the fuel and absorber dimensions, as well as the fuel, moderator, and absorber densities. The results are compared to those obtained from conventional finite difference sensitivity calculations. A figure of merit, defined as the product inverse of runtime and square of relative error, is used to assess method efficiencies. For a couple of the calculated sensitivities, well-configured finite difference calculations are the most efficient, followed closely by DSM. However, since seeking such well-configured finite difference is not always practical, in the rest of the cases, including the all-parameter simultaneous sensitivity calculation, DSM has the highest efficiency, demonstrating its robustness.

physics.comp-ph↗

arXiv2025

A Batch Power Iteration Approach for the Iterative Quasi-Monte Carlo Method Using a Randomized-Halton Sequence

The Iterative Quasi-Monte Carlo (iQMC) method is a recently developed hybrid method for neutron transport simulations. iQMC replaces standard quadrature techniques used in deterministic linear solvers with Quasi-Monte Carlo simulation for accurate and efficient solutions to the neutron transport equation. Previous iQMC studies utilized a fixed-seed approach wherein particles were reset to the same initial position and direction of travel at the start of every transport sweep. While the QMC samples offered greatly improved uniformity compared to pseudo-random samples, the fixed-seed approach meant that some regions of the problem were under-sampled and resulted in errors similar to ray effects observed in discrete ordinates methods. This work explores using randomized-Quasi Monte Carlo techniques (RQMC) to generate unique sets of QMC samples for each transport sweep and gain a much-improved sampling of the phase space. The use of RQMC introduces some stochastic noise to iQMC's iterative process, which was previously absent. To compensate, we adopt a ``batch'' approach similar to typical Monte Carlo k-eigenvalue problems, where the iQMC source is converged over $N_\text{inactive}$ batches, then results from $N_\text{active}$ batches are recorded and used to calculate the average and standard deviation of the solution. The RQMC batch method was implemented in the Monte Carlo Dynamic Code (MC/DC) and is shown to be a large improvement over the fixed-seed method. The batch method was able to provide iteratively stable and more accurate solutions with nearly two orders of magnitude reduction in the number of particle histories per batch. Notably, despite introducing some stochastic noise to the solution, the RQMC batch approach converges both the k-effective and mean scalar flux error at the theoretical QMC convergence rate of $O(N^{-1})$.

physics.comp-ph↗

arXiv2024

Numerical Robustness of PINNs for Multiscale Transport Equations

We investigate the numerical solution of multiscale transport equations using Physics Informed Neural Networks (PINNs) with ReLU activation functions. Therefore, we study the analogy between PINNs and Least-Squares Finite Elements (LSFE) which lies in the shared approach to reformulate the PDE solution as a minimization of a quadratic functional. We prove that in the diffusive regime, the correct limit is not reached, in agreement with known results for first-order LSFE. A diffusive scaling is introduced that can be applied to overcome this, again in full agreement with theoretical results for LSFE. We provide numerical results in the case of slab geometry that support our theoretical findings.

math.NA↗

arXiv2024

The non-equilibrium Marshak wave problem in non-homogeneous media

We derive a family of similarity solutions to the nonlinear non-equilibrium Marshak wave problem for an inhomogeneous planar medium which is coupled to a time dependent radiation driving source. We employ the non-equilibrium gray diffusion approximation in the supersonic regime. The solutions constitute a generalization of the non-equilibrium nonlinear solutions that were developed recently for homogeneous media. Self-similar solutions are constructed for a power law time dependent surface temperature, a spatial power law density profile and a material model with power law temperature and density dependent opacities and specific energy density. The extension of the problem to non-homogeneous media enables the existence of similarity solutions for a general power law specific material energy. It is shown that the solutions exist for specific values of the temporal temperature drive and spatial density exponents, which depend on the material exponents. We also illustrate how the similarity solutions take various qualitatively different forms which are analyzed with respect to various parameters. Based on the solutions, we define a set of non-trivial benchmarks for supersonic non-equilibrium radiative heat transfer. The similarity solutions are compared to gray diffusion simulations as well as to detailed implicit Monte-Carlo and discrete-ordinate transport simulations in the optically-thick regime, showing a great agreement, which highlights the benefit of these solutions as a code verification test problem.

astro-ph.HE↗

arXiv2024

A unified theory of the self-similar supersonic Marshak wave problem

We present a systematic study of the similarity solutions for the Marshak wave problem, in the local thermodynamic equilibrium (LTE) diffusion approximation and in the supersonic regime. Self-similar solutions exist for a temporal power law surface temperature drive and a material model with power law temperature dependent opacity and energy density. The properties of the solutions in both linear and nonlinear conduction regimes are studied as a function of the temporal drive, opacity and energy density exponents. We show that there exists a range of the temporal exponent for which the total energy in the system decreases, and the solution has a local maxima. For nonlinear conduction, we specify the conditions on the opacity and energy density exponents under which the heat front is linear or even flat, and does posses its common sharp character; this character is independent of the drive exponent. We specify the values of the temporal exponents for which analytical solutions exist and employ the Hammer-Rosen perturbation theory to obtain highly accurate approximate solutions, which are parameterized using only two numerically fitted quantities. The solutions are used to construct a set of benchmarks for supersonic LTE radiative heat transfer, including some with unusual and interesting properties such as local maxima and non sharp fronts. The solutions are compared in detail to implicit Monte-Carlo and discrete-ordinate transport simulations as well gray diffusion simulations, showing a good agreement, which highlights their usefulness as a verification test problem for radiative transfer simulations.

astro-ph.HE↗

arXiv2024

Self-similar solutions for the non-equilibrium nonlinear supersonic Marshak wave problem

Similarity solutions to the nonlinear non-equilibrium Marshak wave problem with a time dependent radiation driving source are presented. The radiation transfer model used is the gray, non-equilibrium diffusion approximation in the supersonic regime. These solutions constitute an extension of existing non-equilibrium supersonic Marshak wave solutions which are linear, to the nonlinear regime, which prevails in realistic high energy density systems. The generalized solutions assume a material model with power law temperature dependent opacities and a material energy density which is proportional to the radiation energy density, as well as a surface radiation temperature drive which obeys a temporal power-law. The solutions are analyzed in detail and it is shown that they take various qualitatively different forms according to the values of the opacity exponents. The solutions are used to construct a set of standardized benchmarks for supersonic non-equilibrium radiative heat transfer, which are nontrivial but straightforward to implement. These solutions are compared in detail to implicit Monte-Carlo and discrete-ordinate transport simulations as well gray diffusion simulations, showing a good agreement, which demonstrates the usefulness of these solutions as a code verification test problem.

astro-ph.HE↗

arXiv2024

Mitigating Spatial Error in the iterative-Quasi-Monte Carlo (iQMC) Method for Neutron Transport Simulations with Linear Discontinuous Source Tilting and Effective Scattering and Fission Rate Tallies

The iterative Quasi-Monte Carlo (iQMC) method is a recently proposed method for multigroup neutron transport simulations. iQMC can be viewed as a hybrid between deterministic iterative techniques, Monte Carlo simulation, and Quasi-Monte Carlo techniques. iQMC holds several algorithmic characteristics that make it desirable for high performance computing environments including a $O(N^{-1})$ convergence scheme, ray tracing transport sweep, and highly parallelizable nature similar to analog Monte Carlo. While there are many potential advantages of using iQMC there are also inherent disadvantages, namely the spatial discretization error introduced from the use of a mesh across the domain. This work introduces two significant modifications to iQMC to help reduce the spatial discretization error. The first is an effective source transport sweep, whereby the source strength is updated on-the-fly via an additional tally. This version of the transport sweep is essentially agnostic to the mesh, material, and geometry. The second is the addition of a history-based linear discontinuous source tilting method. Traditionally, iQMC utilizes a piecewise-constant source in each cell of the mesh. However, through the proposed source tilting technique iQMC can utilize a piecewise-linear source in each cell and reduce spatial error without refining the mesh. Numerical results are presented from the 2D C5G7 and Takeda-1 k-eigenvalue benchmark problems. Results show that the history-based source tilting significantly reduces error in global tallies and the eigenvalue solution in both benchmarks. Through the effective source transport sweep and linear source tilting iQMC was able to converge the eigenvalue from the 2D C5G7 problem to less than $0.04\%$ error on a uniform Cartesian mesh with only $204\times204$ cells.

physics.comp-ph↗

arXiv2023

Uncertainty benchmarks for time-dependent transport problems

Verification solutions for uncertainty quantification are presented for time dependent transport problems where $c$, the scattering ratio, is uncertain. The method of polynomial chaos expansions is employed for quick and accurate calculation of the quantities of interest and uncollided solutions are used to treat part of the uncertainty calculation analytically. We find that approximately six moments in the polynomial expansion are required to represent the solutions to these problems accurately. Additionally, the results show that if the uncertainty interval spans c=1, which means it is uncertain whether the system is multiplying or not, the confidence interval will grow in time. Finally, since the QoI is a strictly increasing function, the percentile values are known and can be used to verify the accuracy of the expansion. These results can be used to test UQ methods for time-dependent transport problems.

cs.CE↗