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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Quantum Hamilton-Jacobi theory, spectral path integrals, and exact WKB analysis

We propose a new way to perform path integrals in quantum mechanics by using a quantum version of Hamilton-Jacobi (HJ) theory. In classical mechanics, Hamilton-Jacobi theory is a powerful formalism, however, its utility is not explored in quantum theory beyond approximation schemes. The canonical transformation enables one to set the new Hamiltonian to constant or zero, but keeps the information about solution in Hamilton’s characteristic function. To benefit from this in quantum theory, one must work with a formulation in which classical Hamiltonian is used. This uniquely points to phase space path integral. However, the main variable in HJ formalism is energy, not time. Thus, we are led to consider the Fourier transform of the path integral, the spectral path integral Z ˜ ( E ) . The evaluation of path integrals reduces to determining the quantum Hamilton characteristic functions (which can be achieved via an asymptotic analysis) and a discrete sum over the quantum period lattice, generalizing Gutzwiller’s sum. Published by the American Physical Society 2025

Türe, Mustafa (ORCID:0009000975968618)↗
DOE OSTI2024

Modifying the Asynchronous Jacobi Method for Data Corruption Resilience

Moving scientific computation from high-performance computing (HPC) and cloud computing (CC) environments to devices on the edge, i.e., physically near instruments of interest, has received tremendous interest in recent years. Such edge computing environments can operate on data in situ, offering enticing benefits over data aggregation to HPC and CC facilities that include avoiding costs of transmission, increased data privacy, and real-time data analysis. Because of the inherent unreliability of edge computing environments, new fault-tolerant approaches must be developed before the benefits of edge computing can be realized. Motivated by algorithm-based fault tolerance, a variant of the asynchronous Jacobi (ASJ) method is developed that achieves resilience to data corruption by rejecting solution approximations from neighbor devices according to a bound derived from convergence theory. Numerical results on a two-dimensional Poisson problem show that the new rejection criterion, along with a novel approximation to the shortest path length on which the criterion depends, restores convergence for the ASJ variant in the presence of certain types data corruption. Numerical results are obtained for when the singular values in the analytic bound are approximated. Additional linear systems are also explored, one with a more dense sparsity pattern and one that includes advection. All results indicate that successful resilience to data corruption depends on whether the bound tightens fast enough to reject corrupted data before the iteration evolution deviates significantly from that predicted by the convergence theory defining the bound. This observation generalizes to future work on algorithm-based fault tolerance for other asynchronous algorithms, including upcoming approaches that leverage Krylov subspaces.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

A scalable multidimensional fully implicit solver for Hall magnetohydrodynamics

We propose an optimally performant fully implicit algorithm for the Hall magnetohydrodynamics (HMHD) equations based on multigrid-preconditioned Jacobian-free Newton-Krylov methods. HMHD is a challenging system to solve numerically because it supports stiff fast dispersive waves. The preconditioner is formulated using an operator-split approximate block factorization (Schur complement), informed by physics insight. We use a vector-potential formulation (instead of a magnetic field one) to allow a clean segregation of the problematic $\nabla$ x $\nabla$ x operator in the electron Ohm's law subsystem. This segregation allows the formulation of an effective damped block-Jacobi smoother for multigrid. We demonstrate by analysis that our proposed block-Jacobi iteration is convergent and has the smoothing property. The resulting HMHD solver is verified linearly with wave propagation examples, and nonlinearly with the GEM challenge reconnection problem by comparison against another HMHD code. We demonstrate the excellent algorithmic and parallel performance of the algorithm up to 16384 MPI tasks in two dimensions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Large language model evaluation for high–performance computing software development

We apply AI-assisted large language model (LLM) capabilities of GPT-3 targeting high-performance computing (HPC) kernels for (i) code generation, and (ii) auto-parallelization of serial code in C ++, Fortran, Python and Julia. Our scope includes the following fundamental numerical kernels: AXPY, GEMV, GEMM, SpMV, Jacobi Stencil, and CG, and language/programming models: (1) C++ (e.g., OpenMP [including offload], OpenACC, Kokkos, SyCL, CUDA, and HIP), (2) Fortran (e.g., OpenMP [including offload] and OpenACC), (3) Python (e.g., numpy, Numba, cuPy, and pyCUDA), and (4) Julia (e.g., Threads, CUDA.jl, AMDGPU.jl, and KernelAbstractions.jl). Kernel implementations are generated using GitHub Copilot capabilities powered by the GPT-based OpenAI Codex available in Visual Studio Code given simple + + prompt variants. To quantify and compare the generated results, we propose a proficiency metric around the initial 10 suggestions given for each prompt. For auto-parallelization, we use ChatGPT interactively giving simple prompts as in a dialogue with another human including simple “prompt engineering” follow ups. Results suggest that correct outputs for C++ correlate with the adoption and maturity of programming models. For example, OpenMP and CUDA score really high, whereas HIP is still lacking. We found that prompts from either a targeted language such as Fortran or the more general-purpose Python can benefit from adding language keywords, while Julia prompts perform acceptably well for its Threads and CUDA.jl programming models. Finally, we expect to provide an initial quantifiable point of reference for code generation in each programming model using a state-of-the-art LLM. Overall, understanding the convergence of LLMs, AI, and HPC is crucial due to its rapidly evolving nature and how it is redefining human-computer interactions.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Optimal Polynomial Smoothers and One‐Sided V‐Cycles for Poisson Problems

The solution to the Poisson equation arising from the spectral element discretization of the incompressible Navier‐Stokes equations needs robust preconditioning strategies. One such strategy is multigrid. To realize the potential of multigrid methods, effective smoothing strategies are needed. Chebyshev polynomial smoothers, in conjunction with pointwise Jacobi or additive Schwarz methods (ASMs), prove to be an effective smoother. Other polynomial smoothers, however, may provide superior convergence to the multigrid preconditioner. The authors compare the standard Chebyshev polynomial smoothers to both the novel fourth‐kind Chebyshev polynomial smoothers proposed by Lottes as well as smoothers based on the polynomial of best uniform approximation to as proposed by Kraus, Vassilevski, and Zikatanov. At the cost of symmetry, further improvements may be made. For example, a order polynomial smoother on both sides of the V‐cycle may be substituted with an order polynomial smoother on one side at no additional cost. The choice of omitting the postsmoother in favor of higher‐order polynomial presmoothing is advantageous in cases where the multigrid approximation property constant is large. The authors consider a 2D model problem based on finite differences to motivate the choice of polynomial smoother, order, and whether to apply postsmoothing for the target application of high‐order ‐geometric multigrid methods for GPU architectures. Results from both domains demonstrate the substantial improvement of these approaches over the standard Chebyshev polynomial smoother with a symmetric V‐cycle.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Exact Ground State of Interacting Electrons in Magic Angle Graphene

One of the most remarkable theoretical findings in magic angle twisted bilayer graphene (TBG) is the emergence of ferromagnetic Slater determinants as exact ground states for the interacting Hamiltonian at the chiral limit. This discovery provides an explanation for the correlated insulating phase which has been experimentally observed at half filling. This work is the first mathematical study of interacting models in magic angle graphene systems. These include not only TBG but also TBG-like systems featuring four flat bands per valley, and twisted trilayer graphene systems with equal twist angles. We identify symmetries of the chiral limit of the Bistritzer-MacDonald Hamiltonian that are responsible for characterizing the Hartree-Fock ground states as zero energy many-body ground states. Furthermore, for a general class of Hamiltonian, we establish criteria that the ferromagnetic Slater determinants are the unique ground states within the class of uniformly half-filled, translation invariant Slater determinants. We then demonstrate that these criteria can be explicitly verified for TBG and TBG-like systems at the chiral limit, using properties of Jacobi-θ$${\theta }$$ and Weierstrass-℘$${\wp }$$ functions.

Becker, Simon↗
DOE OSTI2025

ZERNIPAX: A fast and accurate Zernike polynomial calculator in Python

Zernike polynomials serve as an orthogonal basis on the unit disc, and have proven to be effective in optics simulations, astrophysics, and more recently in plasma simulations. Unlike Bessel functions, Zernike polynomials are inherently finite and smooth at the disc center (r=0), ensuring continuous differentiability along the axis. This property makes them particularly suitable for simulations, requiring no additional handling at the origin. We developed ZERNIPAX, an open-source Python package capable of utilizing CPU/GPUs, leveraging Google's JAX package and available on GitHub as well as the Python software repository PyPI. Furthermore, our implementation of the recursion relation between Jacobi polynomials significantly improves computation time compared to alternative methods by use of parallel computing while still performing more accurately for high-mode numbers.

Astrophysics↗
DOE OSTI2025

Fast meta-solvers for 3D complex-shape scatterers using neural operators trained on a non-scattering problem

Three-dimensional target identification using scattering techniques requires high accuracy solutions and very fast computations for real-time predictions in some critical applications. We first train a deep neural operator (DeepONet) to solve wave propagation problems described by the Helmholtz equation in a domain without scatterers but at different wavenumbers and with a complex absorbing boundary condition. We then design two classes of fast meta-solvers by combining DeepONet with either relaxation methods, such as Jacobi and Gauss-Seidel, or with Krylov methods, such as GMRES and BiCGStab, using the trunk basis of DeepONet as a coarse-scale preconditioner. We leverage the spectral bias of neural networks to account for the lower part of the spectrum in the error distribution while the upper part is handled inexpensively using relaxation methods or fine-scale preconditioners. The meta-solvers are then applied to solve scattering problems with different shape of scatterers, at no extra training cost. We first demonstrate that the resulting meta-solvers are shape-agnostic, fast, and robust, whereas the standard standalone solvers may even fail to converge without the DeepONet. We then apply both classes of meta-solvers to scattering from a submarine, a complex three-dimensional problem. We achieve very fast solutions, especially with the DeepONet-Krylov methods, which require orders of magnitude fewer iterations than any of the standalone solvers.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Fast solvers for tokamak fluid models with PETSc

Multigrid (MG) is widely recognized as a highly effective solver for the model problem, the Laplacian, but textbook MG fails on most problems of interest. MG methods have been applied to complex, real-world applications with careful consideration of the physical model and discretization. In this work we develop the first step in applying MG methods to science and engineering relevant magnetohydrodynamics (MHD) tokamak models in the M3D-C1 (https://m3dc1.pppl.gov) fusion energy science code. The semi-implicit time integrator in M3D-C1 is composed of many linear solves. The implicit advance of the momentum equation is the most challenging and is the focus of this work. The current production solver in M3D-C1 is a block Jacobi (BJ) preconditioner within a Krylov solver, where blocks group degrees of freedom on planes of constant toroidal coordinate. BJ convergence degrades as the number of planes increases due to the spectral properties of the matrix preconditioned with BJ. The partially magnetic field-aligned, regular toroidal grid structure in M3D-C1 is amenable to semi-coarsening geometric MG in the toroidal direction. This paper develops such a solver and demonstrates competitive performance on a runaway electron model of a SPARC (https://cfs.energy/technology/sparc) disruption, and superior robustness on a stellarator model on which the BJ solver fails to converge.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

An FFT-based micromechanical model for gradient enhanced brittle fracture

Damage models incorporated within FFT-based micromechanical methods have received much attention recently because of the need to better understand and predict brittle and ductile fracture. An important aspect of a damage model is non-local regularization, which removes the mesh dependence of the predictions that otherwise become physically unacceptable upon grid refinement. In this work, the Helmholtz-type equation for non-local gradient regularization of a damage model on a distorted grid is solved using an FFT-based approach. Further, the resulting system of equations is solved using the Jacobi iterative method. The model is applied to simulate brittle fracture of an intermetallic. The influence of the time and space discretization, the length-scale parameter, and intermetallic crystallographic orientation on crack evolution is studied.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Rotational excitation and de-excitation of magnesium mono-sulphide (MgS) by He collisions

ABSTRACT Magnesium mono-sulphide (MgS) plays a crucial role in astrochemical processes, particularly in the interstellar medium (ISM), where metal-sulphur chemistry influences molecular formation and evolution. This work presents a detailed study of the rotational excitation and de-excitation of MgS through collisions with helium (He) atoms, the second most abundant species in the ISM after hydrogen, which includes both atomic (H) and molecular forms (H2). The focus on MgS–He collisions arises from He's high abundance, chemical inertness, and simpler electronic structure, which make it well suited for quantum scattering calculations. These characteristics establish He as an ideal candidate for initial studies, providing fundamental data for future investigations involving H2. The study uses quantum scattering methods to calculate the collisional rate coefficients over a broad temperature range. These rates are critical for interpreting observational data on MgS and predicting its abundance in space. The interaction potential between MgS and He is calculated using the rigid rotor approximation and the Jacobi coordinate system, employing the CCSD(T)-F12a/aug-cc-pVTZ method for accurate two-dimensional potential energy surface. The study explores the anisotropic nature of the MgS–He interaction, which favours odd Δj rotational transitions at low collision energies. The inelastic cross-sections for rotational transitions involving up to 16 rotational levels of MgS were computed up to 1000 cm−1, enabling the calculation of rate coefficients up to 150 K for Δj = ±1, ±2, and ± 3 rotational transitions. The results show that Δj = 1 transitions dominate at low temperatures, while Δj = 2 transitions become more significant at higher temperatures. This study provides valuable data for interpreting future astrophysical observations of MgS. The findings also propose new rotational transitions for MgS detection in space, enhancing our ability to track and study this molecule in various cosmic environments.

Hendaoui, Hamza (ORCID:0000000218641872)↗
DOE OSTI2024

Native frames: An approach for separating sequential and concerted three-body fragmentation

Disentangling sequential and concerted three-body fragmentation has been a longstanding endeavor in studies of molecular dynamics. To accomplish this goal, we recently introduced a novel method to separate sequential from concerted breakup, where the sequential events are analyzed in their native frames [J. Rajput et al., Phys. Rev. Lett. 120, 103001 (2018)]. The essence of this method is the use of the Jacobi coordinates' conjugate momenta to reduce the dimensionality of the multibody fragmentation in combination with a clear signature for sequential breakup. To demonstrate this method, we employ coincidence momentum imaging to study the strong-field dissociative ionization of OCS into O + +C + + S + , which typically undergoes concerted fragmentation or sequential breakup involving either a metastable CO 2+ or CS 2+ molecule. We identify sequential breakup using a uniform distribution as a function of the angle between the conjugate momenta, associated with the first and second fragmentation steps, which is due to the rotation of the intermediate molecule in the fragmentation plane. Furthermore, by exploiting this uniform distribution, we separate the sequential and concerted distributions in any plot created from the measured momenta.

Atomic & molecular processes in external fields↗
DOE OSTI2025

Pentaquarks made of light quarks and their admixture to baryons

This paper is a continuation of our studies of multiquark hadrons. The antisymmetrization of their wave functions required by Fermi statistics is nontrivial, as it mixes orbital, color, spin, and flavor structures. In our previous papers we developed a method to find them based on the representations of the permutation group, and derived the explicit wave functions for baryons excited to the first and second shells (L = 1, 2), tetraquarks $qq$$\overline{q}$$\overline{q}$ and hexaquarks (6q). Now we apply it to light pentaquarks ($qqq$$\overline{q}$), in the S- and P-shells (L = 0, 1). Using Jacobi coordinates, one can use the hyperdistance approximation in 12-dimensional space. We further address the issue of “unquenching” of baryons, by considering their mixing with pentaquarks, via two channels, through the addition of σ-like or π-like $\overline{q}$$q$ pairs. This mixing is central for understanding of the observed flavor asymmetry of the antiquark sea, the amount of orbital motion issue as well as other nucleon properties.

Baryons↗
DOE OSTI2024

A Performance and Energy Study of GPU-Resident Preconditioners for Conjugate Gradient Solvers: In the Context of Existing and Novel Approaches

Optimizing a particular subprogram out of the set of Basic (sparse) Linear Algebra Subprograms (BLAS) for a given architecture is a common topic of research. In applications, however, these BLAS functions rarely appear in isolation; usually, many of them are used together, in various combinations and with varying inputs. As the need to solve a large, sparse linear system is ubiquitous throughout HPC applications, linear solvers constitute a realistic, sufficiently complex and well-defined representative use case for composite BLAS routines. To this end, based on a representative set of matrices drawn from a diverse set of fields, we present a framework to study, from the performance and energy perspective, the efficacy of GPU- resident parallel Conjugate Gradient (CG) linear solver with different preconditioner options, including Gauss-Seidel, Jacobi, and incomplete Cholesky. We also propose a novel GPU-based preconditioner, in which the triangular solves are approximated by an iterative process. The development of this preconditioner was motivated by solving large graph Laplacian linear systems, for which the existing preconditioners either perform slow on GPU-based platforms or are not applicable. We compare the performance of these preconditioners on different hardware accelerator architectures, i.e., AMD MI250X, MI100, Nvidia A100, V100, and Jetson. Our experiments reveal performance trade-offs and provide information on how to select the best strategy for the given linear system, dictated by its properties, and the platform of interest. We demonstrate the application of our novel preconditioner for solving CG and graph Laplacian systems. Overall, the framework can be utilized as a benchmark to guide informed decisions in choosing a specific preconditioner, i.e., whether it is better to rely on the performance of a triangular solver or on the performance of sparse matrix-vector product. Finally, by considering power consumption to solve the linear systems, we report the energy footprint for the solvers.

Preconditioned Conjugate Gradient, GPUs, iterative↗
DOE OSTI2025

Bias-Variance Trade-Off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs

Physics-Informed Neural Networks (PINNs) have triggered a paradigm shift in scientific computing, leveraging mesh-free properties and robust approximation capabilities. While proving effective for low-dimensional partial differential equations (PDEs), the computational cost of PINNs remains a hurdle in high-dimensional scenarios. This is particularly pronounced when computing high-order and high-dimensional derivatives in the physics-informed loss. Randomized Smoothing PINN (RS-PINN) introduces Gaussian noise for stochastic smoothing of the original neural net model, enabling the use of Monte Carlo methods for derivative approximation, which eliminates the need for costly automatic differentiation. Despite its computational efficiency, especially in the approximation of high-dimensional derivatives, RS-PINN introduces biases in both loss and gradients, negatively impacting convergence, especially when coupled with stochastic gradient descent (SGD) algorithms. We present a comprehensive analysis of biases in RS-PINN, attributing them to the nonlinearity of the Mean Squared Error (MSE) loss as well as the intrinsic nonlinearity of the PDE itself. We propose tailored bias correction techniques, delineating their application based on the order of PDE nonlinearity. The derivation of an unbiased RS-PINN allows for a detailed examination of its advantages and disadvantages compared to the biased version. Specifically, the biased version has a lower variance and runs faster than the unbiased version, but it is less accurate due to the bias. To optimize the bias-variance trade-off, we combine the two approaches in a hybrid method that balances the rapid convergence of the biased version with the high accuracy of the unbiased version. In addition to methodological contributions, we present an enhanced implementation of RS-PINN. Extensive experiments on diverse high-dimensional PDEs, including Fokker-Planck, Hamilton-Jacobi-Bellman (HJB), viscous Burgers’, Allen-Cahn, and Sine-Gordon equations, illustrate the bias-variance trade-off and highlight the effectiveness of the hybrid RS-PINN. Empirical guidelines are provided for selecting biased, unbiased, or hybrid versions, depending on the dimensionality and nonlinearity of the specific PDE problem.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker–Planck Equations

The Fokker-Planck (FP) equation is a foundational partial differential equation (PDE) in stochastic processes involving Brownian motions. However, the curse of dimensionality (CoD) poses a formidable challenge when dealing with high-dimensional FP equations. Although Monte Carlo simulation and (vanilla) Physics-Informed Neural Networks (PINNs) have shown the potential to tackle CoD, both methods exhibit significant numerical errors in high dimensions when dealing with the probability density function (PDF) associated with Brownian motion. The point-wise PDF values tend to decrease exponentially as dimensionality increases, surpassing the precision of numerical simulations and resulting in substantial errors. In addition, due to its massive sampling, Monte Carlo fails to offer fast sampling. Modeling the logarithm likelihood (LL) via vanilla PINNs transforms the FP equation into a notoriously difficult Hamilton-Jacobi-Bellman (HJB) equation, which is impractical for PINN learning, whose error grows rapidly with dimension. To this end, we propose a novel approach utilizing a score-based solver to fit the score function in stochastic differential equations (SDEs). The score function, defined as the gradient of the LL, plays a fundamental role in inferring LL and PDF and enables fast SDE sampling, offering an effective means to overcome the CoD. Three fitting methods, Score Matching (SM), Sliced Score Matching (SSM), and Score-PINN, are introduced, each contributing unique advantages in computational complexity, accuracy, and generality. The proposed score-based SDE solver operates in two stages: first, employing score matching or Score-PINN to acquire the score function; and second, solving the LL via an ordinary differential equation (ODE) using the obtained score function. Comparative evaluations across these methods showcase varying trade-offs. The proposed methodology is evaluated across diverse SDEs, including anisotropic Ornstein-Uhlenbeck processes, geometric Brownian motion, and Brownian motion with varying eigenspace. We also test various distributions, including Gaussian, Log-normal, Laplace, and Cauchy distributions. The numerical results demonstrate the score-based SDE solver’s stability, speed, and performance across different experimental settings, solidifying its potential as a solution to CoD for high-dimensional FP equations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

pnnl/LAP

A software framework to study, from the performance and energy perspective, the efficacy of GPU-resident parallel Conjugate Gradient (CG) linear solver with different preconditioner options, including Gauss-Seidel, Jacobi, and incomplete Cholesky. We also propose a novel GPU-based preconditioner, in which the triangular solves are approximated by an iterative process

Swirydowicz, Kasia↗