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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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At least 127 records · Page 7
All sourcesNASA NTRSDOE OSTI
DOE OSTI2014

MO-E-18C-04: Advanced Computer Simulation and Visualization Tools for Enhanced Understanding of Core Medical Physics Concepts

Purpose: Most medical physics programs emphasize proficiency in routine clinical calculations and QA. The formulaic aspect of these calculations and prescriptive nature of measurement protocols obviate the need to frequently apply basic physical principles, which, therefore, gradually decay away from memory. E.g. few students appreciate the role of electron transport in photon dose, making it difficult to understand key concepts such as dose buildup, electronic disequilibrium effects and Bragg-Gray theory. These conceptual deficiencies manifest when the physicist encounters a new system, requiring knowledge beyond routine activities. Methods: Two interactive computer simulation tools are developed to facilitate deeper learning of physical principles. One is a Monte Carlo code written with a strong educational aspect. The code can “label” regions and interactions to highlight specific aspects of the physics, e.g., certain regions can be designated as “starters” or “crossers,” and any interaction type can be turned on and off. Full 3D tracks with specific portions highlighted further enhance the visualization of radiation transport problems. The second code calculates and displays trajectories of a collection electrons under arbitrary space/time dependent Lorentz force using relativistic kinematics. Results: Using the Monte Carlo code, the student can interactively study photon and electron transport through visualization of dose components, particle tracks, and interaction types. The code can, for instance, be used to study kerma-dose relationship, explore electronic disequilibrium near interfaces, or visualize kernels by using interaction forcing. The electromagnetic simulator enables the student to explore accelerating mechanisms and particle optics in devices such as cyclotrons and linacs. Conclusion: The proposed tools are designed to enhance understanding of abstract concepts by highlighting various aspects of the physics. The simulations serve as virtual experiments that give deeper and long lasting understanding of core principles. The student can then make sound judgements in novel situations encountered beyond routine clinical activities.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2024

DNS of ignition and flame stabilization in a simplified gas turbine premixer

With the increasing need for fuel flexibility, mitigation of auto-ignition (AI) inside gas turbine (GT) premixers becomes crucial. They must be designed to yield a sufficiently homogeneous fuel-air mixture to achieve low emissions while at the same time avoiding the occurrence of AI and subsequent flame stabilization. This challenge requires a detailed understanding of turbulent mixing and chemistry interactions. In the present work, a direct numerical simulation (DNS) of an array of jets in crossflow (JICF), representative of an industrial GT premixer, is reported to shed light on these complex phenomena. It is found that AI kernels form in the aft part of the premixer and coalesce into a flame front that then propagates upstream, mainly through the boundary layer, and successively engulfs the jets. This, therefore, suggests a significant role of the jet array pattern on the flame stabilization. It is noted that AI kernels continue to form independently during the whole time of the simulation. To clarify the contribution of AI and diffusion in the ignition kernels and the main flame, chemical explosive mode analysis (CEMA) is employed jointly with a kernel tracking algorithm. It is found that during the initial formation of the flame, many ignition kernels form in mixtures with low scalar dissipation rate and large contribution from AI mode. As they quickly grow, they merge into a single flame front that becomes increasingly more diffusion-assisted over time, balancing the AI mode. Turbulence is shown to have a significant enhancing effect in lean premixed flames, but further analysis is required to fully characterize it. These findings are relevant for the industrial premixer studied, and also for novel micromix concepts that may be used in the next generation of GT combustion systems.

ADVANCED PROPULSION SYSTEMS↗
DOE OSTI2024

MONTE CARLO CROSS SECTION LOOKUP KERNEL FOR THE CEREBRAS WSE-2 IN CSL

This is a small kernel that was used to collect data for an upcoming paper. We would like to have the code be open source so that the reviewers (and then readers) of the paper can see the whole code, and can reproduce/verify our results. This is not a fully featured application, it cannot produce any useful simulation results, it just executes a small abstracted kernel using synthetic data. The purpose of the kernel is to understand the basic performance characteristics of an HPC kernel on novel AI accelerator architectures. The main kernel is written in the CSL coding language for use with the Cerebras WSE-2 AI accelerator. The kernel represented is the Monte Carlo cross section lookup kernel, which is a small kernel used by the Monte Carlo neutral particle transport algorithm. There is also a baseline kernel written in CUDA that we will include in the repository to form a basis for comparing the WSE-2 to GPU.

Tramm, John↗
DOE OSTI2021

Time-dependent orbital-free density functional theory: Background and Pauli kernel approximations

Time-dependent orbital-free density functional theory (DFT) is an efficient method for calculating the dynamic properties of large-scale quantum systems due to the low computational cost compared to standard time-dependent DFT. In this work, we formalize this method by mapping the real system of interacting fermions onto a fictitious system of noninteracting bosons. The dynamic Pauli potential and associated kernel emerge as key ingredients of time-dependent orbital-free DFT. Using the uniform electron gas as a model system, we derive an approximate frequency-dependent Pauli kernel. Pilot calculations suggest that space nonlocality is a key feature for this kernel. Nonlocal terms arise already in the second-order expansion with respect to unitless frequency and reciprocal space variable ($\frac{ω}{qk_F}$ and $\frac{q}{2k_F}$, respectively). Given the encouraging performance of the proposed kernel, we expect it will lead to more accurate orbital-free DFT simulations of nanoscale systems out of equilibrium. Additionally, the proposed path to formulate nonadiabatic Pauli kernels presents several avenues for further improvements which can be exploited in future work to improve the results.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2022

Representation Learning via Quantum Neural Tangent Kernels

Variational quantum circuits are used in quantum machine learning and variational quantum simulation tasks. Designing good variational circuits or predicting how well they perform for given learning or optimization tasks is still unclear. Here we discuss these problems, analyzing variational quantum circuits using the theory of neural tangent kernels. We define quantum neural tangent kernels, and derive dynamical equations for their associated loss function in optimization and learning tasks. We analytically solve the dynamics in the frozen limit, or lazy training regime, where variational angles change slowly and a linear perturbation is good enough. We extend the analysis to a dynamical setting, including quadratic corrections in the variational angles. We then consider a hybrid quantum classical architecture and define a large-width limit for hybrid kernels, showing that a hybrid quantum classical neural network can be approximately Gaussian. The results presented here show limits for which analytical understandings of the training dynamics for variational quantum circuits, used for quantum machine learning and optimization problems, are possible. These analytical results are supported by numerical simulations of quantum machine-learning experiments.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Three-dimensional super line-localization in low signal-to-noise microscope images via prior-apprised unsupervised learning (PAUL)

Biological processes such as processive enzyme turnover and intracellular cargo tracking involve the dynamic motion of a small "article" along a curvilinear biopolymer track. To understand these processes that occur across multiple length and time scales, one must acquire both the trajectory of the particle and the position of the track along which it moves, possibly by combining high-resolution single-particle tracking with conventional microscopy. Yet, usually there is a significant resolution mismatch between these modalities: while the tracked particle is localized with a precision of 10 nm, the image of the surroundings is limited by optical difraction, with 200 nm lateral and 500 nm axial resolutions. Compared to the particle's trajectory, the surrounding curvilinear structure appears as a blurred and noisy image. This disparity in the spatial resolutions of the particle trajectory and the surrounding curvilinear structure image makes data reconstruction, as well as interpretation, particularly challenging. Analysis is further complicated when the curvilinear structures are oriented arbitrarily in 3D space. Here, we present a prior-apprised unsupervised learning (PAUL) approach to extract information from 3D images where the underlying features resemble a curved line such as a filament or microtubule. This three-stage framework starts with a Hessian-based feature enhancement, which is followed by feature registration, where local line segments are detected on repetitively sampled subimage tiles. In the final stage, statistical learning, segments are clustered based on their geometric relationships. Principal curves are then approximated from each segment group via statistical tools including principal component analysis, bootstrap and kernel transformation. This procedure is characterized on simulated images, where sub-voxel medium deviations from true curves have been achieved. The 3D PAUL approach has also been implemented for successful line localization in experimental 3D images of gold nanowires obtained using a multifocal microscope. Lastly, this work not only bridges the resolution gap between two microscopy modalities, but also allows us to conduct 3D super line-localization imaging experiments, without using super-resolution techniques.

42 ENGINEERING↗
DOE OSTI2023

Contributions to the mechanistic understanding of the microstructural evolution in irradiated U-Mo dispersion fuel

Here, advanced microstructural characterization techniques, such as scanning electron microscopy (SEM) and scanning transmission electron microscopy - energy dispersive x-ray spectroscopy (STEM-EDS), were used to interpret the fuel microstructure evolution and fission products behavior in U-Mo dispersion fuel irradiated in the Advanced Test Reactor (ATR) as part of the European Mini-Plate Irradiation Experiment (EMPIrE) test. The larger as-fabricated fuel grain size achieved by heat-treating the U-Mo powder resulted in slower high burnup structure (HBS) development and reduced fission gas porosity. Slower HBS kinetics was observed at the fuel kernels’ periphery, which contained smaller and less fission gas bubbles at all fission densities (FDs) investigated and was attributed to a locally reduced damage density and fission products concentration, as corroborated with Monte Carlo simulations. The non-refined grains at the fuel kernel periphery hosted a perfectly ordered fission Gas Bubble Superlattice (GBS) up to 6.3 × 10 21 fissions/cm 3 . Nano-scale STEM-EDS analysis presented in this study provided useful information on the GBS characteristic morphology and evolution in U-Mo fuel. The concentration of fission gas in the GBS progressively increased with FD, pointing to an evolution of the nanobubble pressure status with irradiation. A possible connection between the GBS collapse and HBS onset is proposed for which there exists a threshold in the misorientation of the refined sub-grains above which the GBS stability during irradiation is no longer preserved, resulting in the GBS collapse.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2018

Microstructure and Fission Product Distribution Examination in the UCO kernel of TRISO Fuel Particles

The effect of irradiation and fabrication process on the UCO fuel kernel, kernel-buffer interlayer and fission products, has not been fully studied for UCO fuel kernels of TRISO coated particles. Specifically, the potential correlation(s) with fission product retention, composition and distribution within these structures for the tristructural isotropic (TRISO) fuel performance at normal and accident conditions, are not well reported in literature. Structural and chemical information in the irradiated UCO fuel kernel, e.g., the stoichiometry variations, may further contribute to fuel performance modeling under the Nuclear Energy Advanced Modeling and Simulation (NEAMS) program. The microstructural and micro-chemical evolution of selected irradiated UCO fuel kernels, kernel buffer interfaces and recoil zones were characterized using Scanning Transmission Electron Microscopy (STEM) and Energy Dispersive Spectroscopy (EDS) elemental mapping. The fuel kernel of particles from Advanced Gas Reactor (AGR)-2 irradiated to 10.8% fissions per initial metal atom (FIMA), were examined with the following main findings: (1) large fission gas bubbles were mainly identified in the zirconium and molybdenum enriched but oxygen depleted phases in the center of fuel kernel, (2) significant inter-diffusion between the UCO fuel kernel and pyro-carbon buffer layer was observed as a high density of sub-micron sized carbon inclusions was identified at near the interface in the kernel side and some isolated UCO particles were located in the buffer layer, (3) pores filled with free carbon were also observed in the recoil zone specimen. Fuel kernels from two AGR-1 compacts irradiated to 11.4 and 17.4% FIMA respectively, were examined with the following main findings: (1) two primary phases consisting of a “high-Z (atomic mass)” UC(O) phase (rocksalt; a=0.496 nm) that also contained Zr, Mo, and/or Ru and a “low-Z” UO2(C) phase (fluorite structure; a = 0.547nm) in which trace amounts of I, Nd, Pr, and/or Er could be detected. This latter phase contained many precipitates of the “high-Z” phase. Conversely, this latter phase did not contain precipitates of the UO2-type material, (2) Xe bubbles were found almost exclusively in the “high-Z” phase and in the particular example shown, the Zr-containing variant. Further work in this area should address whether this potential Xe preference for just one of the high-Z phases is indeed a real effect.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2016

A fast band–Krylov eigensolver for macromolecular functional motion simulation on multicore architectures and graphics processors

We introduce a new iterative Krylov subspace-based eigensolver for the simulation of macromolecular motions on desktop multithreaded platforms equipped with multicore processors and, possibly, a graphics accelerator (GPU). The method consists of two stages, with the original problem first reduced into a simpler band-structured form by means of a high-performance compute-intensive procedure. This is followed by a memory-intensive but low-cost Krylov iteration, which is off-loaded to be computed on the GPU by means of an efficient data-parallel kernel. The experimental results reveal the performance of the new eigensolver. Concretely, when applied to the simulation of macromolecules with a few thousands degrees of freedom and the number of eigenpairs to be computed is small to moderate, the new solver outperforms other methods implemented as part of high-performance numerical linear algebra packages for multithreaded architectures.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Cylindrical Kernel Density Estimators For Monte Carlo Neutron Transport Reactor Physics Problems

Kernel Density Estimators (KDEs) have been investigated as an alternative to histogram tallies in Monte Carlo radiation transport simulations for capturing spatially-resolved quantities such as scalar flux and reaction rate densities. KDEs are non-parametric estimators that show potential for producing smooth estimates of quantities of interest with reduced variance compared to traditional histogram tallies. Furthermore, KDEs obtain estimates of the underlying distribution at user-defined points without requiring an underlying mesh, thus making KDEs an attractive potential alternative to histogram tallies. Recent work has introduced the Mean Free Path based KDE (MFP KDE) in order to capture distributions with large curvature at material interfaces, e.g. absorption reaction rates in UO{sub 2} fuel in light water reactors. However, the multivariate MFP KDE produces inaccuracies in the form of localized spikes in problems with material cross sections that contain large resonances. Since the MFP KDE works well in one dimension, it is hypothesized that a different MFP KDE formulation where the normalization coefficient only contains one factor of the cross section would be able to capture these distributions with large curvature without producing spikes in the estimate. This can be accomplished by using a cylindrical KDE whose 2-D support area is wedge-shaped and applying the MFP KDE to the radial component of the cylindrical KDE. These cylindrical KDEs use 2-D (r,θ) cylindrical coordinates with no dependence in axial dimension z. This paper derives the cylindrical KDE and the cylindrical MFP KDE and tests them on a reactor physics pin-cell benchmark problem, producing accurate estimates of flux and reaction rate densities with differences less than 0.5% when compared to a reference histogram. (authors)

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2014

Using Monte Carlo ray tracing simulations to model the quantum harmonic oscillator modes observed in uranium nitride

Recently an extended series of equally spaced vibrational modes was observed in uranium nitride (UN) by performing neutron spectroscopy measurements using the ARCS and SEQUOIA time-of- flight chopper spectrometers [A.A. Aczel et al, Nature Communications 3, 1124 (2012)]. These modes are well described by 3D isotropic quantum harmonic oscillator (QHO) behavior of the nitrogen atoms, but there are additional contributions to the scattering that complicate the measured response. In an effort to better characterize the observed neutron scattering spectrum of UN, we have performed Monte Carlo ray tracing simulations of the ARCS and SEQUOIA experiments with various sample kernels, accounting for the nitrogen QHO scattering, contributions that arise from the acoustic portion of the partial phonon density of states (PDOS), and multiple scattering. These simulations demonstrate that the U and N motions can be treated independently, and show that multiple scattering contributes an approximate Q-independent background to the spectrum at the oscillator mode positions. Temperature dependent studies of the lowest few oscillator modes have also been made with SEQUOIA, and our simulations indicate that the T-dependence of the scattering from these modes is strongly influenced by the uranium lattice.

Lin, J. Y. Y.↗
DOE OSTI2016

Reducing Disk Storage of Full-3D Seismic Waveform Tomography (F3DT) Through Lossy Online Compression

Full-3D seismic waveform tomography (F3DT) is the latest seismic tomography technique that can assimilate broadband, multi-component seismic waveform observations into high-resolution 3D subsurface seismic structure models. The main drawback in the current F3DT implementation, in particular the scattering-integral implementation (F3DT-SI), is the high disk storage cost and the associated I/O overhead of archiving the 4D space-time wavefields of the receiver- or source-side strain tensors. The strain tensor fields are needed for computing the data sensitivity kernels, which are used for constructing the Jacobian matrix in the Gauss-Newton optimization algorithm. In this study, we have successfully integrated a lossy compression algorithm into our F3DT SI workflow to significantly reduce the disk space for storing the strain tensor fields. The compressor supports a user-specified tolerance for bounding the error, and can be integrated into our finite-difference wave-propagation simulation code used for computing the strain fields. The decompressor can be integrated into the kernel calculation code that reads the strain fields from the disk and compute the data sensitivity kernels. During the wave-propagation simulations, we compress the strain fields before writing them to the disk. To compute the data sensitivity kernels, we read the compressed strain fields from the disk and decompress them before using them in kernel calculations. Experiments using a realistic dataset in our California statewide F3DT project have shown that we can reduce the strain-field disk storage by at least an order of magnitude with acceptable loss, and also improve the overall I/O performance of the entire F3DT-SI workflow significantly. The integration of the lossy online compressor may potentially open up the possibilities of the wide adoption of F3DT-SI in routine seismic tomography practices in the near future.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Balancing Particle and Mesh Computation in a Particle-In-Cell Code

The XGC1 plasma microturbulence particle-in-cell simulation code has both particle-based and mesh-based computational kernels that dominate performance. Both of these are subject to load imbalances that can degrade performance and that evolve during a simulation. Each separately can be addressed adequately, but optimizing just for one can introduce significant load imbalances in the other, degrading overall performance. A technique has been developed based on Golden Section Search that minimizes wallclock time given prior information on wallclock time, and on current particle distribution and mesh cost per cell, and also adapts to evolution in load imbalance in both particle and mesh work. In problems of interest this doubled the performance on full system runs on the XK7 at the Oak Ridge Leadership Computing Facility compared to load balancing only one of the kernels.

Worley, Patrick H↗
DOE OSTI2017

Comparative Implementation of High Performance Computing for Power System Dynamic Simulations

Dynamic simulation for transient stability assessment is one of the most important, but intensive, computations for power system planning and operation. Present commercial software is mainly designed for sequential computation to run a single simulation, which is very time consuming with a single processer. The application of High Performance Computing (HPC) to dynamic simulations is very promising in accelerating the computing process by parallelizing its kernel algorithms while maintaining the same level of computation accuracy. This paper describes the comparative implementation of four parallel dynamic simulation schemes in two state-of-the-art HPC environments: Message Passing Interface (MPI) and Open Multi-Processing (OpenMP). These implementations serve to match the application with dedicated multi-processor computing hardware and maximize the utilization and benefits of HPC during the development process.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Multilevel summation with B-spline interpolation for pairwise interactions in molecular dynamics simulations

The multilevel summation method for calculating electrostatic interactions in molecular dynamics simulations constructs an approximation to a pairwise interaction kernel and its gradient, which can be evaluated at a cost that scales linearly with the number of atoms. The method smoothly splits the kernel into a sum of partial kernels of increasing range and decreasing variability with the longer-range parts interpolated from grids of increasing coarseness. Multilevel summation is especially appropriate in the context of dynamics and minimization, because it can produce continuous gradients. This article explores the use of B-splines to increase the accuracy of the multilevel summation method (for nonperiodic boundaries) without incurring additional computation other than a preprocessing step (whose cost also scales linearly). To obtain accurate results efficiently involves technical difficulties, which are overcome by a novel preprocessing algorithm. Numerical experiments demonstrate that the resulting method offers substantial improvements in accuracy and that its performance is competitive with an implementation of the fast multipole method in general and markedly better for Hamiltonian formulations of molecular dynamics. The improvement is great enough to establish multilevel summation as a serious contender for calculating pairwise interactions in molecular dynamics simulations. In particular, the method appears to be uniquely capable for molecular dynamics in two situations, nonperiodic boundary conditions and massively parallel computation, where the fast Fourier transform employed in the particle–mesh Ewald method falls short.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

kynema-fmb [SWR-23-07]

Kynema-FMB (FKA: Kynema) is an open-source performance portable flexible multibody (FMB) dynamics solver designed for time-domain simulations. While originally tailored for wind turbine structural dynamics, the formulation and implementation are those of a general flexible-multidbody dynamics solver that can readily be applied to a wide range of systems. Kynema was designed with a narrow focus, namely to provide a lightweight, fast, accurate FMD solver for coupling to computational-fluid-dynamics (CFD) codes, especially the CFD codes in the Kynema suite, for fluid-structure-interaction (FSI) simulations. Kynema-FMB is equipped to model systems that can be represented as a collection of beams and rigid bodies that are connected through constraints. Degrees of freedom are defined in the inertial/global frame of reference and include displacements and rotations (formally as rotation matrices, but stored as quaternions). The underlying formulation is built on a Lie-group time integrator designed for index-3 differential-algebraic equations, which is second-order accurate in time (Bruls et al., 2012). Beam models are based on geometrically exact beam theory and are discretized as high-order spectral finite elements similar to those in BeamDyn (Wang et al., 2017). The governing equations for a FMD system like a wind turbine constitute a highly nonlinear system of constrained partial-differential equations. Kynema-FMB uses analytical Jacobians in the nonlinear-system solves in each time step. Linear systems use sparse storage and several third-party sparse-linear-system solvers are enabled. Ill conditioning of linear systems is mitigated with preconditioning described in Bottasso et al, 2008. Kynema-FMB is integrated with a simple open-source controller (ROSCO). There is an application programming interface (API) for coupling to geometry-resolved CFD (like that in Sharma et al., 2023) and actuator-force CFD (like that in Kuhn et al., 2025). In the latter, for actuator-line models, Kynema-FMB includes an internal blade-element solver that depends on user-provided lookup tables for coefficients of lift and drag, i.e., aerodynamic polars. Kynema-FMB is written in C++ and leverages Kokkos and Kokkos-Kernels (KokkosEcosystem) as its performance portability layer enabling simulations on both CPU and GPU systems. The repository is equipped with extensive automated testing at the unit and regression/system levels. The following describes the high-level development objectives conceived for Kynema: *Kynema will follow modern software development best practices, including test-driven development (TDD), version control, hierarchical automated testing, and continuous integration (CI) for a robust development environment. *The core data structures are memory efficient and enable vectorization and parallelization at multiple levels. *Data structures are data-oriented to exploit methods for accelerated computing including high utilization of chip resources (e.g., single instruction multiple data (SIMD) instruction sets) and parallelization using GP-GPUs. *The computational algorithms incorporate robust open-source libraries for mathematical operations, resource allocation, and data management. *The API design considers multiple stakeholder needs and ensure integration with existing and future ecosystems for data science, machine learning, and AI. *Kynema-FMB is written in modern C++ and leverages Kokkos as its performance-portability library with inspiration from the kynema stack.

Sprague, MichaelA.↗
DOE OSTI2020

Extended Lagrangian Born–Oppenheimer molecular dynamics using a Krylov subspace approximation

It is shown how the electronic equations of motion in extended Lagrangian Born–Oppenheimer molecular dynamics simulations can be integrated using low-rank approximations of the inverse Jacobian kernel. This kernel determines the metric tensor in the harmonic oscillator extension of the Lagrangian that drives the evolution of the electronic degrees of freedom. The proposed kernel approximation is derived from a pseudoinverse of a low-rank estimate of the Jacobian, which is expressed in terms of a generalized set of directional derivatives with directions that are given from a Krylov subspace approximation. The approach allows a tunable and adaptive approximation that can take advantage of efficient preconditioning techniques. The proposed kernel approximation for the integration of the electronic equations of motion makes it possible to apply extended Lagrangian first-principles molecular dynamics simulations to a broader range of problems, including reactive chemical systems with numerically sensitive and unsteady charge solutions. This can be achieved without requiring exact full calculations of the inverse Jacobian kernel in each time step or relying on iterative non-linear self-consistent field optimization of the electronic ground state prior to the force evaluations as in regular direct Born–Oppenheimer molecular dynamics. We note the low-rank approximation of the Jacobian is directly related to Broyden’s class of quasi-Newton algorithms and Jacobian-free Newton–Krylov methods and provides a complementary formulation for the solution of nonlinear systems of equations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗