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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 361 records · Page 20
All sourcesNASA NTRSDOE OSTI
DOE OSTI2016

Investigation of Natural Circulation Instability and Transients in Passively Safe Small Modular Reactors

The NEUP funded project, NEUP-3496, aims to experimentally investigate two-phase natural circulation flow instability that could occur in Small Modular Reactors (SMRs), especially for natural circulation SMRs. The objective has been achieved by systematically performing tests to study the general natural circulation instability characteristics and the natural circulation behavior under start-up or design basis accident conditions. Experimental data sets highlighting the effect of void reactivity feedback as well as the effect of power ramp-up rate and system pressure have been used to develop a comprehensive stability map. The safety analysis code, RELAP5, has been used to evaluate experimental results and models. Improvements to the constitutive relations for flashing have been made in order to develop a reliable analysis tool. This research has been focusing on two generic SMR designs, i.e. a small modular Simplified Boiling Water Reactor (SBWR) like design and a small integral Pressurized Water Reactor (PWR) like design. A BWR-type natural circulation test facility was firstly built based on the three-level scaling analysis of the Purdue Novel Modular Reactor (NMR) with an electric output of 50 MWe, namely NMR-50, which represents a BWR-type SMR with a significantly reduced reactor pressure vessel (RPV) height. The experimental facility was installed with various equipment to measure thermalhydraulic parameters such as pressure, temperature, mass flow rate and void fraction. Characterization tests were performed before the startup transient tests and quasi-steady tests to determine the loop flow resistance. The control system and data acquisition system were programmed with LabVIEW to realize the realtime control and data storage. The thermal-hydraulic and nuclear coupled startup transients were performed to investigate the flow instabilities at low pressure and low power conditions for NMR-50. Two different power ramps were chosen to study the effect of startup power density on the flow instability. The experimental startup transient results showed the existence of three different flow instability mechanisms, i.e., flashing instability, condensation induced flow instability, and density wave oscillations. In addition, the void-reactivity feedback did not have significant effects on the flow instability during the startup transients for NMR-50. ii Several initial startup procedures with different power ramp rates were experimentally investigated to eliminate the flow instabilities observed from the startup transients. Particularly, the very slow startup transient and pressurized startup transient tests were performed and compared. It was found that the very slow startup transients by applying very small power density can eliminate the flashing oscillations in the single-phase natural circulation and stabilize the flow oscillations in the phase of net vapor generation. The initially pressurized startup procedure was tested to eliminate the flashing instability during the startup transients as well. The pressurized startup procedure included the initial pressurization, heat-up, and venting process. The startup transient tests showed that the pressurized startup procedure could eliminate the flow instability during the transition from single-phase flow to two-phase flow at low pressure conditions. The experimental results indicated that both startup procedures were applicable to the initial startup of NMR. However, the pressurized startup procedures might be preferred due to short operating hours required. In order to have a deeper understanding of natural circulation flow instability, the quasi-steady tests were performed using the test facility installed with preheater and subcooler. The effect of system pressure, core inlet subcooling, core power density, inlet flow resistance coefficient, and void reactivity feedback were investigated in the quasi-steady state tests. The experimental stability boundaries were determined between unstable and stable flow conditions in the dimensionless stability plane of inlet subcooling number and Zuber number. To predict the stability boundary theoretically, linear stability analysis in the frequency domain was performed at four sections of the natural circulation test loop. The flashing phenomena in the chimney section was considered as an axially uniform heat source. And the dimensionless characteristic equation of the pressure drop perturbation was obtained by considering the void fraction effect and outlet flow resistance in the core section. The theoretical flashing boundary showed some discrepancies with previous experimental data from the quasi-steady state tests. In the future, thermal non-equilibrium was recommended to improve the accuracy of flashing instability boundary. As another part of the funded research, flow instabilities of a PWR-type SMR under low pressure and low power conditions were investigated experimentally as well. The NuScale reactor design was selected as the prototype for the PWR-type SMR. In order to experimentally study the natural circulation behavior of NuScale iii reactor during accidental scenarios, detailed scaling analyses are necessary to ensure that the scaled phenomena could be obtained in a laboratory test facility. The three-level scaling method is used as well to obtain the scaling ratios derived from various non-dimensional numbers. The design of the ideally scaled facility (ISF) was initially accomplished based on these scaling ratios. Then the engineering scaled facility (ESF) was designed and constructed based on the ISF by considering engineering limitations including laboratory space, pipe size, and pipe connections etc. PWR-type SMR experiments were performed in this well-scaled test facility to investigate the potential thermal hydraulic flow instability during the blowdown events, which might occur during the loss of coolant accident (LOCA) and loss of heat sink accident (LOHS) of the prototype PWR-type SMR. Two kinds of experiments, normal blowdown event and cold blowdown event, were experimentally investigated and compared with code predictions. The normal blowdown event was experimentally simulated since an initial condition where the pressure was lower than the designed pressure of the experiment facility, while the code prediction of blowdown started from the normal operation condition. Important thermal hydraulic parameters including reactor pressure vessel (RPV) pressure, containment pressure, local void fraction and temperature, pressure drop and natural circulation flow rate were measured and analyzed during the blowdown event. The pressure and water level transients are similar to the experimental results published by NuScale [51], which proves the capability of current loop in simulating the thermal hydraulic transient of real PWR-type SMR. During the 20000s blowdown experiment, water level in the core was always above the active fuel assemble during the experiment and proved the safety of natural circulation cooling and water recycling design of PWR-type SMR. Besides, pressure, temperature, and water level transient can be accurately predicted by RELAP5 code. However, the oscillations of natural circulation flow rate, water level and pressure drops were observed during the blowdown transients. This kind of flow oscillations are related to the water level and the location upper plenum, which is a path for coolant flow from chimney to steam generator and down comer. In order to investigate the transients start from the opening of ADS valve in both experimental and numerical way, the cold blow-down experiment is conducted. For the cold blowdown event, different from setting both reactor iv pressure vessel (RPV) and containment at high temperature and pressure, only RPV was heated close to the highest designed pressure and then open the ADS valve, same process was predicted using RELAP5 code. By doing cold blowdown experiment, the entire transients from the opening of ADS can be investigated by code and benchmarked with experimental data. Similar flow instability observed in the cold blowdown experiment. The comparison between code prediction and experiment data showed that the RELAP5 code can successfully predict the pressure void fraction and temperature transient during the cold blowdown event with limited error, but numerical instability exists in predicting natural circulation flow rate. Besides, the code is lack of capability in predicting the water level related flow instability observed in experiments.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2025

A provably stable numerical method for the anisotropic diffusion equation in confined magnetic fields

We present a novel numerical method for solving the anisotropic diffusion equation in magnetic fields confined to a periodic box which is accurate and provably stable. We derive energy estimates of the solution of the continuous initial boundary value problem. A discrete formulation is presented using operator splitting in time with the summation by parts finite difference approximation of spatial derivatives for the perpendicular diffusion operator. Weak penalty procedures are derived for implementing both boundary conditions and parallel diffusion operator obtained by field line tracing. We prove that the fully-discrete approximation is unconditionally stable. Discrete energy estimates are shown to match the continuous energy estimate given the correct choice of penalty parameters. A nonlinear penalty parameter is shown to provide an effective method for tuning the parallel diffusion penalty and significantly minimises rounding errors. Several numerical experiments, using manufactured solutions, the “NIMROD benchmark” problem and a single island problem, are presented to verify numerical accuracy, convergence, and asymptotic preserving properties of the method. Finally, we present a magnetic field with chaotic regions and islands and show the contours of the anisotropic diffusion equation reproduce key features in the field.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2018

A family of second-order fully explicit time integration schemes

A new family of fully explicit time integration methods is proposed, which has second-order accuracy for structures with and without damping. Using a diagonal mass matrix, the suggested scheme remains fully explicit not only for structures with a non-diagonal damping matrix, but also when the internal force vector is a nonlinear function of velocity. The present algorithm has an acceptable domain of stability, and it is self-starting. This technique introduces effectively numerical dissipation to suppress the high-frequency spurious modes, while at the same time the lower modes are not affected too much. In addition, numerical dispersion error of the scheme is considerably smaller than that of the central difference method. Solving several linear and nonlinear problems highlights the superior performance of the authors’ approach. Findings demonstrate that solution time for the suggested scheme is much less than that of the central difference technique. The related algorithm can be easily implemented into programs, which already contain the central difference method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Efficient sensitivity analysis of the thermal profile in powder bed fusion of metals using hypercomplex automatic differentiation finite element method

Rapid cyclic temperature fluctuation occurring in powder bed fusion of metals using a laser beam (PBF-LB/M) influences the formation of flaws in printed parts. Consequently, there is a pressing need to enhance the quality of printed parts by developing innovative methodologies that can predict thermal histories and help uncover the intricate relationships between process parameters and thermal profiles. Sensitivity Analysis (SA) emerges as an essential tool for this, offering the potential for process optimization and enhanced quality control. Nonetheless, conventional SA methodologies often incur in excessive computational costs and potential numerical approximation errors. Here, to address this technical challenge, we present a novel method for SA that integrates the HYPercomplex-based Automatic Differentiation (HYPAD) technique with transient thermal simulations conducted via the finite element method (FEM). Leveraging this methodology, we efficiently and accurately perform SA for PBF-LB/M processes in a post-processing step. Compared to traditional methods like Finite Differences (FD), HYPAD-FEM required 96 % less computational time for obtaining sensitivities for 22 process parameters, under a comparative study conducted within the context of the 2018–02 AM benchmark of the National Institute of Standards and Technology. In summary, HYPAD-FEM offers superior efficiency and accuracy in SA over conventional methods, delivering the best sensitivity of a model without the need for step-size selection and problem or parameter-based implementations.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Computationally efficient and error aware surrogate construction for numerical solutions of subsurface flow through porous media

Limiting the injection rate to restrict the pressure below a threshold at a critical location can be an important goal of simulations that model the subsurface pressure between injection and extraction wells. The pressure is approximated by the solution of Darcy’s partial differential equation for a given permeability field. The subsurface permeability is modeled as a random field since it is known only up to statistical properties. This induces uncertainty in the computed pressure. Solving the partial differential equation for an ensemble of random permeability simulations enables estimating a probability distribution for the pressure at the critical location. These simulations are computationally expensive, and practitioners often need rapid online guidance for real-time pressure management. An ensemble of numerical partial differential equation solutions is used to construct a Gaussian process regression model that can quickly predict the pressure at the critical location as a function of the extraction rate and permeability realization. The Gaussian process surrogate analyzes the ensemble of numerical pressure solutions at the critical location as noisy observations of the true pressure solution, enabling robust inference using the conditional Gaussian process distribution. Our first novel contribution is to identify a sampling methodology for the random environment and matching kernel technology for which fitting the Gaussian process regression model scales as O ( n log n ) instead of the typical O ( n 3 ) rate in the number of samples n used to fit the surrogate. The surrogate model allows almost instantaneous predictions for the pressure at the critical location as a function of the extraction rate and permeability realization. Our second contribution is a novel algorithm to calibrate the uncertainty in the surrogate model to the discrepancy between the true pressure solution of Darcy’s equation and the numerical solution. Finally, although our method is derived for building a surrogate for the solution of Darcy’s equation with a random permeability field, the framework broadly applies to solutions of other partial differential equations with random coefficients.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Multilevel Monte Carlo methods for the Grad-Shafranov free boundary problem

The equilibrium configuration of a plasma in an axially symmetric reactor is described mathematically by a free boundary problem associated with the celebrated Grad-Shafranov equation. The presence of uncertainty in the model parameters introduces the need to quantify the variability in the predictions. This is often done by computing a large number of model solutions on a computational grid for an ensemble of parameter values and then obtaining estimates for the statistical properties of solutions. In this study, we explore the savings that can be obtained using multilevel Monte Carlo methods, which reduce costs by performing the bulk of the computations on a sequence of spatial grids that are coarser than the one that would typically be used for a simple Monte Carlo simulation. We examine this approach using both a set of uniformly refined grids and a set of adaptively refined grids guided by a discrete error estimator. Numerical experiments show that multilevel methods dramatically reduce the cost of simulation, with cost reductions typically on the order of 60 or more and possibly as large as 200. Furthermore, adaptive griding results in more accurate computation of geometric quantities such as x-points associated with the model.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

Magnus method for electronic structure calculations at extreme conditions

We present the application of Magnus based methods to the solution of first order coupled ordinary differential equations in High Energy Density (HED) physics applications. Our focus is on the application to quantum mechanical methods, specifically on the solution of the radial Dirac equation for real and complex energies. HED applications require accurate solutions across a wide range of spatial and energy domains, including regimes where the solutions exhibit pronounced oscillatory behavior. Such cases pose significant computational challenges. We demonstrate that Magnus-based integrators can efficiently and accurately address these challenges. We discuss the implementation of the Magnus method for the solution of the radial Dirac equation, including practical considerations such as the evaluation of matrix exponentials, numerical integration, error estimation, and adaptive step size control. We also discuss the application of these methods to complex energy Green’s function techniques and the efficient approximation of integrals of the solutions relevant to HED electronic structure calculations. Here, we demonstrate the accuracy and robustness of the resulting method in applications to the free-particle case, for which analytic solutions are available for comparison, as well as the challenging case of gold at HED conditions.

general physics↗
DOE OSTI2016

LNTgate: How scientific misconduct by the U.S. NAS led to governments adopting LNT for cancer risk assessment

This paper provides a detailed rebuttal to the letter of Beyea (2016) which offered a series of alternative interpretations to those offered in my article in Environmental Research (Calabrese, 2015a) concerning the role of the U.S. National Academy of Sciences (NAS) Biological Effects of Atomic Radiation (BEAR) I Committee Genetics Panel in the adoption of the linear dose response model for cancer risk assessment. Significant newly uncovered evidence is presented which supports and extends the findings of Calabrese (2015a), reaffirming the conclusion that the Genetics Panel should be evaluated for scientific misconduct for deliberate misrepresentation of the research record in order to enhance an ideological agenda. This critique documents numerous factual errors along with extensive and deliberate filtering of information in the Beyea letter (2016) that leads to consistently incorrect conclusions and an invalid general perspective.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2017

An equilibrium-preserving discretization for the nonlinear Rosenbluth–Fokker–Planck operator in arbitrary multi-dimensional geometry

Here, the Fokker–Planck collision operator is an advection-diffusion operator which describe dynamical systems such as weakly coupled plasmas, photonics in high temperature environment biological, and even social systems. For plasmas in the continuum, the Fokker–Planck collision operator supports such important physical properties as conservation of number, momentum, and energy, as well as positivity. It also obeys the Boltzmann's H-theorem, i.e., the operator increases the system entropy while simultaneously driving the distribution function towards a Maxwellian. In the discrete, when these properties are not ensured, numerical simulations can either fail catastrophically or suffer from significant numerical pollution. There is strong emphasis in the literature on developing numerical techniques to solve the Fokker–Planck equation while preserving these properties. In this short note, we focus on the analytical equilibrium preserving property, meaning that the Fokker–Planck collision operator vanishes when acting on an analytical Maxwellian distribution function. The equilibrium preservation property is especially important, for example, when one is attempting to capture subtle transport physics. Since transport arises from small Ο(ϵ) corrections to the equilibrium (where ϵ is a small expansion parameter), numerical truncation error present in the equilibrium solution may dominate, overwhelming transport dynamics.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Impoverishing Roots Will Improve Wheat Yield and Profitability Through Increased Water and Nitrogen Use Efficiencies

We report that more than a 60% increase in crop production is required by the 2050's to feed a growing world population. Understanding how plant functional traits and field management affect crop yields has the potential to improve agricultural productivity, minimize economic and environmental losses, and maximize food security. We explored the influence of winter wheat root characteristics and management on winter wheat growth, yield, and profit using a mechanistic and well-tested ecosystem and crop model, ecosys. We applied and further tested ecosys at an agricultural farm growing winter wheat in Ardmore, Oklahoma, United States. The model accurately predicted observed shoot carbon (R 2 = 0.95), soil moisture (R 2 = 0.67), soil temperature (R 2 = 0.91), and yield (percent error = 17%). Numerical optimization experiments were conducted to explore potential improvements of winter wheat yield and profit by modifying root characteristics, including root radius and root:shoot carbon transfer conductance, and fertilizer inputs. Our results show the potential for simultaneously improving winter wheat yields and profits. The optimum conditions were found to be in the range of root radius between 0.1 and 0.3 mm, carbon transfer conductance between 0.004 and 0.01 h -1 , and the currently applied fertilizer rate of 112 kg ha -1 . Under these conditions, improvements in yields and profits of up to approximately 25% and 110%, respectively, were modeled compared to those under baseline root traits. These improvements were achieved by impoverishing root structures, thereby increasing nutrient allocation to grains. Our results also demonstrate and motivate model structures that integrate the complex network of plant physiology, soil nutrient biogeochemistry, hydrology, and management.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2014

Effects of electron inertia in collisionless magnetic reconnection

We present a study of collisionless magnetic reconnection within the framework of full two-fluid MHD for a completely ionized hydrogen plasma, retaining the effects of the Hall current, electron pressure and electron inertia. We performed 2.5D simulations using a pseudo-spectral code with no dissipative effects. We check that the ideal invariants of the problem are conserved down to round-off errors. Our numerical results confirm that the change in the topology of the magnetic field lines is exclusively due to the presence of electron inertia. The computed reconnection rates remain a fair fraction of the Alfvén velocity, which therefore qualifies as fast reconnection.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2012

HIGH-PRECISION ASTROMETRY WITH A DIFFRACTIVE PUPIL TELESCOPE

Astrometric detection and mass determination of Earth-mass exoplanets require sub-{mu}as accuracy, which is theoretically possible with an imaging space telescope using field stars as an astrometric reference. The measurement must, however, overcome astrometric distortions, which are much larger than the photon noise limit. To address this issue, we propose to generate faint stellar diffraction spikes using a two-dimensional grid of regularly spaced small dark spots added to the surface of the primary mirror (PM). Accurate astrometric motion of the host star is obtained by comparing the position of the spikes to the background field stars. The spikes do not contribute to scattered light in the central part of the field and therefore allow unperturbed coronagraphic observation of the star's immediate surroundings. Because the diffraction spikes are created on the PM and imaged on the same focal plane detector as the background stars, astrometric distortions affect equally the diffraction spikes and the background stars and are therefore calibrated. We describe the technique, detail how the data collected by the wide-field camera are used to derive astrometric motion, and identify the main sources of astrometric error using numerical simulations and analytical derivations. We find that the 1.4 m diameter telescope, 0.3 deg{sup 2} field we adopt as a baseline design achieves 0.2 {mu}as single measurement astrometric accuracy. The diffractive pupil concept thus enables sub-{mu}as astrometry without relying on the accurate pointing, external metrology, or high-stability hardware required with previously proposed high-precision astrometry concepts.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Uniform convergence of an upwind discontinuous Galerkin method for solving scaled discrete-ordinate radiative transfer equations with isotropic scattering

Here we present an error analysis for the discontinuous Galerkin (DG) method applied to the discrete-ordinate discretization of the steady-state radiative transfer equation with isotropic scattering. Under some mild assumptions, we show that the DG method converges uniformly with respect to a scaling parameter which characterizes the strength of scattering in the system. However, the rate is not optimal and can be polluted by the presence of boundary layers. In one-dimensional slab geometries, we demonstrate optimal convergence when boundary layers are not present and analyze a simple strategy for balance interior and boundary layer errors. Some numerical tests are also provided in this reduced setting.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Spectral partitioning with blends of eigenvectors

We report many common methods for data analysis rely on linear algebra. We provide new results connecting data analysis error to numerical accuracy in the context of spectral graph partitioning. We provide pointwise convergence guarantees so that spectral blends (linear combinations of eigenvectors) can be employed to solve data analysis problems with confidence in their accuracy. We apply this theory to an accessible model problem, the ring of cliques, by deriving the relevant eigenpairs and finding necessary and sufficient solver tolerances. Analysis of the ring of cliques provides an upper bound on eigensolver tolerances for graph partitioning problems. These results bridge the gap between linear algebra based data analysis methods and the convergence theory of iterative approximation methods. These results explain how the combinatorial structure of a problem can be recovered much faster than numerically accurate solutions to the associated linear algebra problem.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

A simple solution to the Rietveld refinement recipe problem

Rietveld refinements are widely used for many purposes in the physical sciences. Conducting a Rietveld refinement typically requires expert input because correct results may require that parameters be added to the fit in the proper order. This order will depend on the nature of the data and the initial parameter values. A mechanism for computing the next parameter to add to the refinement is shown. The fitting function is evaluated with the current parameter value set and each parameter incremented and decremented by a small offset. This provides the partial derivatives with respect to each parameter, along with information to discriminate meaningful values from numerical computational errors. The implementation of this mechanism in the open-source GSAS-II program is discussed. This new method is discussed as an important step towards the development of automated Rietveld refinement technology.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Rapid Commissioning of Large Machine Tools Using Finite Element-Based Correction of Geometric Errors

Large computer numerical control (CNC) machine tools derive their stiffness from monolithic cast iron bases or weldments that are sometimes integral to machine motion systems like box ways or guideways. However, the sheer size of castings and even floor flatness deviations result in dimensional errors in these systems, which manifest as machine motion errors. Typical geometric alignment processes rely on an iterative approach, where measurements are taken to assess alignment (straightness, squareness, and parallelism), followed by adjustment of the machine supports (fixators or leveling pads), which can take weeks even for an experienced operator. Conversely, a novel method is proposed to shorten the correction time by eliminating the trial-and-error process in favor of a more deterministic approach guided by a finite element (FE) method. A feasibility study is conducted on a CNC polymer hybrid machine, with a steel weldment frame, supported by six leveling pads. An FE model of the frame is utilized to obtain recommended leveling pad adjustments, based on measurement of machine errors taken using a laser tracker. After a single adjustment cycle, measurements reveal that geometric errors of the machine tool are reduced from 2.22 mm of flatness deviation to 0.32 mm, achieving an 85.6% reduction. Furthermore, the entire process including measurement, adjustment, and assessment is completed in just 6 h by two operators who are not professional service engineers. In conclusion, this methodology demonstrates feasibility for scaling up, especially to large, high-precision CNC machine tools with bases mounted by fixators, offering the capability for bidirectional adjustment.

42 ENGINEERING↗
DOE OSTI2018

A Low-Rank Multigrid Method for the Stochastic Steady-State Diffusion Problem

We study a multigrid method for solving large linear systems of equations with tensor product structure here. Such systems are obtained from stochastic finite element discretization of stochastic partial differential equations such as the steady-state diffusion problem with random coefficients. When the variance in the problem is not too large, the solution can be well approximated by a low-rank object. In the proposed multigrid algorithm, the matrix iterates are truncated to low rank to reduce memory requirements and computational effort. The technique is proved convergent with an analytic error bound. Numerical experiments show its effectiveness in solving the Galerkin systems compared to the original multigrid solver, especially when the number of degrees of freedom associated with the spatial discretization is large.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

Parrallel Implementation of Fast Randomized Algorithms for Low Rank Matrix Decomposition

We analyze the parallel performance of randomized interpolative decomposition by de- composing low rank complex-valued Gaussian random matrices larger than 100 GB. We chose a Cray XMT supercomputer as it provides an almost ideal PRAM model permitting quick investigation of parallel algorithms without obfuscation from hardware idiosyncrasies. We obtain that on non-square matrices performance scales almost linearly with runtime about 100 times faster on 128 processors. We also verify that numerically discovered error bounds still hold on matrices two orders of magnitude larger than those previously tested.

Lucas, Andrew J.↗