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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 235 records · Page 13
All sourcesNASA NTRSDOE OSTI
NASA NTRS2013

Reducing Pointing Errors During Cassini Reaction Control System Orbit Trim Maneuvers

The effect of altering a gain parameter in the Cassini reaction control system (RCS) delta-V controller on the maneuver execution errors during orbit trim maneuvers (OTMs) is explored. Cassini consists of two reaction control thruster branches (A & B) each with eight thrusters. Currently, the B-branch is operational while the A-branch serves as a back-up. The four Z-thrusters control the X and Y-axes, while the four Y-thrusters control the Z-axis. During an OTM, the Z-thrusters fire to maintain the X and Y-axes pointing within an attitude control dead-zone (-10 to 10 milliradians). The errors do not remain at zero due to pointing error sources such as spacecraft center of mass offset from the geometric center of the Z-facing thrusters, and variability in the thruster forces due to the thruster hardware differences. The delta-V reaction control system (RCS) controller ensures that the attitude error remains within this dead-zone. Gain parameters within the RCS delta-V controller affect the maneuver execution errors. Different parameter values are used to explore effect on these errors. It is found that pointing error decreases and magnitude error increases rapidly for gain parameters 10 times greater than the current parameter values used in the flight software.

European Space Agency (ESA)↗
NASA NTRS2014

Data Analysis & Statistical Methods for Command File Errors

This paper explains current work on modeling for managing the risk of command file errors. It is focused on analyzing actual data from a JPL spaceflight mission to build models for evaluating and predicting error rates as a function of several key variables. We constructed a rich dataset by considering the number of errors, the number of files radiated, including the number commands and blocks in each file, as well as subjective estimates of workload and operational novelty. We have assessed these data using different curve fitting and distribution fitting techniques, such as multiple regression analysis, and maximum likelihood estimation to see how much of the variability in the error rates can be explained with these. We have also used goodness of fit testing strategies and principal component analysis to further assess our data. Finally, we constructed a model of expected error rates based on the what these statistics bore out as critical drivers to the error rate. This model allows project management to evaluate the error rate against a theoretically expected rate as well as anticipate future error rates.

Correlation Analysis↗
NASA NTRS2016

Effect of Vertical Rate Error on Recovery from Loss of Well Clear Between UAS and Non-Cooperative Intruders

When an Unmanned Aircraft System (UAS) encounters an intruder and is unable to maintain required temporal and spatial separation between the two vehicles, it is referred to as a loss of well-clear. In this state, the UAS must make its best attempt to regain separation while maximizing the minimum separation between itself and the intruder. When encountering a non-cooperative intruder (an aircraft operating under visual flight rules without ADS-B or an active transponder) the UAS must rely on the radar system to provide the intruders location, velocity, and heading information. As many UAS have limited climb and descent performance, vertical position andor vertical rate errors make it difficult to determine whether an intruder will pass above or below them. To account for that, there is a proposal by RTCA Special Committee 228 to prohibit guidance systems from providing vertical guidance to regain well-clear to UAS in an encounter with a non-cooperative intruder unless their radar system has vertical position error below 175 feet (95) and vertical velocity errors below 200 fpm (95). Two sets of fast-time parametric studies was conducted, each with 54000 pairwise encounters between a UAS and non-cooperative intruder to determine the suitability of offering vertical guidance to regain well clear to a UAS in the presence of radar sensor noise. The UAS was not allowed to maneuver until it received well-clear recovery guidance. The maximum severity of the loss of well-clear was logged and used as the primary indicator of the separation achieved by the UAS. One set of 54000 encounters allowed the UAS to maneuver either vertically or horizontally, while the second permitted horizontal maneuvers, only. Comparing the two data sets allowed researchers to see the effect of allowing vertical guidance to a UAS for a particular encounter and vertical rate error. Study results show there is a small reduction in the average severity of a loss of well-clear when vertical maneuvers are suppressed, for all vertical error rate thresholds examined. However, results also show that in roughly 35 of the encounters where a vertical maneuver was selected, forcing the UAS to do a horizontal maneuver instead increased the severity of the loss of well-clear for that encounter. Finally, results showed a small reduction in the number of severe losses of well-clear when the high performance UAS (2000 fpm climb and descent rate) was allowed to maneuver vertically, and the vertical rate error was below 500 fpm. Overall, the results show that using a single vertical rate threshold is not advisable, and that limiting a UAS to horizontal maneuvers when vertical rate errors are above 175 fpm can make a UAS less safe about a third of the time. It is suggested that the hard limit be removed, and system manufacturers instructed to account for their own UAS performance, as well as vertical rate error and encounter geometry, when determining whether or not to provide vertical guidance to regain well-clear.

detect and avoid↗
NASA NTRS2022

Using an Explainable Machine Learning Approach to Characterize Earth System Model Errors: Application of SHAP Analysis to Modeling Lightning Flash Occurrence

Computational models of the Earth System are critical tools for modern scientific inquiry. Effortstoward evaluating and improving errors in representations of physical and chemical processes inthese large computational systems are commonly stymied by highly nonlinear and complexerror behavior. Recent work has shown that these errors can be effectively predicted usingmodern Artificial Intelligence (A.I.) techniques. In this work, we go beyond these previousstudies to apply an interpretable A.I. technique to not only predict model errors but also movetoward understanding the underlying reasons for successful error prediction. We use XGBoostclassification trees and SHapley Additive exPlanations (SHAP) analysis to explore the errors inthe prediction of lightning occurrence in the NASA GEOS model, a widely used Earth SystemModel. This explainable error prediction system can effectively predict the model error andindicates that the errors are strongly related to convective processes and the characteristics ofthe land surface.

Artificial intelligence↗
DOE OSTI2021

TQEA: Temporal Quantum Error Analysis

For the last few years quantum computers have captured researchers’ interest in solving complicated problems in finance, chemistry, machine learning, and many other NP problems. Quantum computing benefits from parallel computation, entanglement, and calculation speedup compared to classical computing. Even though the operations and number of qubits in quantum computers are limited, previous research showed that using quantum computers can be beneficial. These benefits gained researchers’ interest to work on the improvement of quantum computers and overcome the limitations by building more advanced quantum computers with fault tolerance capability, more physical qubits, and better operational gates. In practice, Quantum applications would be translated into Quantum Circuits and each circuit can contain one or many entanglements (s), single qubits operations, and readouts. The importance of running Quantum applications with controlled errors or without errors is critical to the desired result from Quantum readout. In this paper, we study the characteristics of superconducting quantum machines along with the previously proposed methods of Quantum error detection/correction and in order to build a better error model we propose a temporal method for analyzing the the behavior of qubits errors (T1 and T2), frequency of operation, entanglement errors, and readout errors. In order to detect and predict the behavior of each machine, we use simple exponential smoothing analysis on heuristic calibration data. Finally, based on observation, we show that based on frequency, qubits, readout and single operation errors we can build a temporal based model to study and predict the outcome of runs on each Quantum machine.

Baheri, Betis↗
NASA NTRS1994

A New Class of Single Event Hard Errors

This paper reports on hard errors induced by single ions in dynamic memories. For ions with atomic number below 80, hard errors in DRAMs appear to be similar to the hard errors reported in previous work on SRAMs. One feature of these hard errors is that they tend to recover gradually with time, because of annealing, and are thus partially recoverable. However, for gold ions, a second type of hard error was discovered which is not recoverable, and appears to be due to catastrophic internal shorting rather than small changes in leakage current. Thus, nonrecoverable errors will likely occur even in devices which eliminate the extreme sensitivity to leakage current that is inherent in 4-T SRAMs and DRAMs. It is important to understand the mechanism that is responsible for nonrecoverable errors, and investigate the effect of device scaling.

single event hard errors DRAM SRAM ions gold ions ↗
DOE OSTI2020

Time-series machine-learning error models for approximate solutions to parameterized dynamical systems

This work proposes a machine-learning framework for modeling the error incurred by approximate solutions to parameterized dynamical systems. In particular, we extend the machine-learning error models (MLEM) framework proposed in Ref. Freno and Carlberg (2019) to dynamical systems. The proposed Time-Series Machine-Learning Error Modeling (T-MLEM) method constructs a regression model that maps features – which comprise error indicators that are derived from standard a posteriori error-quantification techniques – to a random variable for the approximate-solution error at each time instance. The proposed framework considers a wide range of candidate features, regression methods, and additive noise models. We consider primarily recursive regression techniques developed for time-series modeling, including both classical time-series models (e.g., autoregressive models) and recurrent neural networks (RNNs), but also analyze standard non-recursive regression techniques (e.g., feed-forward neural networks) for comparative purposes. Finally, numerical experiments conducted on multiple benchmark problems illustrate that the long short-term memory (LSTM) neural network, which is a type of RNN, outperforms other methods and yields substantial improvements in error predictions over traditional approaches.

42 ENGINEERING↗
DOE OSTI2023

Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems

We explore using neural operators, or neural network representations of nonlinear maps between function spaces, to accelerate infinite-dimensional Bayesian inverse problems (BIPs) with models governed by nonlinear parametric partial differential equations (PDEs). Neural operators have gained significant attention in recent years for their ability to approximate the parameter-to-solution maps defined by PDEs using as training data solutions of PDEs at a limited number of parameter samples. The computational cost of BIPs can be drastically reduced if the large number of PDE solves required for posterior characterization are replaced with evaluations of trained neural operators. However, reducing error in the resulting BIP solutions via reducing the approximation error of the neural operators in training can be challenging and unreliable. We provide an a priori error bound result that implies certain BIPs can be ill-conditioned to the approximation error of neural operators, thus leading to inaccessible accuracy requirements in training. To reliably deploy neural operators in BIPs, we consider a strategy for enhancing the performance of neural operators: correcting the prediction of a trained neural operator by solving a linear variational problem based on the PDE residual. We show that a trained neural operator with error correction can achieve a quadratic reduction of its approximation error, all while retaining substantial computational speedups of posterior sampling when models are governed by highly nonlinear PDEs. The strategy is applied to two numerical examples of BIPs based on a nonlinear reaction–diffusion problem and deformation of hyperelastic materials. We demonstrate that posterior representations of the two BIPs produced using trained neural operators are greatly and consistently enhanced by error correction.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Quantifying UAS Observation Error Variance Used in Data Assimilation Systems and Its Impact on Predictive Skill

Observation error determines the weights of the observations and background state used in data assimilation to generate analyses. Quantifying observation error is critical for the optimal assimilation of observational data sets. Uncrewed Aircraft System (UAS) observations have shown potential benefits in filling observational gaps in the lower atmosphere; however, characterization of their error characteristics has been limited. To optimize the use of UAS observations in numerical weather prediction, UAS observation error is estimated based on the 3‐cornered hat diagnostic approach which uses three independent estimates of the atmospheric state. This approach is applied to data from the 2018 Lower Atmospheric Profiling Studies at Elevation‐a Remotely‐piloted Aircraft Team Experiment field campaign using collocated UAS and rawinsonde observations along with output from a set of convection‐permitting model simulations. The estimated observation error values for UAS temperature, wind, and relative humidity measurements were found to be only weakly dependent on height AGL with mean values equal to 0.5°C, 0.8 m s −1 , and 3%, respectively. Only the newly estimated observation error for temperature differed from that previously used to assimilate commercial aircraft observations into global models (1.0°C). However, using this reduced temperature observation error produced more accurate mesoscale analyses and forecasts of both terrain‐driven flows and convection initiation generated by colliding outflow boundaries within the San Luis Valley of Colorado.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

A Framework for Error-Bounded Approximate Computing, with an Application to Dot Products

Approximate computing techniques, which trade off the computation accuracy of an algorithm for better performance and energy efficiency, have been successful in reducing computation and power costs in several domains. However, error sensitive applications in high-performance computing are unable to benefit from existing approximate computing strategies that are not developed with guaranteed error bounds. While approximate computing techniques can be developed for individual high-performance computing applications by domain specialists, this often requires additional theoretical analysis and potentially extensive software modification. Hence, the development of low-level error-bounded approximate computing strategies that can be introduced into any high-performance computing application without requiring additional analysis or significant software alterations is desirable. In this paper, we provide a contribution in this direction by proposing a general framework for designing error-bounded approximate computing strategies and apply it to the dot product kernel to develop \bf qdot---an error-bounded approximate dot product kernel. Following the introduction of qdot, here we perform a theoretical analysis that yields a deterministic bound on the relative approximation error introduced by qdot. Empirical tests are performed to illustrate the tightness of the derived error bound and to demonstrate the effectiveness of qdot on a synthetic dataset, as well as two scientific benchmarks---the conjugate gradient (CG) and power methods. In some instances, using qdot for the dot products in CG can result in many components being quantized to half precision without increasing the iteration count required for convergence to the same solution as CG using a double precision dot product.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Better Than Worst-Case Decoding for Quantum Error Correction

The overheads of classical decoding for quantum error correction in cryogenic quantum systems grow rapidly with the number of logical qubits and their correction code distance. Decoding at room temperature is bottlenecked by refrigerator I/O bandwidth while cryogenic on-chip decoding is limited by area/power/thermal budget. To overcome these overheads, we are motivated by the observation that in the common case (over 90% of the time), error correction 'syndromes' are fairly trivial with high redundancy / sparsity, since the error correction codes are over-provisioned to be able to correct for uncommon worst-case complex scenarios (to ensure substantially low logical error rates). If suitably exploited, these trivial scenarios can be handled with insignificant overhead, thereby alleviating any bottlenecks towards handling the worst-case scenarios by state-of-the-art means. Here, we propose Better Than Worst-Case Decoding for Quantum Error Correction, targeting cryogenic quantum systems and Surface Code, consisting of: On-chip Clique Decoder: An extremely lightweight decoder for correcting trivial common-case errors, designed for the cryogenic domain. The decoder is implemented and evaluated for SFQ logic. Statistical Off-chip Bandwidth Allocation: A statistical confidence-based technique for allocation of off-chip decoding bandwidth, to efficiently handle the rare complex decodes that are not covered by the Clique Decoder. Decode-Overflow Execution Stalling: A method to stall circuit execution, for the worst-case scenarios in which the provisioned off-chip bandwidth is insufficient to complete all requested off-chip decodes. In all, BTWC decoding achieves 70-99+% off-chip bandwidth elimination across a range of logical and physical error rates, without significantly sacrificing the accuracy of a state-of-the-art off-chip decoder. Further, it achieves 10-1000x bandwidth reduction over prior bandwidth reduction techniques, as well as 15-37x resource overhead reduction compared to prior on-chip decoding.

cryogenic optimization↗
DOE OSTI2023

diffReplication - An Energy-Aware Fault Tolerance Model for Silent Error Detection and Mitigation in Heterogeneous Extreme-scale Computing Environment

At extreme scale, the frequency of silent errors – a class of errors that remain undetected by low-level error detection mechanisms – increases significantly with the computational complexity of the application and the scale of the computing infrastructure. As hardware and software advances are made to usher in the next scientific era of computing, developing new approaches to mitigate the impact of silent errors remains a challenging problem. In this work, we propose an energy-aware fault-tolerance model, referred to diffReplication to overcome silent errors. In the proposed model, the main process is associated with one replica that executes at the same rate as the main process, and one diffReplica that is executed at a fraction of the main process' execution rate. If the main and its replica reach consensus at the end of a computation phase, the state of the diffReplica is updated and computation is resumed. If the synchronization attempt results in a disagreement, however, the diffReplica increases its execution speed to complete the computation and quickly reach the synchronization barrier. Assuming a single error over any given synchronization interval, a majority voting is used to reach consensus and tolerate silent errors. To further enhance its performance, diffReplication is augmented with speculative execution, whereby the main or its fast replica is selected to continue execution without waiting for the diffReplica. The selection process is based on the previous behaviour of the main and its replica. A performance analysis study is carried out to assess the performance of diffReplication, in terms of the energy saving and time-to-completion reduction achieved by the diffReplication scheme. The experiment shows that speculative execution reduces the time to completion with additional energy, and dynamic decision-making balances the energy consumption and time to completion.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2013

Interpreting SBUV Smoothing Errors: an Example Using the Quasi-biennial Oscillation

The Solar Backscattered Ultraviolet (SBUV) observing system consists of a series of instruments that have been measuring both total ozone and the ozone profile since 1970. SBUV measures the profile in the upper stratosphere with a resolution that is adequate to resolve most of the important features of that region. In the lower stratosphere the limited vertical resolution of the SBUV system means that there are components of the profile variability that SBUV cannot measure. The smoothing error, as defined in the optimal estimation retrieval method, describes the components of the profile variability that the SBUV observing system cannot measure. In this paper we provide a simple visual interpretation of the SBUV smoothing error by comparing SBUV ozone anomalies in the lower tropical stratosphere associated with the quasi-biennial oscillation (QBO) to anomalies obtained from the Aura Microwave Limb Sounder (MLS). We describe a methodology for estimating the SBUV smoothing error for monthly zonal mean (mzm) profiles. We construct covariance matrices that describe the statistics of the inter-annual ozone variability using a 6 yr record of Aura MLS and ozonesonde data. We find that the smoothing error is of the order of 1percent between 10 and 1 hPa, increasing up to 15-20 percent in the troposphere and up to 5 percent in the mesosphere. The smoothing error for total ozone columns is small, mostly less than 0.5 percent. We demonstrate that by merging the partial ozone columns from several layers in the lower stratosphere/troposphere into one thick layer, we can minimize the smoothing error. We recommend using the following layer combinations to reduce the smoothing error to about 1 percent: surface to 25 hPa (16 hPa) outside (inside) of the narrow equatorial zone 20 S-20 N.

ozone profile retrievals↗
NASA NTRS2012

Achieving Accuracy Requirements for Forest Biomass Mapping: A Spaceborne Data Fusion Method for Estimating Forest Biomass and Lidar Sampling Error

The synergistic use of active and passive remote sensing (i.e., data fusion) demonstrates the ability of spaceborne light detection and ranging (LiDAR), synthetic aperture radar (SAR) and multispectral imagery for achieving the accuracy requirements of a global forest biomass mapping mission (+/-20 Mg/ha or 20%, the greater of the two, for at least 80% of grid cells). A data fusion approach also provides a means to extend 3D information from discrete spaceborne LiDAR measurements of forest structure across scales much larger than that of the LiDAR footprint. For estimating biomass, these measurements mix a number of errors including those associated with LiDAR footprint sampling over regional-global extents. A general framework for mapping above ground live forest biomass density (AGB) with a data fusion approach is presented and verified using data from NASA field campaigns near Howland, ME, USA, to assess AGB and LiDAR sampling errors across a regionally representative landscape. We combined SAR and Landsat-derived optical (passive optical) image data to identify contiguous areas (>0.5 ha) that are relatively homogenous in remote sensing metrics (forest patches). We used this image-derived data with simulated spaceborne LiDAR derived from orbit and cloud cover simulations and airborne data from NASA's Laser Vegetation Imaging Sensor (LVIS) to compute AGB and estimate LiDAR sampling error for forest patches and 100 m, 250 m, 500 m, and 1 km grid cells. At both the patch and grid scales, we evaluated differences in AGB estimation and sampling error from the combined use of LiDAR with both SAR and passive optical and with either SAR or passive optical alone. First, this data fusion approach demonstrates that incorporating forest patches into the AGB mapping framework can provide sub-grid forest information for coarser grid-level AGB reporting. Second, a data fusion approach for estimating AGB using simulated spaceborne LiDAR with SAR and passive optical image combinations reduced forest AGB sampling errors 12%-38% from those where LiDAR is used with SAR or passive optical alone. In absolute terms, sampling errors were reduced from 14-40 Mg/ha to 11-28 Mg/ha across all grid scales and prediction methods, where minimum sampling errors were 11, 15, 18, and 22 Mg/ha for 1 km, 500 m, 250 m, and 100 m grid scales, respectively. Third, spaceborne global scale accuracy requirements were achieved whereby at least 80% of the grid cells at 100 m, 250 m, 500 m, and 1 km grid levels met AGB accuracy requirements using a combination of passive optical and SAR along with machine learning methods to predict vegetation structure metrics for forested areas without LiDAR samples. Finally, using either passive optical or SAR, accuracy requirements were met at the 500 m and 250 m grid level, respectively..

LiDAR↗
DOE OSTI2022

Experimental demonstration of continuous quantum error correction

The storage and processing of quantum information are susceptible to external noise, resulting in computational errors. A powerful method to suppress these effects is quantum error correction. Typically, quantum error correction is executed in discrete rounds, using entangling gates and projective measurement on ancillary qubits to complete each round of error correction. Here we use direct parity measurements to implement a continuous quantum bit-flip correction code in a resource-efficient manner, eliminating entangling gates, ancillary qubits, and their associated errors. An FPGA controller actively corrects errors as they are detected, achieving an average bit-flip detection efficiency of up to 91%. Furthermore, the protocol increases the relaxation time of the protected logical qubit by a factor of 2.7 over the relaxation times of the bare comprising qubits. Our results showcase resource-efficient stabilizer measurements in a multi-qubit architecture and demonstrate how continuous error correction codes can address challenges in realizing a fault-tolerant system.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Classifying handedness in chiral nanomaterials using label error robust deep learning

Abstract High-throughput scanning electron microscopy (SEM) coupled with classification using neural networks is an ideal method to determine the morphological handedness of large populations of chiral nanoparticles. Automated labeling removes the time-consuming manual labeling of training data, but introduces label error, and subsequently classification error in the trained neural network. Here, we evaluate methods to minimize classification error when training from automated labels of SEM datasets of chiral Tellurium nanoparticles. Using the mirror relationship between images of opposite handed particles, we artificially create populations of varying label error. We analyze the impact of label error rate and training method on the classification error of neural networks on an ideal dataset and on a practical dataset. Of the three training methods considered, we find that a pretraining approach yields the most accurate results across label error rates on ideal datasets, where size and other morphological variables are held constant, but that a co-teaching approach performs the best in practical application.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Group-theoretic error mitigation enabled by classical shadows and symmetries

Abstract Estimating expectation values is a key subroutine in quantum algorithms. Near-term implementations face two major challenges: a limited number of samples required to learn a large collection of observables, and the accumulation of errors in devices without quantum error correction. To address these challenges simultaneously, we develop a quantum error-mitigation strategy called symmetry-adjusted classical shadows , by adjusting classical-shadow tomography according to how symmetries are corrupted by device errors. As a concrete example, we highlight global U(1) symmetry, which manifests in fermions as particle number and in spins as total magnetization, and illustrate their group-theoretic unification with respective classical-shadow protocols. We establish rigorous sampling bounds under readout errors obeying minimal assumptions, and perform numerical experiments with a more comprehensive model of gate-level errors derived from existing quantum processors. Our results reveal symmetry-adjusted classical shadows as a low-cost strategy to mitigate errors from noisy quantum experiments in the ubiquitous presence of symmetry.

Zhao, Andrew (ORCID:0000000202990277)↗
DOE OSTI2023

Error-Field Penetration Thresholds in Ohmically Heated ITER and SPARC Plasmas

The critical n=1, n=2, and n=3 error-field amplitudes needed to trigger error-field penetration in ITER and SPARC, steady-state, ohmically heated plasmas are calculated using a standard asymptotic matching approach. The calculation incorporates plasma impurities, trapped particles, the bootstrap current, and neoclassical poloidal flow-damping. The energy confinement time is specified by the neo-Alcator scaling law in the low-density linear ohmic confinement (LOC) regime, and by the ITER-89P L-mode scaling law in the high-density saturated ohmic confinement regime (SOC) regime. The response of the plasma in the inner region is calculated using a linearized version of the four-field model. At the normal operating electron number density, diamagnetic levels of rotation are found to be sufficient to protect ITER and SPARC ohmically heated plasmas from m=2/n=1 error-field penetration. On other hand, SPARC, and especially ITER, ohmically heated plasmas may be vulnerable to n>1 error-field penetration. ITER and SPARC ohmically heated plasmas are also slightly more susceptible to error-field penetration when the electron fluid at the rational surface rotates in the ion diamagnetic direction, rather than the electron diamagnetic direction. At electron number densities that are sufficiently low that the plasma lies in the LOC confinement regime, the error-field penetration threshold increases with increasing density. However, as soon as the electron number density becomes large enough that the plasma enters the SOC regime, the increase in the error-field penetration threshold with increasing density levels off.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗