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DOE OSTI2016

Efficient estimators for likelihood ratio sensitivity indices of complex stochastic dynamics

We demonstrate that centered likelihood ratio estimators for the sensitivity indices of complex stochastic dynamics are highly efficient with low, constant in time variance and consequently they are suitable for sensitivity analysis in long-time and steady-state regimes. These estimators rely on a new covariance formulation of the likelihood ratio that includes as a submatrix a Fisher information matrix for stochastic dynamics and can also be used for fast screening of insensitive parameters and parameter combinations. The proposed methods are applicable to broad classes of stochastic dynamics such as chemical reaction networks, Langevin-type equations and stochastic models in finance, including systems with a high dimensional parameter space and/or disparate decorrelation times between different observables. Furthermore, they are simple to implement as a standard observable in any existing simulation algorithm without additional modifications.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2014

Uncertainty quantification for the impact of injection rate fluctuation on the geomechanical response of geological carbon sequestration

We present an analysis of the geomechanical effects of injection rate fluctuations for geological sequestration of carbon dioxide (CO2). Initially, we present analytical solutions for the effects of injection rate fluctuations on CO2 fluid pressure spatial distribution and temporal evolution for a typical injection scenario. Numerical calculations are performed using a finite element method to investigate the effects of injection rate fluctuations on geomechanical deformation, stresses, and potential failure of the aquifer and caprock layers. The numerical method was first validated by the fluid pressure distribution’s good agreement with the analytical solution. It was shown that for any Gaussian fluctuations of injection rate Q with given mean Q and variance ε_Q, the coefficients of variance for fluid pressure (ϵ_p=ε_p ), deformation (ϵ_u=ε_u ), and stresses (ϵ_σ=ε_σ ) increase linearly with the coefficient of variance for injection rate (ϵ_Q=ε_Q ). The proportional constants are identified, and the fluctuations have the most pronounced effect on the geomechanical stresses, and, therefore, on the potential failure of the aquifer and caprock layers. Instead of expensive computational simulation, this study provides an efficient tool to estimate the geomechanical response variance to injection rate fluctuation. A failure analysis was presented based on the numerical results, where probability of failure was estimated for fluctuating injection rates with different mean and variance during the entire injection period. It was found that with increasing injection rate fluctuation, the failure probability increases significantly. Therefore, the risk associated with injection rate fluctuations should be carefully evaluated.

58 GEOSCIENCES↗

DOE OSTI2021

Characterizing Signatures of Geothermal Exploration Data with Machine Learning Techniques: An Application to the Nevada Play Fairway Analysis

We are introducing machine learning methods to the play fairway analysis to generate geothermal potential maps to support the evaluation of geothermal resource potential and the exploration for undiscovered blind geothermal systems in the Nevada Great Basin region. Our project aims to identify new ways to combine the play fairway data and empirically organize relationships between feature weights and labels in an improved workflow. As a means of doing this, we introduce machine learning methods to evaluate the influence of certain geological and geophysical features/feature sets in predicting geothermal favorability. This report highlights promising approaches based on supervised and unsupervised learning methods. First, we demonstrate a filter method applied to supervised classification modeling. The supervised filter method is based on permutation analysis to evaluate every possible feature combination/drop out scenario and rank feature influence based on the performance variance of supervised classification models. Additionally, we present an unsupervised factor analysis based on principal component analysis coupled with a semi-supervised kmeans clustering algorithm. This analysis allows us to identify the optimal number of groups/clusters for training sites and structural settings to identify feature patterns including correlation, variance, and latent and dominant feature relationships. The results from these methods offer a promising avenue for identifying favorable sources of predictive information to identify the locations of blind geothermal systems and furthering our understanding of complex geothermal feature and label relationships in the Great Basin region and beyond.

15 GEOTHERMAL ENERGY↗

DOE OSTI2026

A hybrid Monte Carlo-deterministic second moment method with efficient variance reduction

In this work, we present a hybrid method that combines Monte Carlo with deterministic finite element methods to solve a linear Boltzmann transport equation. Our hybrid method runs orders of magnitude faster than Monte Carlo, without sacrificing accuracy, for a proxy problem from radiative transfer that contains both optically-thick and optically-thin material. We believe that this is the first demonstration of a hybrid Second Moment Method in more than one spatial dimension, the first to consider more than one material, and the first to use variance reduction. Our variance reduction approach arises from an asymptotic analysis in which we show that the magnitude of the scattering source grows without bound. We transform the problem to compute the deviation of the radiation intensity from isotropy. The magnitude of the source in the transformed problem is bounded, and the quality of the hybrid method solution is dramatically improved by a substantial reduction in the variance.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2017

Ignoring correlation in uncertainty and sensitivity analysis in life cycle assessment: what is the risk?

Life cycle assessment (LCA) is an established tool to quantify the environmental impact of a product. A good assessment of uncertainty is important for making well-informed decisions in comparative LCA, as well as for correctly prioritising data collection efforts. Under- or overestimation of output uncertainty (e.g. output variance) will lead to incorrect decisions in such matters. The presence of correlations between input parameters during uncertainty propagation, can increase or decrease the the output variance. However, most LCA studies that include uncertainty analysis, ignore correlations between input parameters during uncertainty propagation, which may lead to incorrect conclusions. Two approaches to include correlations between input parameters during uncertainty propagation and global sensitivity analysis were studied: an analytical approach and a sampling approach. The use of both approaches is illustrated for an artificial case study of electricity production. Results demonstrate that both approaches yield approximately the same output variance and sensitivity indices for this specific case study. Furthermore, we demonstrate that the analytical approach can be used to quantify the risk of ignoring correlations between input parameters during uncertainty propagation in LCA. We demonstrate that: (1) we can predict if including correlations among input parameters in uncertainty propagation will increase or decrease output variance; (2) we can quantify the risk of ignoring correlations on the output variance and the global sensitivity indices. Moreover, this procedure requires only little data. - Highlights: • Ignoring correlation leads to under- or overestimation of the output variance. • We demonstrated that the risk of ignoring correlation can be quantified. • The procedure proposed is generally applicable in life cycle assessment. • In some cases, ignoring correlation has a minimal effect on decision-making tools.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

Extreme metrics from large ensembles: investigating the effects of ensemble size on their estimates

Abstract. We consider the problem of estimating the ensemble sizes required to characterize the forced component and the internal variability of a number of extreme metrics. While we exploit existing large ensembles, our perspective is that of a modeling center wanting to estimate a priori such sizes on the basis of an existing small ensemble (we assume the availability of only five members here). We therefore ask if such a small-size ensemble is sufficient to estimate accurately the population variance (i.e., the ensemble internal variability) and then apply a well-established formula that quantifies the expected error in the estimation of the population mean (i.e., the forced component) as a function of the sample size n, here taken to mean the ensemble size. We find that indeed we can anticipate errors in the estimation of the forced component for temperature and precipitation extremes as a function of n by plugging into the formula an estimate of the population variance derived on the basis of five members. For a range of spatial and temporal scales, forcing levels (we use simulations under Representative Concentration Pathway 8.5) and two models considered here as our proof of concept, it appears that an ensemble size of 20 or 25 members can provide estimates of the forced component for the extreme metrics considered that remain within small absolute and percentage errors. Additional members beyond 20 or 25 add only marginal precision to the estimate, and this remains true when statistical inference through extreme value analysis is used. We then ask about the ensemble size required to estimate the ensemble variance (a measure of internal variability) along the length of the simulation and – importantly – about the ensemble size required to detect significant changes in such variance along the simulation with increased external forcings. Using the F test, we find that estimates on the basis of only 5 or 10 ensemble members accurately represent the full ensemble variance even when the analysis is conducted at the grid-point scale. The detection of changes in the variance when comparing different times along the simulation, especially for the precipitation-based metrics, requires larger sizes but not larger than 15 or 20 members. While we recognize that there will always exist applications and metric definitions requiring larger statistical power and therefore ensemble sizes, our results suggest that for a wide range of analysis targets and scales an effective estimate of both forced component and internal variability can be achieved with sizes below 30 members. This invites consideration of the possibility of exploring additional sources of uncertainty, such as physics parameter settings, when designing ensemble simulations.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2023

Conformational space exploration of cryo-EM structures by variability refinement

Cryo-EM observation of biological samples enables visualization of sample heterogeneity, in the form of discrete states that are separable, or continuous heterogeneity as a result of local protein motion before flash freezing. Variability analysis of this continuous heterogeneity describes the variance between a particle stack and a volume, and results in a map series describing the various steps undertaken by the sample in the particle stack. While this observation is absolutely stunning, it is very hard to pinpoint structural details to elements of the maps. Here, in order to bridge the gap between observation and explanation, we designed a tool that refines an ensemble of structures into all the maps from variability analysis. Using this bundle of structures, it is easy to spot variable parts of the structure, as well as the parts that are not moving. Comparison with molecular dynamics simulations highlights the fact that the movements follow the same directions, albeit with different amplitudes. Ligand can also be investigated using this method. Variability refinement is available in the Phenix software suite, accessible under the program name phenix.varref.

59 BASIC BIOLOGICAL SCIENCES↗

NASA NTRS1992

Estimation of surface heat and moisture fluxes over a prairie grassland. II - Two-dimensional time filtering and site variability

The behavior of in situ measurements of surface fluxes obtained during FIFE 1987 is examined by using correlative and spectral techniques in order to assess the significance of fluctuations on various time scales, from subdiurnal up to synoptic, intraseasonal, and annual scales. The objectives of this analysis are: (1) to determine which temporal scales have a significant impact on areal averaged fluxes and (2) to design a procedure for filtering an extended flux time series that preserves the basic diurnal features and longer time scales while removing high frequency noise that cannot be attributed to site-induced variation. These objectives are accomplished through the use of a two-dimensional cross-time Fourier transform, which serves to separate processes inherently related to diurnal and subdiurnal variability from those which impact flux variations on the longer time scales. A filtering procedure is desirable before the measurements are utilized as input with an experimental biosphere model, to insure that model based intercomparisons at multiple sites are uncontaminated by input variance not related to true site behavior. Analysis of the spectral decomposition indicates that subdiurnal time scales having periods shorter than 6 hours have little site-to-site consistency and therefore little impact on areal integrated fluxes.

Crosson, William L.↗

NASA NTRS2012

The Transport of Density Fluctuations Throughout the Heliosphere

The solar wind is recognized as a turbulent magnetofluid, for which the properties of the turbulent velocity and magnetic field fluctuations are often described by the equations of incompressible magnetohydrodynamics (MHD). However, low-frequency density turbulence is also ubiquitous. On the basis of a nearly incompressible formulation of MHD in the expanding inhomogeneous solar wind, we derive the transport equation for the variance of the density fluctuations (Rho(exp 2)). The transport equation shows that density fluctuations behave as a passive scalar in the supersonic solar wind. In the absence of sources of density turbulence, such as within 1AU, the variance (Rho(exp 2)) approximates r(exp -4). In the outer heliosphere beyond 1 AU, the shear between fast and slow streams, the propagation of shocks, and the creation of interstellar pickup ions all act as sources of density turbulence. The model density fluctuation variance evolves with heliocentric distance within approximately 300 AU as (Rho(exp 2)) approximates r(exp -3.3) after which it flattens and then slowly increases. This is precisely the radial profile for the density fluctuation variance observed by Voyager 2. Using a different analysis technique, we confirm the radial profile for Rho(exp 2) of Bellamy, Cairns, & Smith using Voyager 2 data. We conclude that a passive scalar description for density fluctuations in the supersonic solar wind can explain the density fluctuation variance observed in both the inner and the outer heliosphere.

Zank, G. P.↗

DOE OSTI2021

Proof that Combining the Forced-collision and DXTRAN Monte Carlo Variance-reduction Techniques is Fair

This report provides mathematical proof that the MCNP DXTRAN (also known in other codes as forced-flight) and forced-collision variance-reduction techniques do not bias the expected value of Monte Carlo simulation estimates when combined. Proof is also provided that the techniques are unbiased when used independently. To prove that the techniques are unbiased, this report derives the first History Score Moment Equation (HSME) for non-multiplying media employing only forced collisions, only DXTRAN, and then both the variance-reduction techniques combined, as defined next. The HSMEs are found by forming the History Score Probability Density Functions (HSPDFs) and then taking the first score moment. Following its derivation, each HSME with a variance-reduction technique employed is reduced to the HSME for an analog simulation. Because the HSME represents the expected contribution to estimators in the simulation, reducing the HSME with variance reduction to the analog HSME shows that the simulation is unbiased despite the variance-reduction technique considered in the HSME. Analysis regarding higher-score moments of each technique is the subject of prior work and is not addressed herein. Throughout this work, the phase space p is defined to be the particle position x, direction-of-flight unit vector $\hat{Ω}$, energy E, and statistical weight w, $$p ≡ (x; \hat{Ω}; E; w).$$ A reduced phase-space excluding the statistical weight of the particle, $$r ≡ (x; \hat{Ω}; E),$$ is also used.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2015

Extension and Statistical Analysis of the GACP Aerosol Optical Thickness Record.

The primary product of the Global Aerosol Climatology Project (GACP) is a continuous record of the aerosol optical thickness (AOT) over the oceans. It is based on channel-1 and -2 radiance data from the Advanced Very High Resolution Radiometer (AVHRR) instruments flown on successive National Oceanic and Atmospheric Administration (NOAA) platforms. We extend the previous GACP dataset by four years through the end of 2009 using NOAA-17 and -18 AVHRR radiances recalibrated against MODerate resolution Imaging Spectroradiometer (MODIS) radiance data, thereby making the GACP record almost three decades long. The temporal overlap of over three years of the new NOAA-17 and the previous NOAA-16 record reveals an excellent agreement of the corresponding global monthly mean AOT values, thereby confirming the robustness of the vicarious radiance calibration used in the original GACP product. The temporal overlap of the NOAA-17 and -18 instruments is used to introduce a small additive adjustment to the channel-2 calibration of the latter resulting in a consistent record with increased data density. The Principal Component Analysis (PCA) of the newly extended GACP record shows that most of the volcanic AOT variability can be isolated into one mode responsible for ~12% of the total variance. This conclusion is confirmed by a combined PCA analysis of the GACP, MODIS, andMulti-angle Imaging SpectroRadiometer (MISR) AOTs during the volcano-free period fromFebruary 2000 to December 2009.We show that the modes responsible for the tropospheric AOT variability in the three datasets agree well in terms of correlation and spatial patterns. A previously identified negative AOT trend which started in the late 1980s and continued into the early 2000s is confirmed. Its magnitude and duration indicate that it was caused by changes in tropospheric aerosols. The latest multi-satellite segment of the GACP record shows that this trend tapered off, with no noticeable AOT change after 2002. This result is consistent with the MODIS andMISR AOT records as well as with the recent gradual reversal frombrightening to dimming revealed by surface flux measurements in many aerosol producing regions. Thus the robustness of the GACP record is confirmed, increasing our confidence in the validity of the negative trend. Although the nominal negative GACP AOT trend could partially be an artifact of increasing aerosol absorption, we argue that the time dependence of the GACP record, including the latest flat period, is more consistent with the actual decrease in the tropospheric AOT.

aerosols↗

DOE OSTI2022

The role of cool-flame fluctuations in high-pressure spray flames, studied using high-speed optical diagnostics and Large-Eddy Simulations

Here this work investigates the low- and high-temperature ignition and combustion processes, applied to the Engine Combustion Network Spray A flame, combining advanced optical diagnostics and large-eddy simulations (LES). Simultaneous high-speed (50 kHz) formaldehyde (CH 2 0) planar laser-induced fluorescence (PLIF) and line-of-sight OH* chemiluminescence imaging were used to measure the low- and high-temperature flame, during ignition as well as during quasi-steady combustion. While tracking the cool flame at the laser sheet plane, the present experimental setup allows detection of distinct ignition spots and dynamic fluctuations of the lift-off length over time, which overcomes limitations for flame tracking when using schlieren imaging. After significant development to improve LES prediction of the low-and high-temperature flame position, both during the ignition processes and quasi-steady combustion, the simulations were analyzed to gain understanding of the mixture variance and how this variance affects formation/consumption of CH 2 0. Analysis of the high-temperature ignition period shows that a key improvement in the LES is the ability to predict heterogeneous ignition sites, not only in the head of the jet, but in shear layers at the jet edge close to the position where flame lift-off eventually stabilizes. The LES analysis also shows concentrated pockets of CH 2 0, in the center of jet and at 20 mm downstream of the injector (in regions where the equivalence ratio is greater than 6), that are of similar length scale and frequency as the experiment (approximately 5–6 kHz). The periodic oscillation of CH 2 0 match the frequency of pressure waves generated during auto-ignition and reflected within the constant-volume vessel throughout injection. The ability of LES to capture the periodic appearance and destruction of CH 2 0 is particularly important because these structures travel downstream and become rich premixed flames that affect soot production.

42 ENGINEERING↗

DOE OSTI2025

Investigation of the Effect of Framework Flexibility on CO 2 Adsorption in SIFSIX-3-Cu Using a Machine-Learned Force Field

Metal–organic frameworks (MOFs) offer promise as selective CO 2 sorbents, but successful MOF sorbent materials need high CO 2 binding affinity and selectivity for CO 2 over water. This work focuses on the use of machine-learned force fields (MLFFs) to model CO 2 adsorption in flexible MOFs, with a focus on SIFSIX-3-Cu, an anion-pillared MOF known for its high CO 2 affinity. A preliminary high-throughput screening of over 900 anion-pillared MOFs was performed using rigid UFF+DDEC6 force fields to predict zero-loading heats of adsorption for CO 2 and H 2 O. SIFSIX-3-Cu was selected for further computational study due to its predicted CO 2 heat of adsorption and experimental relevance. A DeePMD-based MLFF was trained to reproduce DFT (PBE+D3) energies and forces, with an iterative sampling scheme combining molecular dynamics, geometry optimization, random geometric insertion, and NVT Monte Carlo-based configuration generation to capture both attractive and repulsive regions of the potential energy surface. Flexibility of the MOF was explicitly included, contrasting with previous models that approximated the MOF as rigid. Hybrid Monte Carlo/molecular dynamics (MC/MD) simulations with the MLFF produced CO 2 adsorption isotherms in good agreement with experimental data at direct air capture (DAC) pressures (e.g., 40 Pa), in contrast to previous overestimations of CO 2 sorption by models with rigid structures. Bond and angle histogram analysis showed that MOF flexibility increased the variance of fluorine–fluorine diagonal distances at adsorption sites, resulting in a lower predicted sorption for flexible, asymmetric SIFSIX-3-Cu pore geometries compared to the rigid, symmetric DFT-optimized SIFSIX-3-Cu pore geometry. A detailed description of flexibility afforded by the MLFF resulted in an accurately predicted CO 2 uptake (0.88 mmol/g) at low pressure (40 Pa) compared to the experimentally measured value (1.24 mmol/g). In conclusion, these results underscore the importance of including framework flexibility when modeling adsorption phenomena in MOFs, particularly for low-pressure applications.

adsorption↗

DOE OSTI2016

Planck 2015 results: XVI. Isotropy and statistics of the CMB

In this paper, we test the statistical isotropy and Gaussianity of the cosmic microwave background (CMB) anisotropies using observations made by the Planck satellite. Our results are based mainly on the full Planck mission for temperature, but also include some polarization measurements. In particular, we consider the CMB anisotropy maps derived from the multi-frequency Planck data by several component-separation methods. For the temperature anisotropies, we find excellent agreement between results based on these sky maps over both a very large fraction of the sky and a broad range of angular scales, establishing that potential foreground residuals do not affect our studies. Tests of skewness, kurtosis, multi-normality, N-point functions, and Minkowski functionals indicate consistency with Gaussianity, while a power deficit at large angular scales is manifested in several ways, for example low map variance. The results of a peak statistics analysis are consistent with the expectations of a Gaussian random field. The “Cold Spot” is detected with several methods, including map kurtosis, peak statistics, and mean temperature profile. We thoroughly probe the large-scale dipolar power asymmetry, detecting it with several independent tests, and address the subject of a posteriori correction. Tests of directionality suggest the presence of angular clustering from large to small scales, but at a significance that is dependent on the details of the approach. We perform the first examination of polarization data, finding the morphology of stacked peaks to be consistent with the expectations of statistically isotropic simulations. Finally, where they overlap, these results are consistent with the Planck 2013 analysis based on the nominal mission data and provide our most thorough view of the statistics of the CMB fluctuations to date.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2018

Optimizing Nuclear Material Accounting and Measurement Systems

The ability to create nuclear weapons from 235 U and 239 Pu makes it imperative to closely account for these materials as they progress through a nuclear fuel cycle. Improved measurement systems provide more accurate estimates of material quantities and material unaccounted for (MUF). This work provides examples of how two safeguards computational toolboxes can optimize and analyze hypothetical nuclear fuel cycle scenarios. The NUclear Measurement System Optimization (NUMSO) toolbox uses operations research techniques to find optimal solutions to safeguards measurement problems based on minimizing the variance of the estimated MUF. The SafeGuards Analysis (SGA) toolbox employs Monte Carlo techniques to analyze a given configuration of measurement methods and material flows to determine the probabilities of Type I (false detection) and Type II (missed detection) errors. Applying these toolboxes to a realistic fuel cycle scenario demonstrates the capability of NUMSO and SGA to address nuclear safeguards problems. Working in tandem, both toolboxes are able to determine how to quickly improve upon an existing safeguards measurement system and to calculate the resulting improvement in the error probabilities of the system. This information shows engineers not only how to develop new measurement systems but also how to improve existing systems in the most efficient manner.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

DOE OSTI2017

Multiscale analysis of the CMB temperature derivatives

We study the Planck CMB temperature at different scales through its derivatives up to second order, which allows one to characterize the local shape and isotropy of the field. The problem of having an incomplete sky in the calculation and statistical characterization of the derivatives is addressed in the paper. The analysis confirms the existence of a low variance in the CMB at large scales, which is also noticeable in the derivatives. Moreover, deviations from the standard model in the gradient, curvature and the eccentricity tensor are studied in terms of extreme values on the data. As it is expected, the Cold Spot is detected as one of the most prominent peaks in terms of curvature, but additionally, when the information of the temperature and its Laplacian are combined, another feature with similar probability at the scale of 10{sup o} is also observed. However, the p -value of these two deviations increase above the 6% when they are referred to the variance calculated from the theoretical fiducial model, indicating that these deviations can be associated to the low variance anomaly. Finally, an estimator of the directional anisotropy for spinorial quantities is introduced, which is applied to the spinors derived from the field derivatives. An anisotropic direction whose probability is <1% is detected in the eccentricity tensor.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2018

Model for scattering with proliferating resonances: Many coupled square wells

We present a multichannel model for elastic interactions, composed of an arbitrary number of coupled finite square-well potentials, and derive semianalytic solutions for its scattering behavior. Despite the model's simplicity, it is flexible enough to include many coupled short-ranged resonances in the vicinity of the collision threshold, as is necessary to describe ongoing experiments in ultracold molecules and lanthanide atoms. Here, we also introduce a simple but physically realistic statistical ensemble for parameters in this model. We compute the resulting probability distributions of nearest-neighbor resonance spacings and analyze them by fitting to the Brody distribution. We quantify the ability of alternative distribution functions, for resonance spacing and resonance number variance, to describe the crossover regime. The analysis demonstrates that the multichannel square-well model with the chosen ensemble of parameters naturally captures the crossover from integrable to chaotic scattering as a function of closed-channel coupling strength.

74 ATOMIC AND MOLECULAR PHYSICS↗

DOE OSTI2015

Stochastic Richardson Extrapolation Based Numerical Error Estimation for Kinetic Plasma Simulations

We present a numerical error estimation technique specifically formulated to deal with stochastic code output with multiple discretization parameters. This method is based on multiple fits to an error model with arbitrary convergence rates and cross-coupling terms, performed using nonlinear optimization. The fitting approach varies by the type of residual norm which influences the importance of outliers, and weights which influences the relative importance of data points in the coarse and refined regions of discretization parameter space. To account for the influence of stochastic noise, these fits are performed on multiple bootstrap values based on the underlying response data set. Using an automated discretization domain selection scheme, the fits are performed on a series of reduced sets of discretization levels in order to find an optimal fully-converged result estimate in the minimum variance sense; this automated approach enables straightforward analysis of multiple quantities of interest and/or time and spatially-dependent response data. The overall numerical error analysis method is useful for verification and validation problems for stochastic simulation methods and forms a key component in the overall uncertainty quantification process. The method was demonstrated for steady and unsteady electron diode problems simulated using a particle-in-cell kinetic plasma code, demonstrating excellent results.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗