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All sourcesNASA NTRSDOE OSTI
NASA NTRS2006

Detection of Water Hazards for Autonomous Robotic Vehicles

Four methods of detection of bodies of water are under development as means to enable autonomous robotic ground vehicles to avoid water hazards when traversing off-road terrain. The methods involve processing of digitized outputs of optoelectronic sensors aboard the vehicles. It is planned to implement these methods in hardware and software that would operate in conjunction with the hardware and software for navigation and for avoidance of solid terrain obstacles and hazards. The first method, intended for use during the day, is based on the observation that, under most off-road conditions, reflections of sky from water are easily discriminated from the adjacent terrain by their color and brightness, regardless of the weather and of the state of surface waves on the water. Accordingly, this method involves collection of color imagery by a video camera and processing of the image data by an algorithm that classifies each pixel as soil, water, or vegetation according to its color and brightness values (see figure). Among the issues that arise is the fact that in the presence of reflections of objects on the opposite shore, it is difficult to distinguish water by color and brightness alone. Another issue is that once a body of water has been identified by means of color and brightness, its boundary must be mapped for use in navigation. Techniques for addressing these issues are under investigation. The second method, which is not limited by time of day, is based on the observation that ladar returns from bodies of water are usually too weak to be detected. In this method, ladar scans of the terrain are analyzed for returns and the absence thereof. In appropriate regions, the presence of water can be inferred from the absence of returns. Under some conditions in which reflections from the bottom are detectable, ladar returns could, in principle, be used to determine depth. The third method involves the recognition of bodies of water as dark areas in short-wavelength infrared (SWIR) images. This method is based on the fact, well known among experts in remote sensing, that water bodies of any appreciable depth appear very dark in near-infrared, overhead imagery. Even under a thick layer of marine fog, SWIR illumination is present. Hence, this method may work even in the presence of clouds, though it is unlikely to work at night. Snow and ice also exhibit very strong absorption at wavelengths greater than about 1.4 m. Hence, the wavelength range of about 1.5 to 1.6 m might be useable in this method for recognizing water, snow, and ice. One notable drawback of this method is that useful look-ahead distance could be limited by surface reflections. The fourth method, intended for use at night, involves the contrast between water and terrain in thermal-infrared (medium-wavelength infrared) imagery. This method is based on the fact that at night, water is usually warmer than the adjacent terrain. Look-ahead distance could be limited in this method because, for reasons not yet fully understood, water appears to darken in the thermal infrared with increasing distance.

Matthes, Larry↗
DOE OSTI2016

Comparisons of Wilks’ and Monte Carlo Methods in Response to the 10CFR50.46(c) Proposed Rulemaking

The Nuclear Regulatory Commission (NRC) is proposing a new rulemaking on emergency core system/loss-of-coolant accident (LOCA) performance analysis. In the proposed rulemaking, designated as 10CFR50.46(c), the US NRC put forward an equivalent cladding oxidation criterion as a function of cladding pre-transient hydrogen content. The proposed rulemaking imposes more restrictive and burnup-dependent cladding embrittlement criteria; consequently nearly all the fuel rods in a reactor core need to be analyzed under LOCA conditions to demonstrate compliance to the safety limits. New analysis methods are required to provide a thorough characterization of the reactor core in order to identify the locations of the limiting rods as well as to quantify the safety margins under LOCA conditions. With the new analysis method presented in this work, the limiting transient case and the limiting rods can be easily identified to quantify the safety margins in response to the proposed new rulemaking. In this work, the best-estimate plus uncertainty (BEPU) analysis capability for large break LOCA with the new cladding embrittlement criteria using the RELAP5-3D code is established and demonstrated with a reduced set of uncertainty parameters. Both the direct Monte Carlo method and the Wilks’ nonparametric statistical method can be used to perform uncertainty quantification. Wilks’ method has become the de-facto industry standard to perform uncertainty quantification in BEPU LOCA analyses. Despite its widespread adoption by the industry, the use of small sample sizes to infer statement of compliance to the existing 10CFR50.46 rule, has been a major cause of unrealized operational margin in today’s BEPU methods. Moreover the debate on the proper interpretation of the Wilks’ theorem in the context of safety analyses is not fully resolved yet, even more than two decades after its introduction in the frame of safety analyses in the nuclear industry. This represents both a regulatory and application risk in rolling out new methods. With the 10CFR50.46(c) proposed rulemaking, the deficiencies of the Wilks’ approach are further exacerbated. The direct Monte Carlo approach offers a robust alternative to perform uncertainty quantification within the context of BEPU analyses. In this work, the Monte Carlo method is compared with the Wilks’ method in response to the NRC 10CFR50.46(c) proposed rulemaking.

10CFR50.46(c)↗
DOE OSTI2018

Progress Update on Segmentation of Neutron Tomographic Images, 2018

The present report documents the techniques developed to segment fast-neutron tomographic images of objects consisting of assemblies of machined parts with distinct boundaries. This report satisfies the fiscal year 2018 technical deliverable for project OR16-3DTomography-PD3Jb, “3D Tomography and Image Processing Using Fast Neutrons,” to report on the extension of geometric shape-finding algorithms to three dimensions. The project has two overall goals. The first of these goals is to extend associated-particle fast-neutron transmission and, particularly, induced-reaction tomographic imaging algorithms to three dimensions. This aspect of the project is beyond the scope of this report. The second goal is to automatically segment the resultant tomographic images into constituent parts and then extract information about the parts, such as the class of shape and potentially the shape parameters. This report describes progress for the image segmentation part of the project. Imaging techniques have been developed as high-confidence methods for confirming the presence and configuration of special nuclear materials (SNM). This high confidence is achieved at the cost of revealing considerable information that may be undesirable to share with the operator of the equipment. One potential way to make use of the high-confidence of imaging methods without revealing imaging data is to employ automated analysis that can extract meaningful attributes of the SNM without showing imaging data to the operator. An essential step of this automated analysis is the segmentation of the image into its constituent parts. For example, a three-dimensional (3D) fast-neutron tomographic image of an assembly containing uranium could be reconstructed and then segmented into discrete parts. The properties of a constituent part identified as uranium would be of interest and could be extracted. For instance, the volume and density of the part would together determine its mass. Moreover, its shape might be expected to fall into a particular geometric class, such as a cube, cylinder, or sphere, and the parameters of the shape (such as side length, height, or diameter) would be expected to fall in a given range. In such an analysis, automated identification of the boundaries and properties of the uranium part is an essential step. A key goal of the present work is to infer the boundaries of objects in fast-neutron tomographic images via the general method of image segmentation. Development of this capability can be broken into more manageable steps, including 1. operator-guided segmentation of two-dimensional (2D) fast-neutron tomographic images, 2. operator-guided segmentation of 3D fast-neutron tomographic images, 3. development of automated segmentation algorithms, and 4. extraction of shape parameters from constituent volumes in 3D. In the first year, the project team reported progress on the first step. Last year, progress on the second step, the extension into 3D, was reported with application to simulated data. The present report describes additional progress on the second step, and progress toward the third step, automated segmentation algorithms. In particular, a multiphase level set approach for image segmentation in three dimensions has been applied to measured 3D data for the first time. Activities related to the fourth step will also be reported, but it will focus on estimation of 3D volumes rather than shape parameters. In this report, the steps before image segmentation will be described, and the multiphase level sets approach to image segmentation will be briefly reviewed. This will be followed by a discussion of the issues encountered when applying the level set technique to measured 3D transmission images, and then a description of the inclusion of induced-reaction data will be provided. Finally, segmentation results for various measured target assemblies will be presented and the image segmentation portion of this project will be summarized and future directions will be discussed.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

Automated Bayesian high-throughput estimation of plasma temperature and density from emission spectroscopy

Here, this paper introduces a novel approach for automated high-throughput estimation of plasma temperature and density using atomic emission spectroscopy, integrating Bayesian inference with sophisticated physical models. We provide an in-depth examination of Bayesian methods applied to the complexities of plasma diagnostics, supported by a robust framework of physical and measurement models. Our methodology is demonstrated using experimental observations in the field of magneto-inertial fusion, focusing on individual and sequential shot analyses of the Plasma Liner Experiment at LANL. The results demonstrate the effectiveness of our approach in enhancing the accuracy and reliability of plasma parameter estimation and in using the analysis to reveal the deep hidden structure in the data. This study not only offers a new perspective of plasma analysis but also paves the way for further research and applications in nuclear instrumentation and related domains.

Bayesian inference↗
NASA NTRS1990

Calibration of polarimetric radar images using only image parameters and trihedral corner reflector responses

A technique that uses the radar return from natural targets and at least one trihedral corner reflector to calibrate compressed polarimetric radar data is described. Calibration for relative amplitude and phase, absolute amplitude, and system crosstalk is addressed. The crosstalk calibration method does not require any external calibration targets. Because compressed data are used, the transmitting and receiving systems as must be modeled as reciprocal. Even though the inferred transmit and receive matrices are not simply related to the physical systems, the true Stokes maxtrix for each pixel can be determined accurately. The method is illustrated by estimating the crosstalk parameters of the NASA/JPL aircraft for different types of terrain and for two frequencies. For a C-band system, the crosstalk is less than -20 dB for all ranges in the images. The crosstalk of the L-band system is a function of range and may be as poor as -10 dB in the near range, leading to a noticeable distortion of the polarization signatures.

Van Zyl, Jakob J.↗
DOE OSTI2017

On the Bayesian calibration of computer model mixtures through experimental data, and the design of predictive models

For many real systems, several computer models may exist with different physics and predictive abilities. To achieve more accurate simulations/predictions, it is desirable for these models to be properly combined and calibrated. Here, we propose the Bayesian calibration of computer model mixture method which relies on the idea of representing the real system output as a mixture of the available computer model outputs with unknown input dependent weight functions. The method builds a fully Bayesian predictive model as an emulator for the real system output by combining, weighting, and calibrating the available models in the Bayesian framework. Moreover, it fits a mixture of calibrated computer models that can be used by the domain scientist as a mean to combine the available computer models, in a flexible and principled manner, and perform reliable simulations. It can address realistic cases where one model may be more accurate than the others at different input values because the mixture weights, indicating the contribution of each model, are functions of the input. Inference on the calibration parameters can consider multiple computer models associated with different physics. The method does not require knowledge of the fidelity order of the models. We provide a technique able to mitigate the computational overhead due to the consideration of multiple computer models that is suitable to the mixture model framework. We implement the proposed method in a real-world application involving the Weather Research and Forecasting large-scale climate model.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Calorimetric classification of track-like signatures in liquid argon TPCs using MicroBooNE data

The MicroBooNE liquid argon time projection chamber located at Fermilab is a neutrino experiment dedicated to the study of short-baseline oscillations, the measurements of neutrino cross sections in liquid argon, and to the research and development of this novel detector technology. Accurate and precise measurements of calorimetry are essential to the event reconstruction and are achieved by leveraging the TPC to measure deposited energy per unit length along the particle trajectory, with mm resolution. We describe the non-uniform calorimetric reconstruction performance in the detector, showing dependence on the angle of the particle trajectory. Such non-uniform reconstruction directly affects the performance of the particle identification algorithms which infer particle type from calorimetric measurements. This work presents a new particle identification method which accounts for and effectively addresses such non-uniformity. The newly developed method shows improved performance compared to previous algorithms, illustrated by a 93.7% proton selection efficiency and a 10% muon mis-identification rate, with a fairly loose selection of tracks performed on beam data. The performance is further demonstrated by identifying exclusive final states in ν μ CC interactions. While developed using MicroBooNE data and simulation, this method is easily applicable to future LArTPC experiments, such as SBND, ICARUS, and DUNE.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2022

SPANet: Generalized permutationless set assignment for particle physics using symmetry preserving attention

The creation of unstable heavy particles at the Large Hadron Collider is the most direct way to address some of the deepest open questions in physics. Collisions typically produce variable-size sets of observed particles which have inherent ambiguities complicating the assignment of observed particles to the decay products of the heavy particles. Current strategies for tackling these challenges in the physics community ignore the physical symmetries of the decay products and consider all possible assignment permutations and do not scale to complex configurations. Attention based deep learning methods for sequence modelling have achieved state-of-the-art performance in natural language processing, but they lack built-in mechanisms to deal with the unique symmetries found in physical set-assignment problems. We introduce a novel method for constructing symmetry-preserving attention networks which reflect the problem's natural invariances to efficiently find assignments without evaluating all permutations. This general approach is applicable to arbitrarily complex configurations and significantly outperforms current methods, improving reconstruction efficiency between 19% - 35% on typical benchmark problems while decreasing inference time by two to five orders of magnitude on the most complex events, making many important and previously intractable cases tractable. A full code repository containing a general library, the specific configuration used, and a complete dataset release, are available at https://github.com/Alexanders101/SPANet

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Insight into Strong QCD and the Emergence of Mass from N* Experiments

Studies of nucleon resonance (N*) electroexcitation amplitudes (?vpN* electrocouplings) within a broad range of virtual photon four-momentum squared Q2 offer unique information on many facets of the strong interaction in the regime of large QCD running coupling (sQCD regime) seen in the generation of different resonances. The results on the ?vpN* electrocouplings from exclusive meson electroproduction data measured with the CLAS detector at JLab and their impact on understanding of sQCD dynamics will be presented in this talk. These CLAS data have provided the first and only available results on the evolution of the ?vpN* electrocouplings with Q2 to 5 GeV2 for most N* states in the mass range up to 1.8 GeV. A successful description of the CLAS results on the Q2-evolution of the ?(1232)3/2+, N(1440)1/2+, and ?(1600)3/2+ electrocouplings has been achieved within the continuum Schwinger method (CSM) by employing the same momentum dependence of the dressed quark mass inferred from the QCD Lagrangian, which also reproduces the experimental results on the pion elastic electromagnetic form factor and parton distribution function, and the nucleon elastic form factor. This success has demonstrated the capability for gaining insight into the emergence of hadron mass (EHM) from the exploration of the Q2-evolution of the ?vpN* electrocouplings. These studies also allow us to establish either universality or environmental sensitivity of the dressed quark mass function and to explore qq-correlation amplitudes of different spin-parities and the mixing between configurations of different orbital angular momentum. Exploration of the resonance electroexcitation with the CLAS12 detector and in the future with a possible 22 GeV machine at JLab offer the only foreseen opportunity to extend information on the ?vpN* electrocouplings for Q2 from 5-30 GeV2. Analyses of these results will cover the full range of distances where the dominant part of hadron mass and N* structure emerge from QCD.

Mokeev, Victor↗
DOE OSTI2025

Scalable Algorithms for Inverse Problems With High-Dimensional Parameter Spaces

Inverse problems, which involve inferring unknown parameters from observed data, present significant computational challenges, especially in large-scale settings with high-dimensional unknown parameters and nonlinear relationships between the unknowns and observations. Bayesian inference provides an approach for addressing these problems, often relying on sequential sampling methods like Markov chain Monte Carlo (MCMC) to approximate the posterior distribution of the parameters. However, MCMC methods become computationally demanding as the dimensionality of the problem increases, particularly in large-scale systems where likelihood evaluations rely on solving partial differential equations (PDEs) on large spatial domains with finely resolved meshes. To overcome these limitations, recent advancements have focused on designing scalable computa tional techniques – for both PDE simulations and sampling strategies – to make Bayesian methods feasible for high-dimensional problems.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1989

Global, seasonal cloud variations from satellite radiance measurements. I - Sensitivity of analysis

Global, daily, visible and IR radiance measurements from the NOAA-5 Scanning Radiometer (SR) are analyzed for the months of January, April, July and October, 1977 to infer cloud and surface radiative properties. In this paper, the data and analysis method are described. A unique feature of the method is that it utilizes radiative transfer models that simulate the SR measurements using explicit parameters representing the properties of the surface, atmosphere, and clouds. The accuracy of all the results depends primarily on the proper separation of the total radiance distribution into those parts representing clear and cloudy scenes.

Rossow, William B.↗
NASA NTRS2012

Astrophysical Model Selection in Gravitational Wave Astronomy

Theoretical studies in gravitational wave astronomy have mostly focused on the information that can be extracted from individual detections, such as the mass of a binary system and its location in space. Here we consider how the information from multiple detections can be used to constrain astrophysical population models. This seemingly simple problem is made challenging by the high dimensionality and high degree of correlation in the parameter spaces that describe the signals, and by the complexity of the astrophysical models, which can also depend on a large number of parameters, some of which might not be directly constrained by the observations. We present a method for constraining population models using a hierarchical Bayesian modeling approach which simultaneously infers the source parameters and population model and provides the joint probability distributions for both. We illustrate this approach by considering the constraints that can be placed on population models for galactic white dwarf binaries using a future space-based gravitational wave detector. We find that a mission that is able to resolve approximately 5000 of the shortest period binaries will be able to constrain the population model parameters, including the chirp mass distribution and a characteristic galaxy disk radius to within a few percent. This compares favorably to existing bounds, where electromagnetic observations of stars in the galaxy constrain disk radii to within 20%.

hierarchical↗
NASA NTRS2011

Time-Distance Helioseismology Data-Analysis Pipeline for Helioseismic and Magnetic Imager Onboard Solar Dynamics Observatory (SDO-HMI) and Its Initial Results

The Helioseismic and Magnetic Imager onboard the Solar Dynamics Observatory (SDO/HMI) provides continuous full-disk observations of solar oscillations. We develop a data-analysis pipeline based on the time-distance helioseismology method to measure acoustic travel times using HMI Doppler-shift observations, and infer solar interior properties by inverting these measurements. The pipeline is used for routine production of near-real-time full-disk maps of subsurface wave-speed perturbations and horizontal flow velocities for depths ranging from 0 to 20 Mm, every eight hours. In addition, Carrington synoptic maps for the subsurface properties are made from these full-disk maps. The pipeline can also be used for selected target areas and time periods. We explain details of the pipeline organization and procedures, including processing of the HMI Doppler observations, measurements of the travel times, inversions, and constructions of the full-disk and synoptic maps. Some initial results from the pipeline, including full-disk flow maps, sunspot subsurface flow fields, and the interior rotation and meridional flow speeds, are presented.

Sun: helioseismology↗
NASA NTRS2021

The First Widespread Solar Energetic Particle Event Observed By Solar Orbiter on 2020 November 29

Context. On 2020 November 29, the first widespread solar energetic particle (SEP) event of solar cycle 25 was observed at four widely separated locations in the inner (. 1AU) heliosphere. Relativistic electrons as well as protons with energies > 50MeV were observed by Solar Orbiter (SolO), Parker Solar Probe (PSP), the Solar Terrestrial Relations Observatory (STEREO)-A and multiple near-Earth spacecraft. The SEP event was associated with an M4.4 class X-ray flare and accompanied by a coronal mass ejection (CME) and an extreme ultraviolet (EUV) wave as well as a type II radio burst and multiple type III radio bursts. Aims. We present multi-spacecraft particle observations and place them in context with source observations from remote sensing instruments and discuss how such observations may further our understanding of particle acceleration and transport in this widespread event. Methods. Velocity dispersion analysis (VDA) and time shift analysis (TSA) were used to infer the particle release times at the Sun. Solar wind plasma and magnetic field measurements were examined to identify structures that influence the properties of the energetic particles such as their intensity. Pitch angle distributions and first-order anisotropies were analyzed in order to characterize the particle propagation in the interplanetary medium. Results. We find that during the 2020 November 29 SEP event, particles spread over more than 230° in longitude close to 1AU. The particle onset delays observed at the different spacecraft are larger as the flare–footpoint angle increases and are consistent with those from previous STEREO observations. Comparing the timing when the EUV wave intersects the estimated magnetic footpoints of each spacecraft with particle release times from TSA and VDA, we conclude that a simple scenario where the particle release is only determined by the EUV wave propagation is unlikely for this event. Observations of anisotropic particle distributions at SolO, Wind, and STEREO-A do not rule out that particles are injected over a wide longitudinal range close to the Sun. However, the low values of the first-order anisotropy observed by near-Earth spacecraft suggest that diffusive propagation processes are likely involved.

A Kollhoff↗
DOE OSTI2017

Facility Activity Inference Using Radiation Networks

We consider the problem of inferring the operational status of a reactor facility using measurements from a radiation sensor network deployed around the facility’s ventilation off-gas stack. The intensity of stack emissions decays with distance, and the sensor counts or measurements are inherently random with parameters determined by the intensity at the sensor’s location. We utilize the measurements to estimate the intensity at the stack, and use it in a one-sided Sequential Probability Ratio Test (SPRT) to infer on/off status of the reactor. We demonstrate the superior performance of this method over conventional majority fusers and individual sensors using (i) test measurements from a network of 21 NaI detectors, and (ii) effluence measurements collected at the stack of a reactor facility. We also analytically establish the superior detection performance of the network over individual sensors with fixed and adaptive thresholds by utilizing the Poisson distribution of the counts. We quantify the performance improvements of the network detection over individual sensors using the packing number of the intensity space.

Rao, Nageswara S.↗
DOE OSTI2026

Getting Warmer: IceCube Nears Freeze Out

IceCube has recently detected a diffuse population of high-energy neutrinos arising from the Milky Way. We use this high-significance detection to place new limits on dark matter (DM) annihilation to neutrinos with two complementary approaches. The first method uses the background-subtracted Galactic longitude distribution of shower events to place a conservative bound on the DM annihilation cross section that does not rely on any assumed Galactic cosmic ray emission model; the resulting limits on the velocity-averaged annihilation cross section improve upon existing bounds by factors of a few. The second method uses the template-dependent neutrino energy spectra from the Inner Galaxy, inferred under different Galactic cosmic ray emission models. This complementary approach shows that the inferred Galactic neutrino intensities are already sensitive to DM contributions near the thermal-relic benchmark for a range of TeV-scale DM masses, though this comparison is more model-dependent. Our results demonstrate that measurements of diffuse Galactic neutrino emission can be used as a powerful probe of DM annihilation into neutrinos. Future observations with IceCube-Gen2 and KM3NeT will substantially extend this sensitivity, potentially allowing a decisive test of the thermal freeze-out mechanism with Galactic neutrino observations.

Mukhopadhyay, Mainak [Fermilab; Chicago U., KICP; ↗
DOE OSTI2022

PyApprox: Enabling efficient model analysis

PyApprox is a Python-based one-stop-shop for probabilistic analysis of scientific numerical models. Easy to use and extendable tools are provided for constructing surrogates, sensitivity analysis, Bayesian inference, experimental design, and forward uncertainty quantification. The algorithms implemented represent the most popular methods for model analysis developed over the past two decades, including recent advances in multi-fidelity approaches that use multiple model discretizations and/or simplified physics to significantly reduce the computational cost of various types of analyses. Simple interfaces are provided for the most commonly-used algorithms to limit a user’s need to tune the various hyper-parameters of each algorithm. However, more advanced work flows that require customization of hyper-parameters is also supported. An extensive set of Benchmarks from the literature is also provided to facilitate the easy comparison of different algorithms for a wide range of model analyses. This paper introduces PyApprox and its various features, and presents results demonstrating the utility of PyApprox on a benchmark problem modeling the advection of a tracer in ground water.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey

Uncertainty quantification (UQ) includes the characterization, integration, and propagation of uncertainties that result from stochastic variations and a lack of knowledge or data in the natural world. Monte Carlo (MC) method is a sampling-based approach that has widely used for quantification and propagation of uncertainties. However, the standard MC method is often time-consuming if the simulation-based model is computationally intensive. This article gives an overview of modern MC methods to address the existing challenges of the standard MC in the context of UQ. Specifically, multilevel Monte Carlo (MLMC) extending the concept of control variates achieves a significant reduction of the computational cost by performing most evaluations with low accuracy and corresponding low cost, and relatively few evaluations at high accuracy and corresponding high cost. Multifidelity Monte Carlo (MFMC) accelerates the convergence of standard Monte Carlo by generalizing the control variates with different models having varying fidelities and varying computational costs. Multimodel Monte Carlo method (MMMC), having a different setting of MLMC and MFMC, aims to address the issue of UQ and propagation when data for characterizing probability distributions are limited. Multimodel inference combined with importance sampling is proposed for quantifying and efficiently propagating the uncertainties resulting from small data sets. All of these three modern MC methods achieve a significant improvement of computational efficiency for probabilistic UQ, particularly uncertainty propagation. Finally, an algorithm summary and the corresponding code implementation are provided for each of the modern MC methods. The extension and application of these methods are discussed in detail.

97 MATHEMATICS AND COMPUTING↗