SEARCH · Engineering Papers

Results for “Implicit”

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.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 1,153 records · Page 64

All sources NASA NTRS DOE OSTI

DOE OSTI2017

Peridynamics using Irregular Domain Discretization with MOOSE-Based Implementation

In this paper, reformulation of classical bond-based peridynamic thermomechanical model for irregular domain decomposition and its MOOSE-based implicit formulation are presented. First, the irregular grid based peridynamic thermomechanical model is formulated and model parameters are derived. Following this, an implicit formulation for the solution of static or quasi-static problems is presented. Some aspects of the MOOSE-based implementation are given. After that, the formulation is verified against benchmark solutions for thermomechanic problems. Crack initiation and propagation in circular (2D) and cylindrical (3D) nuclear fuels at high temperature are studied using irregular grids.

42 ENGINEERING↗

DOE OSTI2013

Experimental and computational results on exciton/free-carrier ratio, hot/thermalized carrier diffusion, and linear/nonlinear rate constants affecting scintillator proportionality

Models of nonproportional response in scintillators have highlighted the importance of parameters such as branching ratios, carrier thermalization times, diffusion, kinetic order of quenching, associated rate constants, and radius of the electron track. For example, the fraction ηeh of excitations that are free carriers versus excitons was shown by Payne and coworkers to have strong correlation with the shape of electron energy response curves from Compton-coincidence studies. Rate constants for nonlinear quenching are implicit in almost all models of nonproportionality, and some assumption about track radius must invariably be made if one is to relate linear energy deposition dE/dx to volume-based excitation density n (eh/cm3) in terms of which the rates are defined. Diffusion, affecting time-dependent track radius and thus density of excitations, has been implicated as an important factor in nonlinear light yield. Several groups have recently highlighted diffusion of hot electrons in addition to thermalized carriers and excitons in scintillators. However, experimental determination of many of these parameters in the insulating crystals used as scintillators has seemed difficult. Subpicosecond laser techniques including interband z scan light yield, fluence-dependent decay time, and transient optical absorption are now yielding experimental values for some of the missing rates and ratios needed for modeling scintillator response. First principles calculations and Monte Carlo simulations can fill in additional parameters still unavailable from experiment. As a result, quantitative modeling of scintillator electron energy response from independently determined material parameters is becoming possible on an increasingly firmer data base. This paper describes recent laser experiments, calculations, and numerical modeling of scintillator response.

Williams, Richard↗

DOE OSTI2014

Task-based detectability in CT image reconstruction by filtered backprojection and penalized likelihood estimation

Purpose: Nonstationarity is an important aspect of imaging performance in CT and cone-beam CT (CBCT), especially for systems employing iterative reconstruction. This work presents a theoretical framework for both filtered-backprojection (FBP) and penalized-likelihood (PL) reconstruction that includes explicit descriptions of nonstationary noise, spatial resolution, and task-based detectability index. Potential utility of the model was demonstrated in the optimal selection of regularization parameters in PL reconstruction. Methods: Analytical models for local modulation transfer function (MTF) and noise-power spectrum (NPS) were investigated for both FBP and PL reconstruction, including explicit dependence on the object and spatial location. For FBP, a cascaded systems analysis framework was adapted to account for nonstationarity by separately calculating fluence and system gains for each ray passing through any given voxel. For PL, the point-spread function and covariance were derived using the implicit function theorem and first-order Taylor expansion according toFessler [“Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): Applications to tomography,” IEEE Trans. Image Process. 5(3), 493–506 (1996)]. Detectability index was calculated for a variety of simple tasks. The model for PL was used in selecting the regularization strength parameter to optimize task-based performance, with both a constant and a spatially varying regularization map. Results: Theoretical models of FBP and PL were validated in 2D simulated fan-beam data and found to yield accurate predictions of local MTF and NPS as a function of the object and the spatial location. The NPS for both FBP and PL exhibit similar anisotropic nature depending on the pathlength (and therefore, the object and spatial location within the object) traversed by each ray, with the PL NPS experiencing greater smoothing along directions with higher noise. The MTF of FBP is isotropic and independent of location to a first order approximation, whereas the MTF of PL is anisotropic in a manner complementary to the NPS. Task-based detectability demonstrates dependence on the task, object, spatial location, and smoothing parameters. A spatially varying regularization “map” designed from locally optimal regularization can improve overall detectability beyond that achievable with the commonly used constant regularization parameter. Conclusions: Analytical models for task-based FBP and PL reconstruction are predictive of nonstationary noise and resolution characteristics, providing a valuable framework for understanding and optimizing system performance in CT and CBCT.

62 RADIOLOGY AND NUCLEAR MEDICINE↗

DOE OSTI2014

SU-D-16A-07: Photobleaching Predicts Necrosis in Interstitial PDT

Purpose: Dosimetry for PDT has proven to be a challenge thus far, and for prediction of PDT outcome, a singlet oxygen model based on fundamental photophysical parameters has been developed. Previously, the photobleaching effect of photosensitizers was taken into account in the singlet oxygen explicit dosimetry model; here we report of direct measurements of photobleaching in the same model to assess the conditions under which implicit dosimetry using photobleaching can serve as an intermediate surrogate for PDT damage. Methods: Fluorescence spectra were measured interstitially in sensitized mouse tumors prior to after irradiation via a cylindrical diffuser. Photobleaching was determined by the relative decrease in fluorescence amplitude from the initial pre-treatment measurement. Spectra were analyzed by singular value decomposition to determine the photosensitizer concentration. Different photosensitizers were used to see the effect of photobleaching on PDT outcome and the impact of fluence on photobleaching. The drugs used were BPD (at two drug-light intervals), HPPH, and Photofrin. PDT outcome was determined by tumor necrosis radii measured upon sectioning and staining of treated tumors. Results: Post-PDT photosentizer concentrations were compared to initial pre-PDT photosensitizer concentrations, and the decrease was greater with a higher fluence measured during treatment. Furthermore, photobleaching and necrosis radius were found to be positively correlated. The relationship between photobleaching and necrosis radius is sensitizer-dependent, however the differences among sensitizers can be understood in terms of their respective photophysical parameters. Conclusions: Photobleaching is predictive of PDT outcome, but a comprehensive singlet oxygen model, has the potential to further improve the prediction of PDT outcome and the understanding of implicit dosimetry.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2016

Contour interpolated radial basis functions with spline boundary correction for fast 3D reconstruction of the human articular cartilage from MR images

Purpose: The aim of this work is to demonstrate a new image processing technique that can provide a “near real-time” 3D reconstruction of the articular cartilage of the human knee from MR images which is user friendly. This would serve as a point-of-care 3D visualization tool which would benefit a consultant radiologist in the visualization of the human articular cartilage. Methods: The authors introduce a novel fusion of an adaptation of the contour method known as “contour interpolation (CI)” with radial basis functions (RBFs) which they describe as “CI-RBFs.” The authors also present a spline boundary correction which further enhances volume estimation of the method. A subject cohort consisting of 17 right nonpathological knees (ten female and seven male) is assessed to validate the quality of the proposed method. The authors demonstrate how the CI-RBF method dramatically reduces the number of data points required for fitting an implicit surface to the entire cartilage, thus, significantly improving the speed of reconstruction over the comparable RBF reconstruction method of Carr. The authors compare the CI-RBF method volume estimation to a typical commercial package (3D DOCTOR), Carr’s RBF method, and a benchmark manual method for the reconstruction of the femoral, tibial, and patellar cartilages. Results: The authors demonstrate how the CI-RBF method significantly reduces the number of data points (p-value < 0.0001) required for fitting an implicit surface to the cartilage, by 48%, 31%, and 44% for the patellar, tibial, and femoral cartilages, respectively. Thus, significantly improving the speed of reconstruction (p-value < 0.0001) by 39%, 40%, and 44% for the patellar, tibial, and femoral cartilages over the comparable RBF model of Carr providing a near real-time reconstruction of 6.49, 8.88, and 9.43 min for the patellar, tibial, and femoral cartilages, respectively. In addition, it is demonstrated how the CI-RBF method matches the volume estimation of a typical commercial package (3D DOCTOR), Carr’s RBF method, and a benchmark manual method for the reconstruction of the femoral, tibial, and patellar cartilages. Furthermore, the performance of the segmentation method used for the extraction of the femoral, tibial, and patellar cartilages is assessed with a Dice similarity coefficient, sensitivity, and specificity measure providing high agreement to manual segmentation. Conclusions: The CI-RBF method provides a fast, accurate, and robust 3D model reconstruction that matches Carr’s RBF method, 3D DOCTOR, and a manual benchmark method in accuracy and significantly improves upon Carr’s RBF method in data requirement and computational speed. In addition, the visualization tool has been designed to quickly segment MR images requiring only four mouse clicks per MR image slice.

62 RADIOLOGY AND NUCLEAR MEDICINE↗

DOE OSTI2016

An integrated model-driven method for in-treatment upper airway motion tracking using cine MRI in head and neck radiation therapy

Purpose: For the first time, MRI-guided radiation therapy systems can acquire cine images to dynamically monitor in-treatment internal organ motion. However, the complex head and neck (H&N) structures and low-contrast/resolution of on-board cine MRI images make automatic motion tracking a very challenging task. In this study, the authors proposed an integrated model-driven method to automatically track the in-treatment motion of the H&N upper airway, a complex and highly deformable region wherein internal motion often occurs in an either voluntary or involuntary manner, from cine MRI images for the analysis of H&N motion patterns. Methods: Considering the complex H&N structures and ensuring automatic and robust upper airway motion tracking, the authors firstly built a set of linked statistical shapes (including face, face-jaw, and face-jaw-palate) using principal component analysis from clinically approved contours delineated on a set of training data. The linked statistical shapes integrate explicit landmarks and implicit shape representation. Then, a hierarchical model-fitting algorithm was developed to align the linked shapes on the first image frame of a to-be-tracked cine sequence and to localize the upper airway region. Finally, a multifeature level set contour propagation scheme was performed to identify the upper airway shape change, frame-by-frame, on the entire image sequence. The multifeature fitting energy, including the information of intensity variations, edge saliency, curve geometry, and temporal shape continuity, was minimized to capture the details of moving airway boundaries. Sagittal cine MR image sequences acquired from three H&N cancer patients were utilized to demonstrate the performance of the proposed motion tracking method. Results: The tracking accuracy was validated by comparing the results to the average of two manual delineations in 50 randomly selected cine image frames from each patient. The resulting average dice similarity coefficient (93.28% ± 1.46%) and margin error (0.49 ± 0.12 mm) showed good agreement between the automatic and manual results. The comparison with three other deformable model-based segmentation methods illustrated the superior shape tracking performance of the proposed method. Large interpatient variations of swallowing frequency, swallowing duration, and upper airway cross-sectional area were observed from the testing cine image sequences. Conclusions: The proposed motion tracking method can provide accurate upper airway motion tracking results, and enable automatic and quantitative identification and analysis of in-treatment H&N upper airway motion. By integrating explicit and implicit linked-shape representations within a hierarchical model-fitting process, the proposed tracking method can process complex H&N structures and low-contrast/resolution cine MRI images. Future research will focus on the improvement of method reliability, patient motion pattern analysis for providing more information on patient-specific prediction of structure displacements, and motion effects on dosimetry for better H&N motion management in radiation therapy.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2014

Implementation of Parallel Adaptive-Krylov Exponential Solvers for Stiff Problems

Recently exponential integrators have been receiving increased attention as a means of solving large stiff systems of ODEs. Preliminary performance analysis demonstrated that exponential integrators hold promise compared to state-of-the-art implicit methods. However, much work remains to be done to understand in detail possible computational advantages these methods may offer in practice. This is particularly true for supercomputer-scale problems as there has been very little work on parallelizing exponential methods. In this paper we describe an implementation of a suite of parallel exponential solvers. Additionally, we present some performance tests on four stiff benchmark problems of a particular adaptive-Krylov exponential propagation iterative Runge--Kutta (EPIRK) method from the suite, and compare efficiency with the Newton--Krylov implicit solver CVODE.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2018

Scalable Preconditioners for Structure Preserving Discretizations of Maxwell Equations in First Order Form

Here, we report multiple physical time-scales can arise in electromagnetic simulations when dissipative effects are introduced through boundary conditions, when currents follow external time-scales, and when material parameters vary spatially. In such scenarios, the time-scales of interest may be much slower than the fastest time-scales supported by the Maxwell equations, therefore making implicit time integration an efficient approach. The use of implicit temporal discretizations results in linear systems in which fast time-scales, which severely constrain the stability of an explicit method, can manifest as so-called stiff modes. This study proposes a new block preconditioner for structure preserving (also termed physics compatible) discretizations of the Maxwell equations in first order form. The intent of the preconditioner is to enable the efficient solution of multiple-time-scale Maxwell type systems. An additional benefit of the developed preconditioner is that it requires only a traditional multigrid method for its subsolves and compares well against alternative approaches that rely on specialized edge-based multigrid routines that may not be readily available. Lastly, results demonstrate parallel scalability at large electromagnetic wave CFL numbers on a variety of test problems.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

Parallel-In-Time Multigrid with Adaptive Spatial Coarsening for The Linear Advection and Inviscid Burgers Equations

We apply a multigrid reduction-in-time (MGRIT) algorithm to hyperbolic partial differential equations in one spatial dimension. This study is motivated by the observation that sequential time-stepping is a computational bottleneck when attempting to implement highly concurrent algorithms; thus parallel-in-time methods are desirable. MGRIT adds parallelism by using a hierarchy of successively coarser temporal levels to accelerate the solution on the finest level. In the case of explicit time-stepping, spatial coarsening is a suitable approach to ensure that stability conditions are satisfied on all levels, and it may be useful for implicit time-stepping by producing cheaper multigrid cycles. Unfortunately, uniform spatial coarsening results in extremely slow convergence when the wave speed is near zero, even if only locally. We present an adaptive spatial coarsening strategy that addresses this issue for the variable coefficient linear advection equation and the inviscid Burgers equation using first-order explicit or implicit time-stepping methods. Serial numerical results show this method offers significant improvements over uniform coarsening and is convergent for the inviscid Burgers equation with and without shocks. Parallel scaling tests on up to 128K cores indicate that run-time improvements over serial time-stepping strategies are possible when spatial parallelism alone saturates, and that scalability is robust for oscillatory solutions which change on the scale of the grid spacing.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

A Comparison of Classical and Aggregation-Based Algebraic Multigrid Preconditioners for High-Fidelity Simulation of Wind Turbine Incompressible Flows

This paper presents a comparison of parallel strong scaling performance of classical and aggregation algebraic multigrid (AMG) preconditioners in the context of wind turbine simulations. Fluid motion is governed by the incompressible Navier--Stokes equations, discretized in space with control-volume finite elements and in time with an inexact projection scheme using an implicit integrator. A discontinuous-Galerkin sliding-mesh algorithm captures rotor motion. The momentum equations are solved with iterative Krylov methods, preconditioned by symmetric Gauss--Seidel (SGS) in Trilinos and $\ell_1$ SGS in hypre. The mass-continuity equation is solved with GMRES preconditioned by AMG and can account for the majority of simulation time. Reducing this continuity solve time is crucial. Wind turbine simulations present two unique challenges for AMG preconditioned solvers: the computational meshes include strongly anisotropic elements, and mesh motion requires matrix reinitialization and computation of preconditioners at each time step. Detailed timing profiles are presented and analyzed, and best practices are discussed for both classical and aggregation-based AMG. Results are presented for simulations of two different wind turbines with up to 6 billion grid points on two different computer architectures. For moving mesh problems that require linear-system reinitialization, the well-established strategy of amortizing preconditioner setup costs over a large number of time steps to reduce the solve time is no longer valid. Instead, results show that faster time to solution is achieved by reducing preconditioner setup costs at the expense of linear-system solve costs. Standard smoothed aggregation with Chebyshev relaxation was found to perform poorly when compared with classical AMG in terms of solve time and robustness. However, plain aggregation was comparable to classical AMG.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

A Fourth-Order Embedded Boundary Finite Volume Method for the Unsteady Stokes Equations with Complex Geometries

A fourth-order finite volume embedded boundary (EB) method is presented for the unsteady Stokes equations. The algorithm represents complex geometries on a Cartesian grid using EB, employing a technique to mitigate the ``small cut-cell"" problem without mesh modifications, cell merging, or state redistribution. Spatial discretizations are based on a weighted least-squares technique that has been extended to fourth-order operators and boundary conditions, including an approximate projection to enforce the divergence-free constraint. Solutions are advanced in time using a fourth-order additive implicit-explicit Runge-Kutta method, with the viscous and source terms treated implicitly and explicitly, respectively. Formal accuracy of the method is demonstrated with several grid convergence studies, and results are shown for an application with a complex bio-inspired material. In conclusion, the developed method achieves fourth-order accuracy and is stable despite the pervasive small cells arising from complex geometries.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Coarse-grained explicit-solvent molecular dynamics simulations of semidilute unentangled polyelectrolyte solutions

In this study, we present results from explicit-solvent coarse-grained molecular dynamics (MD) simulations of fully charged, salt-free, and unentangled polyelectrolytes in semidilute solutions. The inclusion of a polar solvent in the model allows for a more physical representation of these solutions at concentrations, where the assumptions of a continuum dielectric medium and screened hydrodynamics break down. The collective dynamic structure factor of polyelectrolytes, S(q, t), showed that at q > q*, where q* = 2π/ξ is the polyelectrolyte peak in the structure factor S(q) and ξ is the correlation length, the relaxation time obtained from fits to stretched exponential was $\tau$ KWW ~ q -3 , which describes unscreened Zimm-like dynamics. This is in contrast to implicit-solvent simulations using a Langevin thermostat where $\tau$ KWW ~ q -2 . At q < q*, a crossover region was observed that eventually transitions to another inflection point $\tau$ KWW ~ q -2 at length scales larger than ξ for both implicit- and explicit-solvent simulations. The simulation results were also compared to scaling predictions for correlation length, ξ ~ c$-½\atop{p}$, specific viscosity, η sp ~ c$½\atop{p}$, and diffusion coefficient, D ~ c$0\atop{p}$, where c p is the polyelectrolyte concentration. The scaling prediction for ξ holds; however, deviations from the predictions for η sp and D were observed for systems at higher c p , which are in qualitative agreements with recent experimental results. This study highlights the importance of explicit-solvent effects in molecular dynamics simulations, particularly in semidilute solutions, for a better understanding of polyelectrolyte solution behavior.

36 MATERIALS SCIENCE↗

DOE OSTI2018

Integrated Model, Batch and Domain Parallelism in Training Neural Networks

We propose a new integrated method of exploiting model, batch and domain parallelism for the training of deep neural networks (DNNs) on large distributed-memory computers using minibatch stochastic gradient descent (SGD). Our goal is to find an efficient parallelization strategy for a fixed batch size using $P$ processes. Our method is inspired by the communication-avoiding algorithms in numerical linear algebra. We see $P$ processes as logically divided into a $P_r \times P_c$ grid where the $P_r$ dimension is implicitly responsible for model/domain parallelism and the $P_c$ dimension is implicitly responsible for batch parallelism. In practice, the integrated matrix-based parallel algorithm encapsulates these types of parallelism automatically. We analyze the communication complexity and analytically demonstrate that the lowest communication costs are often achieved neither with pure model nor with pure data parallelism. We also show how the domain parallel approach can help in extending the theoretical scaling limit of the typical batch parallel method.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

ARKODE: A Flexible IVP Solver Infrastructure for One-step Methods

We describe the ARKODE library of one-step time integration methods for ordinary differential equation (ODE) initial-value problems (IVPs). In addition to providing standard explicit and diagonally implicit Runge–Kutta methods, ARKODE supports one-step methods designed to treat additive splittings of the IVP, including implicit-explicit (ImEx) additive Runge–Kutta methods and multirate infinitesimal (MRI) methods. We present the role of ARKODE within the SUNDIALS suite of time integration and nonlinear solver libraries, the core ARKODE infrastructure for utilities common to large classes of one-step methods, as well as its use of “time stepper” modules enabling easy incorporation of novel algorithms into the library. Numerical results show example problems of increasing complexity, highlighting the algorithmic flexibility afforded through this infrastructure, and include a larger multiphysics application leveraging multiple algorithmic features from ARKODE and SUNDIALS.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2016

Finite Element Simulation and X-Ray Microdiffraction Study of Strain Partitioning in a Layered Nanocomposite

The depth-dependent strain partitioning across the interfaces in the growth direction of the NiAl/Cr(Mo) nanocomposite between the Cr and NiAl lamellae was directly measured experimentally and simulated using a finite element method (FEM). Depth-resolved X-ray microdiffraction demonstrated that in the as-grown state both Cr and NiAl lamellae grow along the 111 direction with the formation of as-grown distinct residual ~0.16% compressive strains for Cr lamellae and ~0.05% tensile strains for NiAl lamellae. Three-dimensional simulations were carried out using an implicit FEM. First simulation was designed to study residual strains in the composite due to cooling resulting in formation of crystals. Strains in the growth direction were computed and compared to those obtained from the microdiffraction experiments. Second simulation was conducted to understand the combined strains resulting from cooling and mechanical indentation of the composite. Numerical results in the growth direction of crystal were compared to experimental results confirming the experimentally observed trends.

Barabash, R. I.↗

DOE OSTI2021

FIERRO V.X

FIERRO is a modern C++ code intended to simulate quasi-static solid mechanics problems and transient, compressible material dynamic problems with Lagrangian methods, which have meshes with constant mass elements that move with the material. FIERRO is designed to aid a) material model research that has historically been done using commercial implicit and explicit finite element codes, b) numerical methods research, and c) computer science research. FIERRO supports user developed material models that adhere to several industry standard formats by using a C++ to Fortran interface to couple the model to the numerical solvers. FIERRO is built on the ELEMENTS library that supports a diverse suite of element types, including high-order elements, and quadrature rules. The mesh class within the ELEMENTS library is designed for efficient calculations on unstructured meshes and to minimize memory usage. FIERRO is designed to readily accommodate a range of numerical methods including continuous finite element, finite volume, and discontinuous Galerkin methods. FIERRO is designed to support explicit and implicit time integration methods. FIERRO only supports a single material in an element. No physical data exists within the code.

Morgan, Nathaniel↗

DOE OSTI2022

Conducting more inclusive solar geoengineering research: A feminist science framework

Solar geoengineering, or deliberate climate modification, has been receiving increased attention in recent years. Given the far-reaching consequences of any potential solar geoengineering deployments, it is prudent to identify inherent biases, blind spots, and other potential issues at all stages of the research process. Here we articulate a feminist science-based framework to concretely describe how solar geoengineering researchers can be more inclusive of different perspectives and potentially contradictory conclusions, in the process illuminating potential implicit bias and enhancing the conclusions that can be gained from their studies. Importantly, this framework is an adoptable method of practice that can be refined, with the aim of conducting better research in solar geoengineering. As an illustration, we retrospectively apply this framework to a well-read solar geoengineering study (also led by the first author of this study), improving transparency by revealing its implicit values, conclusions made from its evidence base, and the methodologies that study pursues. We conclude here with a set of recommendations for the geoengineering research community whereby more inclusive research can become a regular part of practice. Throughout this process, we illustrate how feminist science scholars can use this approach to study climate modeling.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2018

A High-Order Low-Order Algorithm with Exponentially Convergent Monte Carlo for Thermal Radiative Transfer

In this paper, we have implemented a new high-order low-order (HOLO) algorithm for solving thermal radiative transfer problems. The low-order (LO) system is based on the spatial and angular moments of the transport equation and a linear-discontinuous finite-element spatial representation, producing equations similar to the standard S 2 equations. The LO solver is fully implicit in time and efficiently resolves the nonlinear temperature dependence at each time step. The high-order (HO) solver utilizes exponentially convergent Monte Carlo (ECMC) to give a globally accurate solution for the angular intensity to a fixed-source pure-absorber transport problem. This global solution is used to compute consistency terms, which require the HO and LO solutions to converge toward the same solution. The use of ECMC allows for the efficient reduction of statistical noise in the Monte Carlo solution, reducing inaccuracies introduced through the LO consistency terms. Finally, we compare results with an implicit Monte Carlo code for one-dimensional gray test problems and demonstrate the efficiency of ECMC over standard Monte Carlo in this HOLO algorithm.

97 MATHEMATICS AND COMPUTING↗