scholarly journals Analysis of Spatial Discretization Error Estimators Implemented in ARES Transport Code for SN Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Liang Zhang ◽  
Bin Zhang ◽  
Cong Liu ◽  
Yixue Chen
Keyword(s):  
Large Scale ◽  
Inner Product ◽  
Discrete Ordinates ◽  
Test Problems ◽  
Local Error ◽  
Spatial Discretization ◽  
Zeroth Order ◽  
Local Error Estimation ◽  
Error Estimators ◽  
Transport Code

The discrete ordinates method (SN) is one of the mainstream methods for neutral particle transport calculations. Assessing the quality of the numerical solution and controlling the discrete error are essential parts of large-scale high-fidelity simulations of nuclear systems. Three error estimators, a two-mesh estimator, a residual-based estimator, and a dual-weighted residual estimator, are derived and implemented in the ARES transport code to evaluate the error of zeroth-order spatial discretization for SN equations. The difference in scalar fluxes on coarse and fine meshes is adopted to indicate the error in the two-mesh method. To avoid zero residual in zeroth-order discretization, angular fluxes within one cell are reconstructed by Legendre polynomials. The error is estimated by inverting the discrete transport operator using the estimated directional residual as an anisotropic source. The inner product of the forward directional residual and the adjoint angular flux is employed to quantify the error in quantities of interest which can be denoted by a linear functional of forward angular flux. Method of Manufactured Solutions (MMS) is adopted to generate analytical solutions for SN equation with scattering and the determined true error is used to evaluate the effectivity of these estimators. Promising results are obtained in the numerical results for both homogeneous and heterogeneous cases. The larger error region is well captured and the average effectivity index for the local error estimation is less than unity. For the series test problems, the estimated goal quantity error can be contained within an order of magnitude around the exact error.

scholarly journals Application of the Denovo Discrete Ordinates Radiation Transport Code to Large-Scale Fusion Neutronics

2018 ◽  
Vol 74 (4) ◽  
pp. 303-314 ◽  
Author(s):  
Katherine E. Royston ◽  
Seth R. Johnson ◽  
Thomas M. Evans ◽  
Scott W. Mosher ◽  
Jonathan Naish ◽  
...  
Keyword(s):  
Large Scale ◽  
Radiation Transport ◽  
Discrete Ordinates ◽  
Transport Code

Local error estimation for singly-implicit formulas by two-step Runge-Kutta methods

1992 ◽  
Vol 32 (1) ◽  
pp. 104-117 ◽  
Author(s):  
A. Bellen ◽  
Z. Jackiewicz ◽  
M. Zennaro
Keyword(s):  
Error Estimation ◽  
Local Error ◽  
Local Error Estimation ◽  
Runge Kutta

scholarly journals A Boosting Framework of Factorization Machine

2021 ◽  
pp. 2159036
Author(s):  
Jun Zhou ◽  
Longfei Li ◽  
Ziqi Liu ◽  
Chaochao Chen
Keyword(s):  
Large Scale ◽  
State Of The Art ◽  
Experimental Results ◽  
Low Rank ◽  
Inner Product ◽  
Adaptive Boosting ◽  
Rank Matrix ◽  
Factorization Machine ◽  
Fixed Rank ◽  
Low Rank Matrix

Recently, Factorization Machine (FM) has become more and more popular for recommendation systems due to its effectiveness in finding informative interactions between features. Usually, the weights for the interactions are learned as a low rank weight matrix, which is formulated as an inner product of two low rank matrices. This low rank matrix can help improve the generalization ability of Factorization Machine. However, to choose the rank properly, it usually needs to run the algorithm for many times using different ranks, which clearly is inefficient for some large-scale datasets. To alleviate this issue, we propose an Adaptive Boosting framework of Factorization Machine (AdaFM), which can adaptively search for proper ranks for different datasets without re-training. Instead of using a fixed rank for FM, the proposed algorithm will gradually increase its rank according to its performance until the performance does not grow. Extensive experiments are conducted to validate the proposed method on multiple large-scale datasets. The experimental results demonstrate that the proposed method can be more effective than the state-of-the-art Factorization Machines.

scholarly journals Automatic calibration of a large-scale sediment model using suspended sediment concentration, water quality, and remote sensing data

RBRH ◽  
2019 ◽  
Vol 24 ◽  
Author(s):  
Hugo de Oliveira Fagundes ◽  
Fernando Mainardi Fan ◽  
Rodrigo Cauduro Dias de Paiva
Keyword(s):  
Remote Sensing ◽  
Water Quality ◽  
Suspended Sediment ◽  
Large Scale ◽  
Suspended Sediment Concentration ◽  
Remote Sensing Data ◽  
Sediment Concentration ◽  
Spatial Discretization ◽  
Calibration And Validation ◽  
Sediment Model

ABSTRACT Calibration and validation are two important steps in the application of sediment models requiring observed data. This study aims to investigate the potential use of suspended sediment concentration (SSC), water quality and remote sensing data to calibrate and validate a large-scale sediment model. Observed data from across 108 stations located in the Doce River basin was used for the period between 1997-2010. Ten calibration and validation experiments using the MOCOM-UA optimization algorithm coupled with the MGB-SED model were carried out, which, over the same period of time, resulted in 37 calibration and 111 validation tests. The experiments were performed by modifying metrics, spatial discretization, observed data and parameters of the MOCOM-UA algorithm. Results generally demonstrated that the values of correlation presented slight variations and were superior in the calibration step. Additionally, increasing spatial discretization or establishing a background concentration for the model allowed for improved results. In a station with high quantity of SSC data, calibration improved the ENS coefficient from -0.44 to 0.44. The experiments showed that the spectral surface reflectance, total suspended solids and turbidity data have the potential to enhance the performance of sediment models.

A MULTI-CLASS SUPPORT VECTOR MACHINE: THEORY AND MODEL

2013 ◽  
Vol 12 (06) ◽  
pp. 1175-1199 ◽  
Author(s):  
MINGHE SUN
Keyword(s):  
Support Vector Machine ◽  
Quadratic Programming ◽  
Programming Model ◽  
Inner Product ◽  
High Dimensional ◽  
Support Vector ◽  
Test Problems ◽  
Discriminant Functions ◽  
Input Space ◽  
Feature Spaces

A multi-class support vector machine (M-SVM) is developed, its dual is derived, its dual is mapped to high dimensional feature spaces using inner product kernels, and its performance is tested. The M-SVM is formulated as a quadratic programming model. Its dual, also a quadratic programming model, is very elegant and is easier to solve than the primal. The discriminant functions can be directly constructed from the dual solution. By using inner product kernels, the M-SVM can be built and nonlinear discriminant functions can be constructed in high dimensional feature spaces without carrying out the mappings from the input space to the feature spaces. The size of the dual, measured by the number of variables and constraints, is independent of the dimension of the input space and stays the same whether the M-SVM is built in the input space or in a feature space. Compared to other models published in the literature, this M-SVM is equally or more effective. An example is presented to demonstrate the dual formulation and solution in feature spaces. Very good results were obtained on benchmark test problems from the literature.

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