Numerical Error Correction Research Articles

A gas-surface interaction algorithm for discrete velocity methods in predicting rarefied and multi-scale flows: For Maxwell boundary model

The discrete velocity method (DVM) for rarefied flows and the unified methods (based on the DVM framework) for flows in all regimes, from continuum one to free molecular one, have worked well as precise flow solvers over the past decades and have been successfully extended to other important physical fields. Both DVM and unified methods endeavor to model the gas-gas interaction physically. However, for the gas-surface interaction (GSI) at the wall boundary, they have only use the full accommodation boundary up to now, which can be viewed as a rough Maxwell boundary with a fixed accommodation coefficient (AC) at unity, deviating from the real value. For example, the AC for metal materials typically falls in the range of 0.8 to 0.9. To overcome this bottleneck and extend the DVM and unified methods to more physical boundary conditions, an algorithm for Maxwell boundary with an adjustable AC is established into the DVM framework. The Maxwell boundary model splits the distribution of the bounce-back molecules into specular ones and Maxwellian (normal) ones. Since the bounce-back molecules after the spectral reflection does not math with the discrete velocity space (DVS), both macroscopic conservation (from numerical quadrature) and microscopic consistency in the DVS are hard to achieve in the DVM framework. In this work, this problem is addressed by employing a combination of interpolation methods for mismatch points in DVS and an efficient numerical error correction method for micro-macro consistency. On the other hand, the current Maxwell boundary for DVM takes the generality into consideration, accommodating both the recently developed efficient unstructured velocity space and the traditional Cartesian velocity space. Moreover, the proposed algorithm allows for calculations of both monatomic gases and diatomic gases with internal degrees in DVS. Finally, by being integrated with the unified gas-kinetic scheme within the DVM framework, the performance of the present GSI algorithm is validated through a series of benchmark numerical tests across a wide range of Knudsen numbers.

A review of error correction procedures during instruction for children with developmental disabilities.

Error correction procedures are remedial strategies presented following an incorrect response that increases the probability that a correct response will occur in the future. Error correction is commonly used during skill acquisition programs for children with developmental disabilities; however, the specific strategy used may differ considerably. Recent comparative studies have examined the effect of numerous error correction procedures on the efficiency of acquisition for children with developmental disabilities. Despite considerable merit, minor procedural differences and unique terms for similar procedures likely affect comparisons across studies. Here, we clarify the procedures and findings of these studies and suggest areas of future research.

A Bayesian hierarchical model for glacial dynamics based on the shallow ice approximation and its evaluation using analytical solutions

Abstract. Bayesian hierarchical modeling can assist the study of glacial dynamics and ice flow properties. This approach will allow glaciologists to make fully probabilistic predictions for the thickness of a glacier at unobserved spatiotemporal coordinates, and it will also allow for the derivation of posterior probability distributions for key physical parameters such as ice viscosity and basal sliding. The goal of this paper is to develop a proof of concept for a Bayesian hierarchical model constructed, which uses exact analytical solutions for the shallow ice approximation (SIA) introduced by Bueler et al. (2005). A suite of test simulations utilizing these exact solutions suggests that this approach is able to adequately model numerical errors and produce useful physical parameter posterior distributions and predictions. A byproduct of the development of the Bayesian hierarchical model is the derivation of a novel finite difference method for solving the SIA partial differential equation (PDE). An additional novelty of this work is the correction of numerical errors induced through a numerical solution using a statistical model. This error-correcting process models numerical errors that accumulate forward in time and spatial variation of numerical errors between the dome, interior, and margin of a glacier.

An elastic error correction code technique for NAND flash-based consumer electronic devices

Multi-level cell (MLC) NAND flash-based consumer electronic devices suffer from random multiple bit errors that grow exponentially with the increase of program/erase counts. Numerous error correction codes (ECCs) have been developed to detect and correct these multiple erroneous bits within a codeword, such as bose-chaudhuri-hocquenghem (BCH) and reed-solomon (RS) codes. However, most of these existing techniques do not take into account the uneven distribution of bit errors over flash pages, thus they cannot meet varying correction needs of the flash memories during its lifetime. Specifically, weak ECCs are eventually unable to correct some particular pages' bit errors beyond their correction capabilities, while powerful ECCs can protect each page longer yet incur unnecessary computation overhead too early. In this paper, an elastic error correction code (EECC) technique is proposed, which can progressively enhance the error correction capability for each page when performing program operation. In particular, based on a scalable coding mapping model, EECC technique can enhance the ECC level progressively, by allowing each page to employ changeable ECC parity in its own spare out-of-band area according to its own remaining lifetime as well as the hot level of the data in it. In this way, this technique not only meets the changing error correction demands for different page, but also obtains a good reliability-performance tradeoff. Analytically and experimentally, the results demonstrate EECC scheme is efficient in many aspects of performance, and particularly is able to make significant power consumption savings without degrading the error correction capability <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> .

Stable Feature Flow Fields

Feature Flow Fields are a well-accepted approach for extracting and tracking features. In particular, they are often used to track critical points in time-dependent vector fields and to extract and track vortex core lines. The general idea is to extract the feature or its temporal evolution using a stream line integration in a derived vector field-the so-called Feature Flow Field (FFF). Hence, the desired feature line is a stream line of the FFF. As we will carefully analyze in this paper, the stream lines around this feature line may diverge from it. This creates an unstable situation: if the integration moves slightly off the feature line due to numerical errors, then it will be captured by the diverging neighborhood and carried away from the real feature line. The goal of this paper is to define a new FFF with the guarantee that the neighborhood of a feature line has always converging behavior. This way, we have an automatic correction of numerical errors: if the integration moves slightly off the feature line, it automatically moves back to it during the ongoing integration. This yields results which are an order of magnitude more accurate than the results from previous schemes. We present new stable FFF formulations for the main applications of tracking critical points and solving the Parallel Vectors operator. We apply our method to a number of data sets.

Numerical analysis of kinetic rate constants derived from stochastic computer simulation

AbstractA numerical approach has been developed to determine the individual rate constants (kf − s and ks − f), as well as the equilibrium constant from the ratio of the rate constants (kf − s/ks − f), by using stochastic simulation. This approach is illustrated for simulations of the absorption (partition) process between homogeneous fluid and surface phases under diffusion‐limited conditions. A pair of regression methods is derived from the chemical system with the initial distribution of molecules entirely in either the fluid or surface phase. These two methods are complementary in determining the rate constants for distribution coefficients that are lower or higher than unity, respectively. A standard exponential (two‐parameter) regression equation is compared with a biexponential (four‐parameter) equation, which provides for inherent correction of numerical error. The effects of the number of molecules, as well as the time increment and range, are examined in detail. This provides guidance to optimize the computational parameters of the simulation. The proposed approach has been successfully applied for distribution coefficients ranging from 0.01 to 100.0, yielding individual rate constants kf − s and ks − f with ± 0.84% and ± 0.57% average relative standard deviation, and the ratio of the rate constants with ± 1.04% average relative standard deviation and ± 1.14% average relative error.

A Study on the Secular Change of Three-Dimensional Coordinate Measuring Machines.

One of the methods of improving the accuracy of the three-dimensional coordinate measuring machine (cmm) is numerical error correction. For the method, calibration of cmm must be carried out periodically. However, the long term stability of the accuracy cannot be predicted, and it is often difficult to determine the interval time of calibration. For the purpose of determining the periodic time of calibration, the secular change of a cmm has been investigated for over seven years. The same calibration procedure was performed with the cmm under the same environmental conditions. Results of tests show that the change in the accuracy occurs in a short period of one or two months. The accuracy must be frequently measured for the judgment of the time to practice the calibration procedure.

Three-Dimensional Gage. Long-Term Stability of the Gage and Calibration Test of Coordinate Measuring Machine.

To improve the accuracy of CMM (coordinate measuring machine), the numerical error correction of systematic error components ascertained by the calibration procedure is employed. One way to calibrate CMM is to take advantage of the three-dimensional gage made for experimental purposes by the author. It is necessary that the gage is inspected correctly and measured with high accuracy with respect to repeatability and reproducibility. Furthermore, the shape of the gage or the relative positions of steel balls used as measuring points should remain unchanged for a long time. This paper describes the experiments on the accuracy of the gage, repeatability and reproducibility of measuring the gage, and the long-term stability of the measuring points, and gives the results of tests.

A study on the measuring accuracy of three-dimensional coordinate measuring machines. (Influence of measuring type of probe).

The measuring accuracy of three-dimensional coordinate measuring machines (cmm's) is affected by various error factors. To improve the measuring accuracy of cmm's by numerical error correction, it is necessary to give the calibration method careful consideration for several error factors in addition to geometrical errors. For example, as for the air bearing which has been broadly used as the linear guide way of the carriages, it may be considered that the deficiently of bearing stiffness decreases the measuring accuracy of cmm's. In such a case, measured values may vary with the type of probe system. In this paper the error factor which is considered as the influence of the lack of bearing stiffness is experimentally investigated. It is clarified that the measured values of cmm's vary with the measuring type of the probe in relation to the straightness of the linear guide way.

Generalized correction of numerical errors in kinematical differential equations based on euler normalized parameters

It is shown how to correct the solutions of kinematical differential equations for arbitrary two-axis rotations represented by Euler-Rodrigues parameters. Two methods are described: minimizing the errors in the Euler-Rodrigues parameters themselves and minimizing the errors in the Frobenius norm of the direction cosine matrix. Details of the former are developed and numerical experiments are done which show an average improvement of about 17% in the root mean square values of the Euler-Rodrigues parameters. This generalizes the method of [2] and completes the framework developed in [3] for the practical use of these parameters in multiboy dynamic simulation.

Calibration of coordinate measuring machines

For over three years the Physikalisch-Technische Bundesanstalt has been investigating the numerical error correction of cmms, and the relationships between problems in this area and in the calibration of these machines. It has been determined that calibration is possible, for pre-defined conditions, but that the calibration is applicable only to the particular measuring task and conditions. This paper describes the mathematical model used by PTB and gives results of tests on a cmm.

Generalized correction of numerical error in kinematical differential equations based on Euler-Rodrigues parameters

Generalized correction of numerical error in kinematical differential equations based on Euler-Rodrigues parameters

Correction of numerical error in kinematical differential equations when one direction cosine is known

If a direction cosine matrix represents a two-axis rotation, one of the direction cosines remains zero. It is shown how one can take advantage of this knowledge to improve the correction of numerical errors in the parametrizing variables of the matrix. The cases of Euler parameters and six-parameter representations are considered, as well as Euler-Rodrigues parameters for which there is no traditional correction process because they normally do not satisfy any kinematical constraint. Improvement in the matrix estimates can be gotten in all cases, but only for the six-parameter method is it substantial.

Correction of numerical error when two direction cosine columns are used as kinematical variables

Abstract When more than three variables are used to characterize a rotation they must satisfy one or more algebraic constraints. Numerical errors in integrating the kinematical differential equations cause the constraints to be violated. Known correction procedures for four- and nine-variable parametrizations are reviewed. A new correction technique is developed for six-variable methods in which two columns of the direction cosine matrix provide the six variables. Results of numerical tests of the method are given. Baumgarte's method for constraint stabilization also is described briefly.