Discontinuous Galerkin Methods on hp-Anisotropic Meshes II: A Posteriori Error Analysis and Adaptivity

We consider the a posteriori error analysis and hp-adaptation strategies for hp-version interior penalty discontinuous Galerkin methods for second-order partial differential equations with nonnegative characteristic form on anisotropically refined computational meshes with anisotropically enriched elemental polynomial degrees. In particular, we exploit duality based hp-error estimates for linear target functionals of the solution and design and implement the corresponding adaptive algorithms to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local isotropic and anisotropic mesh refinement and isotropic and anisotropic polynomial degree enrichment. The superiority of the proposed algorithm in comparison with standard hp-isotropic mesh refinement algorithms and an h-anisotropic/p-isotropic adaptive procedure is illustrated by a series of numerical experiments.

Model selection and error estimation

We study model selection strategies based on penalized empirical loss minimization. We point out a tight relationship between error estimation and data-based complexity penalization: any good error estimate may be converted into a data-based penalty function and the performance of the estimate is governed by the quality of the error estimate. We consider several penalty functions, involving error estimates on independent test data, empirical VC dimension, empirical VC entropy, and margin-based quantities. We also consider the maximal difference between the error on the first half of the training data and the second half, and the expected maximal discrepancy, a closely related capacity estimate that can be calculated by Monte Carlo integration. Maximal discrepancy penalty functions are appealing for pattern classification problems, since their computation is equivalent to empirical risk minimization over the training data with some labels flipped.

Measurement error of rearfoot kinematics during running between a 100Hz and 30Hz camera

Background: Measurement accuracy is critical for biomechanical gait assessment. Very few studies have determined the accuracy of common clinical rearfoot variables between cameras with different collection frequencies. Research question: What is the measurement error for common rearfoot gait parameters when using a standard 30Hz digital camera compared to 100Hz camera? Type of study: Descriptive. Methods: 100 footfalls were recorded from 10 subjects ( 10 footfalls per subject) running on a treadmill at 2.68m/s. A high-speed digital timer, accurate within 1ms served as an external reference. Markers were placed along the vertical axis of the heel counter and the long axis of the shank. 2D coordinates for the four markers were determined from heel strike to heel lift. Variables of interest included time of heel strike (THS), time of heel lift (THL), time to maximum eversion (TMax), and maximum rearfoot eversion angle (EvMax). Results: THS difference was 29.77ms (+/- 8.77), THL difference was 35.64ms (+/- 6.85), and TMax difference was 16.50ms (+/- 2.54). These temporal values represent a difference equal to 11.9%, 14.3%, and 6.6% of the stance phase of running gait, respectively. EvMax difference was 1.02 degrees (+/- 0.46). Conclusions: A 30Hz camera is accurate, compared to a high-frequency camera, in determining TMax and EvMax during a clinical gait analysis. However, relatively large differences, in excess of 12% of the stance phase of gait, for THS and THL variables were measured.

Error modeling and calibration of exteroceptive sensors for accurate mapping applications

Reliable robotic perception and planning are critical to performing autonomous actions in uncertain, unstructured environments. In field robotic systems, automation is achieved by interpreting exteroceptive sensor information to infer something about the world. This is then mapped to provide a consistent spatial context, so that actions can be planned around the predicted future interaction of the robot and the world. The whole system is as reliable as the weakest link in this chain. In this paper, the term mapping is used broadly to describe the transformation of range-based exteroceptive sensor data (such as LIDAR or stereo vision) to a fixed navigation frame, so that it can be used to form an internal representation of the environment. The coordinate transformation from the sensor frame to the navigation frame is analyzed to produce a spatial error model that captures the dominant geometric and temporal sources of mapping error. This allows the mapping accuracy to be calculated at run time. A generic extrinsic calibration method for exteroceptive range-based sensors is then presented to determine the sensor location and orientation. This allows systematic errors in individual sensors to be minimized, and when multiple sensors are used, it minimizes the systematic contradiction between them to enable reliable multisensor data fusion. The mathematical derivations at the core of this model are not particularly novel or complicated, but the rigorous analysis and application to field robotics seems to be largely absent from the literature to date. The techniques in this paper are simple to implement, and they offer a significant improvement to the accuracy, precision, and integrity of mapped information. Consequently, they should be employed whenever maps are formed from range-based exteroceptive sensor data. © 2009 Wiley Periodicals, Inc.

Error and variance bounds on sigmoidal neurons with weight and input errors

Bounds on the expectation and variance of errors at the output of a multilayer feedforward neural network with perturbed weights and inputs are derived. It is assumed that errors in weights and inputs to the network are statistically independent and small. The bounds obtained are applicable to both digital and analogue network implementations and are shown to be of practical value.

Analysis of physical activity in emmetropic and myopic university students during semester and holiday periods: A pilot study

Background Previous studies (mostly questionnaire-based in children) suggest that outdoor activity is protective against myopia. There are few studies on young adults investigating both the impact of simply being outdoors versus performing physical activity. The aim was to study the relationship between the refractive error of young adults and their physical activity patterns. Methods Twenty-seven university students, aged 18 to 25 years, wore a pedometer (Omron HJ720ITE) for seven days both during the semester and holiday periods. They simultaneously recorded the type of activity performed, its duration, the number of steps taken (from the pedometer) and their location (indoors/outdoors) in a logbook. Mean spherical refractive error was used to divide participants into three groups (emmetropes: +1.00 to -0.50 D, low myopes: -0.62 to -3.00 D, higher myopes: -3.12 D or greater myopia). Results There were no significant differences between the refractive groups during the semester or holiday periods; the average daily times spent outdoors, the duration of physical activity, the ratio of physical activity performed outdoors to indoors and amount of near work performed were similar. The peak exercise intensity was similar across all groups: approximately 100 steps perminute, a brisk walk. Up to one-third of all physical activity was performed outdoors. There were some significant differences in activities performed during semester and holiday times. For example, lowmyopes spent significantly less time outside (49 ± 47 versus 74 ± 41 minutes, p = 0.005) and performed less physical activity (6,388 ± 1,747 versus 6,779 ± 2,746 steps per day; p = 0.03) during the holidays compared to during semester. Conclusions The fact that all groups had similar low exercise intensity butmany were notmyopic suggests that physical activity levels are not critical. There were differences in the activity patterns of lowmyopes during semester and holiday periods. This study highlights the need for a larger longitudinal-based study with particular emphasis on how discretionary time is spent.

Mixed pixel analysis and assessment for flood mapping

The most difficult operation in flood inundation mapping using optical flood images is to map the ‘wet’ areas where trees and houses are partly covered by water. This can be referred to as a typical problem of the presence of mixed pixels in the images. A number of automatic information extracting image classification algorithms have been developed over the years for flood mapping using optical remote sensing images, with most labelling a pixel as a particular class. However, they often fail to generate reliable flood inundation mapping because of the presence of mixed pixels in the images. To solve this problem, spectral unmixing methods have been developed. In this thesis, methods for selecting endmembers and the method to model the primary classes for unmixing, the two most important issues in spectral unmixing, are investigated. We conduct comparative studies of three typical spectral unmixing algorithms, Partial Constrained Linear Spectral unmixing, Multiple Endmember Selection Mixture Analysis and spectral unmixing using the Extended Support Vector Machine method. They are analysed and assessed by error analysis in flood mapping using MODIS, Landsat and World View-2 images. The Conventional Root Mean Square Error Assessment is applied to obtain errors for estimated fractions of each primary class. Moreover, a newly developed Fuzzy Error Matrix is used to obtain a clear picture of error distributions at the pixel level. This thesis shows that the Extended Support Vector Machine method is able to provide a more reliable estimation of fractional abundances and allows the use of a complete set of training samples to model a defined pure class. Furthermore, it can be applied to analysis of both pure and mixed pixels to provide integrated hard-soft classification results. Our research also identifies and explores a serious drawback in relation to endmember selections in current spectral unmixing methods which apply fixed sets of endmember classes or pure classes for mixture analysis of every pixel in an entire image. However, as it is not accurate to assume that every pixel in an image must contain all endmember classes, these methods usually cause an over-estimation of the fractional abundances in a particular pixel. In this thesis, a subset of adaptive endmembers in every pixel is derived using the proposed methods to form an endmember index matrix. The experimental results show that using the pixel-dependent endmembers in unmixing significantly improves performance.

Analysis of an Interior Penalty Method for Fourth Order Problems on Polygonal Domains

Error analysis for a stable C (0) interior penalty method is derived for general fourth order problems on polygonal domains under minimal regularity assumptions on the exact solution. We prove that this method exhibits quasi-optimal order of convergence in the discrete H (2), H (1) and L (2) norms. L (a) norm error estimates are also discussed. Theoretical results are demonstrated by numerical experiments.

A posteriori error estimates of discontinuous Galerkin methods for the Signorini problem

A reliable and efficient a posteriori error estimator is derived for a class of discontinuous Galerkin (DG) methods for the Signorini problem. A common property shared by many DG methods leads to a unified error analysis with the help of a constraint preserving enriching map. The error estimator of DG methods is comparable with the error estimator of the conforming methods. Numerical experiments illustrate the performance of the error estimator. (C) 2015 Elsevier B.V. All rights reserved.

Design and Analysis of Inexact Floating-Point Adders

Power has become a key constraint in nanoscale inte-grated circuit design due to the increasing demands for mobile computing and higher integration density. As an emerging compu-tational paradigm, an inexact circuit offers a promising approach to significantly reduce both dynamic and static power dissipation for error-tolerant applications. In this paper, an inexact floating-point adder is proposed by approximately designing an exponent sub-tractor and mantissa adder. Related operations such as normaliza-tion and rounding are also dealt with in terms of inexact computing. An upper bound error analysis for the average case is presented to guide the inexact design; it shows that the inexact floating-point adder design is dependent on the application data range. High dynamic range images are then processed using the proposed inexact floating-point adders to show the validity of the inexact design; comparison results show that the proposed inexact floating-point adders can improve the power consumption and power-delay product by 29.98% and 39.60%, respectively.

Automated image analysis for experimental investigations of salt water intrusion in coastal aquifers

A novel methodology has been developed to quantify important saltwater intrusion parameters in a sandbox style experiment using image analysis. Existing methods found in the literature are based mainly on visual observations, which are subjective, labour intensive and limits the temporal and spatial resolutions that can be analysed. A robust error analysis was undertaken to determine the optimum methodology to convert image light intensity to concentration. Results showed that defining a relationship on a pixel-wise basis provided the most accurate image to concentration conversion and allowed quantification of the width of mixing zone between the saltwater and freshwater. A large image sample rate was used to investigate the transient dynamics of saltwater intrusion, which rendered analysis by visual observation unsuitable. This paper presents the methodologies developed to minimise human input and promote autonomy, provide high resolution image to concentration conversion and allow the quantification of intrusion parameters under transient conditions.

Correcting Measurement Error Bias in Interaction Models with Small Samples

Several methods have been suggested to estimate non-linear models with interaction terms in the presence of measurement error. Structural equation models eliminate measurement error bias, but require large samples. Ordinary least squares regression on summated scales, regression on factor scores and partial least squares are appropriate for small samples but do not correct measurement error bias. Two stage least squares regression does correct measurement error bias but the results strongly depend on the instrumental variable choice. This article discusses the old disattenuated regression method as an alternative for correcting measurement error in small samples. The method is extended to the case of interaction terms and is illustrated on a model that examines the interaction effect of innovation and style of use of budgets on business performance. Alternative reliability estimates that can be used to disattenuate the estimates are discussed. A comparison is made with the alternative methods. Methods that do not correct for measurement error bias perform very similarly and considerably worse than disattenuated regression

Error and complexity of random walk Monte Carlo radiosity

The author studies the error and complexity of the discrete random walk Monte Carlo technique for radiosity, using both the shooting and gathering methods. The author shows that the shooting method exhibits a lower complexity than the gathering one, and under some constraints, it has a linear complexity. This is an improvement over a previous result that pointed to an O(n log n) complexity. The author gives and compares three unbiased estimators for each method, and obtains closed forms and bounds for their variances. The author also bounds the expected value of the mean square error (MSE). Some of the results obtained are also shown

Selected configuration interaction with truncation energy error and application to the Ne atom

Selected configuration interaction (SCI) for atomic and molecular electronic structure calculations is reformulated in a general framework encompassing all CI methods. The linked cluster expansion is used as an intermediate device to approximate CI coefficients BK of disconnected configurations (those that can be expressed as products of combinations of singly and doubly excited ones) in terms of CI coefficients of lower-excited configurations where each K is a linear combination of configuration-state-functions (CSFs) over all degenerate elements of K. Disconnected configurations up to sextuply excited ones are selected by Brown's energy formula, ΔEK=(E-HKK)BK2/(1-BK2), with BK determined from coefficients of singly and doubly excited configurations. The truncation energy error from disconnected configurations, Δdis, is approximated by the sum of ΔEKS of all discarded Ks. The remaining (connected) configurations are selected by thresholds based on natural orbital concepts. Given a model CI space M, a usual upper bound ES is computed by CI in a selected space S, and EM=E S+ΔEdis+δE, where δE is a residual error which can be calculated by well-defined sensitivity analyses. An SCI calculation on Ne ground state featuring 1077 orbitals is presented. Convergence to within near spectroscopic accuracy (0.5 cm-1) is achieved in a model space M of 1.4× 109 CSFs (1.1 × 1012 determinants) containing up to quadruply excited CSFs. Accurate energy contributions of quintuples and sextuples in a model space of 6.5 × 1012 CSFs are obtained. The impact of SCI on various orbital methods is discussed. Since ΔEdis can readily be calculated for very large basis sets without the need of a CI calculation, it can be used to estimate the orbital basis incompleteness error. A method for precise and efficient evaluation of ES is taken up in a companion paper

A chemical Hamiltonian approach study of the basis set superposition error changes on electron densities and one- and two-center energy components

A comparision of the local effects of the basis set superposition error (BSSE) on the electron densities and energy components of three representative H-bonded complexes was carried out. The electron densities were obtained with Hartee-Fock and density functional theory versions of the chemical Hamiltonian approach (CHA) methodology. It was shown that the effects of the BSSE were common for all complexes studied. The electron density difference maps and the chemical energy component analysis (CECA) analysis confirmed that the local effects of the BSSE were different when diffuse functions were present in the calculations