Order Error Research Articles

Discretization of Fractional Fully Nonlinear Equations by Powers of Discrete Laplacians

AbstractWe study discretizations of fractional fully nonlinear equations by powers of discrete Laplacians. Our problems are parabolic and of order $$\sigma \in (0,2)$$ σ ∈ ( 0 , 2 ) since they involve fractional Laplace operators $$(-\Delta )^{\sigma /2}$$ ( - Δ ) σ / 2 . They arise e.g. in control and game theory as dynamic programming equations – HJB and Isaacs equation – and solutions are non-smooth in general and should be interpreted as viscosity solutions. Our approximations are realized as finite-difference quadrature approximations and are 2nd order accurate for all values of $$\sigma $$ σ . The accuracy of previous approximations of fractional fully nonlinear equations depend on $$\sigma $$ σ and are worse when $$\sigma $$ σ is close to 2. We show that the schemes are monotone, consistent, $$L^\infty $$ L ∞ -stable, and convergent using a priori estimates, viscosity solutions theory, and the method of half-relaxed limits. We also prove a second order error bound for smooth solutions and present many numerical examples.

Toward Utilizing Similarity in Hydrologic Data Assimilation

Similarity to reality is a necessary property of models in earth sciences. Similarity information can thus possess a large potential in advancing geophysical modeling and data assimilation. We present a formalism for utilizing similarity within the existing theoretical data assimilation framework. Two examples illustrate the usefulness of utilizing similarity in data assimilation. The first, theoretical example shows changes in the accuracy of the amplitude estimate in the presence of a phase error in a sine function, where correcting the phase error prior to the assimilation reduces the degree of ill-posedness of the assimilation problem. This signifies the importance of accounting for the phase error in order to reduce the error in the amplitude estimate of the sine function. The second, real-world example illustrates that timing errors in simulated flow degrade the data assimilation performance, and that the flow gradient-informed shifting of rainfall time series improved the assimilation results with less adjusting model states. This demonstrates the benefit of utilizing streamflow gradients in shifting rainfall time series in a way to improve streamflow timing—vital information for flood early warning and preparedness planning. Finally, we discuss the implications, potential issues, and future challenges associated with utilizing similarity in hydrologic data assimilation.

A local projection stabilised HHO method for the Oseen problem

In this article, we consider a local projection stabilisation for a Hybrid High-Order (HHO) approximation of the Oseen problem. We prove an existence-uniqueness result under a stronger SUPG-like norm. We improve the stability and provide error estimation in stronger norm for convection dominated Oseen problem. We also derive an optimal order error estimate under the SUPG-like norm for equal-order polynomial discretisation of velocity and pressure spaces. Numerical experiments are performed to validate the theoretical results.

The spatiotemporal dynamics of bottom-up and top-down processing during at-a-glance reading.

Like all domains of cognition, language processing is affected by top-down knowledge. Classic evidence for this is missing blatant errors in the signal. In sentence comprehension, one instance is failing to notice word order errors, such as transposed words in the middle of a sentence: you that read wrong (Mirault et al., 2018). Our brains seem to fix such errors, since they are incompatible with our grammatical knowledge. But how do our brains do this? Following behavioral work on inner transpositions, we flashed four-word sentences for 300ms using rapid parallel visual presentation (RPVP, Snell and Grainger, 2017). We compared magnetoencephalography responses to fully grammatical and reversed sentences (24 human participants: 21 female, 4 male). Left lateral language cortex robustly distinguished grammatical and reversed sentences starting at 213ms. Thus, the influence of grammatical knowledge begun rapidly after visual word form recognition (Tarkiainen et al., 1999). At the earliest stage of this neural "sentence superiority effect," inner transpositions patterned between grammatical and reversed sentences, showing evidence that the brain initially "noticed" the error. However, 100 milliseconds later, inner transpositions became indistinguishable from grammatical sentences, suggesting at this point, the brain had "fixed" the error. These results show that after a single glance at a sentence, syntax impacts our neural activity almost as quickly as higher-level object recognition is assumed to take place (Cichy et al., 2014). The earliest stage involves detailed comparisons between the bottom-up input and grammatical knowledge, while shortly afterwards, top-down knowledge can override an error in the stimulus.Significance statement Language processing, like all cognitive domains, is profoundly influenced by top-down knowledge. One type of evidence of this is oversight of errors in the signal. For example, in sentence comprehension, individuals often fail to detect word order errors, such as transposed words mid-sentence. Utilizing rapid parallel visual presentation (RPVP), we investigated this phenomenon by exposing participants to four-word sentences for 300ms. Magnetoencephalography revealed robust differentiation between grammatical and reversed sentences in left lateral language cortex starting at 213ms post-presentation. Intriguingly, initial neural responses to inner transpositions treated them as deviant, but 100ms later, neural signals grouped them with grammatical sentence, indicating rapid error correction. These findings reveal the brain's remarkable capacity to reconcile bottom-up input with linguistic knowledge almost instantaneously.

Robust AI Based Bio Inspired Protocol using GANs for Secure and Efficient Data Transmission in IoT to Minimize Data Loss

Objectives: To propose an AI-based protocol to enhance the reliability and security of IoT data transmission in a robust manner. Deep learning with bio-inspired methods is employed to address real-time challenges such as route optimization, data integrity, error recovery, and end-to-end delay in order to minimize data loss and maximize the transmission rate. Methods: Generative Adversarial Networks (GANs) are used to enhance the robustness of the protocol in combination with a bio-inspired Artificial Immune System (AIS) to detect IoT network anomalies and responds to malicious activities by using the adaptive learning capabilities of IS. Hybrid Automatic Repeat Request (HARQ), an error recovery method, is utilized to detect network errors in order to correct and retransmit data to the destination during error-prone network conditions. Queue learning action sets are used to discover the finest path for efficient data transmission. The OMNeT++ simulator is used to assess the performance of the proposed BIP-GANs protocol. The performance results are compared with the prevailing data transmission protocols, such as EAP-IFBA, DNN-CSO, and ALO-DHT. Findings: The suggested BIP-GANs data transmission protocol outperforms with promising results, with an energy consumption rate of 6%, 8 seconds data transmission speed, 6 second less delay rate, 98% robustness (security and confidentiality), 98.5% alive nodes, and 99% network life span, which is higher than the prevailing methods. Novelty: This research study provides a comprehensive solution to secure and efficient data transmission by integrating the AI-based bio-inspired BIP-GANs method to address the security and data transmission challenges of existing methods such as EAP-IFBA, DNN-CSO, and ALO-DHT in terms of energy consumption rate, transmission speed, delay rate, life span of the network, robustness, and number of alive nodes. Keywords: Artificial Intelligence, Generative Adversarial Networks, Error Recovery, Secured Data Transmission, Deep Learning, Bio­Inspired Algorithm

Dynamic three-dimensional reconstruction with phase shift coding division multiplexing

Fringe projection profilometry with temporal phase unwrapping (FPP-TPU) has been gotten more and more attention in reconstructing the object with complex surfaces. However, a general contradiction in the existing FPP-TPUs is that more auxiliary coded patterns may limit sampling rate in dynamic three-dimensional (3D) measurement. In this paper, inheriting the single-frame auxiliary pattern of the phase-shifting temporal phase unwrapping (PS-TPU), an improved phase shift coding temporal phase unwrapping with higher robustness is proposed. And new projection stratagem with phase shift coding division multiplexing (PSC-DM) is also proposed to enhance the reconstructing rate in dynamic 3D scene. Firstly, six codewords instead of the traditional eight codewords are embedded into the single-frame auxiliary coding pattern, which guarantees the decoding robustness and provides a faster decoding process with single-step direction unifying. Then, the new projection sequence stratagem is designed with every two sets of 3-step phase-shifting patterns projection followed by the single-frame phase-shifting coding pattern projection. During phase unwrapping, every adjacent three frames of phase-shifting pattern sequentially resolving a wrapped phase and a highly robust 3D reconstruction can be achieved by using the closest phase-shifting coding pattern so as to achieve phase-shifting coding division multiplexing. It can obtain higher reconstructing rate. In addition, a phase shift segmentation is used to correct mismatched errors of fringe order. The experimental results demonstrate that the proposed method can increase the reconstructing rate from the rate of 30 fps in traditional PS-TPU method to 68.57 fps at the digital light projector (DLP) projection rate of 120 fps, which shows its potential prospective in dynamic 3D shape reconstruction.

Innovative Structural Optimization and Dynamic Performance Enhancement of High-Precision Five-Axis Machine Tools

To satisfy the requirements of five-axis processing quality, this article improves and optimizes the machine tool structure design to produce improved dynamic characteristics. This study focuses on the investigation of five-axis machine tools’ static and dynamic stiffness as well as structural integrity. We also include performance optimization and experimental verification. We use the finite element approach as a structural analysis tool to evaluate and compare the individual parts of the machine created in this study, primarily the saddle, slide table, column, spindle head, and worktable. We discuss the precision of the machine tool model and relative space distortion at each location. To meet the requirements of the actual machine, we optimize the structure of the five-axis machine tool based on the parameters and boundary conditions of each component. The machine’s weight was 15% less than in the original design model, the material it was subjected to was not strained, and the area of the structure where the force was considerably deformed was strengthened. We evaluate the AM machine’s impact resistance to compare the vibrational deformation observed in real time with the analytical findings. During modal analysis, all the order of frequencies were determined to be 97.5, 110.4, 115.6, and 129.6 Hz. The modal test yielded the following orders of frequencies: 104, 118, 125, and 133 Hz. Based on the analytical results, the top three order error percentages are +6.6%, +6.8%, +8.1%, and +2.6%. In ME’scope, the findings of the modal test were compared with the modal assurance criteria (MAC) analysis. According to the static stiffness analysis’s findings, the main shaft and screw have quite substantial major deformations, with a maximum deformation of 33.2 µm. Force flow explore provides the relative deformation amount of 26.98 µm from the rotating base (C) to the tool base, when a force of 1000 N is applied in the X-axis direction, which is more than other relative deformation amounts. We also performed cutting transient analysis, cutting spectrum analysis, steady-state thermal analysis, and analysis of different locations of the machine tool. All of these improvements may effectively increase the stiffness of the machine structure as well improve the machine’s dynamic characteristics and increases its machining accuracy. The topology optimization method checks how the saddle affects the machine’s stability and accuracy. This research will boost smart manufacturing in the machine tool sector, leading to notable advantages and technical innovations.

Virtual element method for solving a viscoelastic contact problem with long memory

In this paper, we use the virtual element method to solve a history-dependent hemivariational inequality arising in contact problems. The contact problem concerns the deformation of a viscoelastic body with long memory, subjected to a contact condition with non-monotone normal compliance and unilateral constraints. A fully discrete scheme based on the trapezoidal rule for the discretization of the time integration and the virtual element method for the spatial discretization are analyzed. We provide a unified priori error analysis for both internal and external approximations. For the linear virtual element method, we obtain the optimal order error estimate. Finally, three numerical examples are reported, providing numerical evidence of the theoretically predicted optimal convergence orders.

Machine-Learning-Based Predictive Models for Punching Shear Strength of FRP-Reinforced Concrete Slabs: A Comparative Study

This study is focused on the punching strength of fiber-reinforced polymer (FRP) concrete slabs. The mechanical properties of reinforced concrete slabs are often constrained by their punching shear strength at the column connection regions. Researchers have explored the use of fiber-reinforced polymer reinforcement as an alternative to traditional steel reinforcement to address this limitation. However, current codes poorly calculate the punching shear strength of FRP-reinforced concrete slabs. The aim of this study was to create a robust model that can accurately predict its punching shear strength, thus improving the analysis and design of composite structures with FRP-reinforced concrete slabs. In this study, 189 sets of experimental data were collected, and six machine learning models, including linear regression, support vector machine, BP neural network, decision tree, random forest, and eXtreme Gradient Boosting, were constructed and evaluated based on goodness of fit, standard deviation, and root-mean-square error in order to select the most suitable model for this study. The optimal model obtained was compared with the models proposed by codes and the researchers. Finally, a model explainability study was conducted using SHapley Additive exPlanations (SHAP). The results showed that random forests performed best among all machine learning models and outperformed existing models suggested by codes and researchers. The effective depth of the FRP-reinforced concrete slabs was the most important and proportional to the punching shear strength. This study not only provides guidance on the design of FRP-reinforced concrete slabs but also informs future engineering practice.

Higher order approximation of option prices in Barndorff-Nielsen and Shephard models

We present an approximation method based on the mixing formula [Hull, J. and White, A., The pricing of options on assets with stochastic volatilities. J. Finance, 1987, 42, 281–300; Romano, M. and Touzi, N., Contingent claims and market completeness in a stochastic volatility model. Math. Finance, 1997, 7, 399–412] for pricing European options in Barndorff-Nielsen and Shephard models. This approximation is based on a Taylor expansion of the option price. It is implemented using a recursive algorithm that allows us to obtain closed form approximations of the option price of any order (subject to technical conditions on the background driving Lévy process). This method can be used for any type of Barndorff-Nielsen and Shephard stochastic volatility model. Explicit results are presented in the case where the stationary distribution of the background driving Lévy process is inverse Gaussian or gamma. In both of these cases, the approximation compares favorably to option prices produced by the characteristic function. In particular, we also perform an error analysis of the approximation, which is partially based on the results of Das and Langrené [Closed-form approximations with respect to the mixing solution for option pricing under stochastic volatility. Stochastics, 2022, 94, 745–788]. We obtain asymptotic results for the error of the N th order approximation and error bounds when the variance process satisfies an inverse Gaussian Ornstein–Uhlenbeck process or a gamma Ornstein–Uhlenbeck process.

Lowest-order nonstandard finite element methods for time-fractional biharmonic problem

In this work, we consider an initial-boundary value problem for a time-fractional biharmonic equation in a bounded polygonal domain with a Lipschitz continuous boundary in R2 with clamped boundary conditions. After stating the well-posedness, we focus on some regularity results of the solution with respect to the regularity of the problem data. The spatially semidiscrete scheme covers several popular lowest-order piecewise-quadratic finite element schemes, namely, Morley, discontinuous Galerkin, and C0 interior penalty methods, and includes both smooth and nonsmooth initial data. Optimal order error bounds with respect to the regularity assumptions on the data are proved for both homogeneous and nonhomogeneous problems. The numerical experiments validate the theoretical convergence rate results.

Puzzle Hi-C: An accurate scaffolding software.

High-quality, chromosome-scale genomes are essential for genomic analyses. Analyses, including 3D genomics, epigenetics, and comparative genomics rely on a high-quality genome assembly, which is often accomplished with the assistance of Hi-C data. Curation of genomes reveal that current Hi-C-assisted scaffolding algorithms either generate ordering and orientation errors or fail to assemble high-quality chromosome-level scaffolds. Here, we offer the software Puzzle Hi-C, which uses Hi-C reads to accurately assign contigs or scaffolds to chromosomes. Puzzle Hi-C uses the triangle region instead of the square region to count interactions in a Hi-C heatmap. This strategy dramatically diminishes scaffolding interference caused by long-range interactions. This software also introduces a dynamic, triangle window strategy during assembly. Initially small, the window expands with interactions to produce more effective clustering. Puzzle Hi-C outperforms available scaffolding tools.

Thermal analysis of V-pad thermocouple used on once-through steam generator tubes

Once-Through Steam Generators (OTSGs) are used in the oil sands industry to produce steam, which is injected into oil sands reservoirs to produce bitumen. OTSGs consist of a combustion zone where natural gas is combusted to generate thermal energy, which boils water within tubes that pass through the OTSG. In a typical OTSG, the temperature at the external wall of the tubes can reach as high as 500 °C, or even higher. Temperature measurement of the tube wall is critical for safe operation and this can be determined by using thermocouples. However, the placement of the thermocouple and its shield to protect it on the tube outer wall interferes with heat transfer and, thus, the measured temperature from the thermocouple may not be equal to the bare tube (no thermocouple) temperature. Here, we examine how the placement of a thermocouple on a tube wall in an OTSG affects the tube wall temperature by using finite-element-based heat transfer modelling. Depending on the foulant thickness, and the placement of the thermocouple, the local tube wall temperature increases by 45 to 55 °C. The results show that using thermocouples with shields can yield a temperature measurement error of order of 15–30 °C above the actual bare tube temperature. The results provide a basis for correcting the thermocouple/shield derived value to the actual temperature of the tube wall.

Enhancements in image quality and block detection performance for Reinforced Soil-Retaining Walls under various illuminance conditions

To ensure continuous monitoring of reinforced soil-retaining walls (RSWs) even under low-illuminance conditions, such as during the night, it is imperative to evaluate the performance of deep learning-based detection. In this study, we constructed a laboratory RSW model and generated a dataset with varying illuminance levels to assess the impact of image enhancement and block detection. Various image enhancement methods were applied to improve image quality and evaluate their effect on deep learning. RGB optimization (RO) was proposed to optimize RGB intensity and compared with gamma correction, histogram equalization, and low-light image enhancement with illumination map estimation. RO demonstrated outstanding image enhancement performance, as evidenced by lightness order error, peak signal-to-noise ratio, and structural similarity index measure, ensuring high image quality. The trained RO model using Mask R-CNN exhibited excellent accuracy, recall, and F1 score, delivering remarkable detection performance under low illuminance conditions, resulting in a 7.44 % improvement in the F1 score. Image enhancement techniques that maintain similarity, such as lightness order error and structural similarity, across varying illuminance conditions contribute to enhancing the block detection performance of Mask R-CNNs.

Machine learning without a processor: Emergent learning in a nonlinear analog network

Standard deep learning algorithms require differentiating large nonlinear networks, a process that is slow and power-hungry. Electronic contrastive local learning networks (CLLNs) offer potentially fast, efficient, and fault-tolerant hardware for analog machine learning, but existing implementations are linear, severely limiting their capabilities. These systems differ significantly from artificial neural networks as well as the brain, so the feasibility and utility of incorporating nonlinear elements have not been explored. Here, we introduce a nonlinear CLLN-an analog electronic network made of self-adjusting nonlinear resistive elements based on transistors. We demonstrate that the system learns tasks unachievable in linear systems, including XOR (exclusive or) and nonlinear regression, without a computer. We find our decentralized system reduces modes of training error in order (mean, slope, curvature), similar to spectral bias in artificial neural networks. The circuitry is robust to damage, retrainable in seconds, and performs learned tasks in microseconds while dissipating only picojoules of energy across each transistor. This suggests enormous potential for fast, low-power computing in edge systems like sensors, robotic controllers, and medical devices, as well as manufacturability at scale for performing and studying emergent learning.

Low regularity error analysis for an H(div)-conforming discontinuous Galerkin approximation of Stokes problem

In this paper, we derive an improved error estimate for the H(div)-conforming discontinuous Galerkin (DG) approximation of the Stokes equations, assuming only minimal regularity on the exact solution. The estimate relies on both a priori and a posteriori analysis, and thus is called a medius error analysis. More precisely, we proved an optimal order error estimate under the assumption (u,p)∈H1+s(Ω)×Hs(Ω) with any s∈(0,1]. Extension to the standard interior penalty DG methods is also explored. Finally, numerical results are provided to verify our theoretical findings.

Error estimates of the direct discontinuous Galerkin method for two-dimensional nonlinear convection–diffusion equations: Superconvergence analysis

This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG) method for two-dimensional nonlinear convection–diffusion equations. To copy with the complexity and difficulty caused by the nonlinear term, we utilize the method of Taylor expansion and correction idea to achieve our superconvergence goal. Specifically, we first adopt Taylor expansion to decompose the error into two parts: a high-order nonlinear term and a lower-order linear term. Then by using the idea of correction function for the linear term, a high order error bound is obtained. By doing so, a unified analysis of DDG method for general nonlinear problems is finally established. We prove that, for any piecewise tensor-product polynomials of degree k≥2, the DDG solution is superconvergent at nodes and Lobatto points, with an order of O(h2k) and O(hk+2), respectively. Moreover, superconvergence properties for the derivative approximation are also studied and the superconvergence points are identified at Gauss points, with an order of O(hk+1). Numerical experiments are presented to confirm the sharpness of all the theoretical findings.

Design of a Website-Based Project Management Information System at Warung Nasi Kak Jum Saridah

Concept, testing, and service are the main priorities in the Warung Nasi Kak Jum Saridah business. Considering the use of paper media in order recording, table monitoring, and product data systems, problems often arise such as data duplication, misplaced documents, and errors in orders. Therefore, to overcome this problem, Warung Nasi Kak Jum Saridah needs a cashier management information system. This system will help business owners manage product data, monitor the dining table, and manage transactions more efficiently. Prototype analysis using the waterfall method in developing the cashier management information system at Warung Nasi Kak Jum Saridah. The hope is that this system will facilitate the overall cashier supervision and management process, increase efficiency in daily operations, and reduce errors that often occur in recording and monitoring data. Thus, the implementation of the cashier management information system will have a positive impact on the productivity and quality of Warung Nasi Kak Jum Saridah services to customers.

Error estimates and numerical simulations of a thermoviscoelastic contact problem with damage and long memory

This paper aims to investigate a thermal frictional contact model with damage and long memory effects. We consider a deformable body made of viscoelastic material and assume the process to be dynamic. The material is expected to adhere to the Kelvin–Voigt constitutive law, with damage and thermal effects incorporated. The variational formulation of the model results in a coupled system comprising a history-dependent hemivariational inequality governing the displacement field, a parabolic variational inequality describing the damage field and an evolution equation for the temperature field. In the analysis of this system, we initially introduce a fully discrete scheme, and then concentrate on deriving error estimates of numerical solutions. An optimal order error estimate is attained under some appropriate solution regularity assumptions. At the tail of this manuscript, numerical simulations are provided for the contact problem to validate the theoretical results.

From low-rank retractions to dynamical low-rank approximation and back

In algorithms for solving optimization problems constrained to a smooth manifold, retractions are a well-established tool to ensure that the iterates stay on the manifold. More recently, it has been demonstrated that retractions are a useful concept for other computational tasks on manifold as well, including interpolation tasks. In this work, we consider the application of retractions to the numerical integration of differential equations on fixed-rank matrix manifolds. This is closely related to dynamical low-rank approximation (DLRA) techniques. In fact, any retraction leads to a numerical integrator and, vice versa, certain DLRA techniques bear a direct relation with retractions. As an example for the latter, we introduce a new retraction, called KLS retraction, that is derived from the so-called unconventional integrator for DLRA. We also illustrate how retractions can be used to recover known DLRA techniques and to design new ones. In particular, this work introduces two novel numerical integration schemes that apply to differential equations on general manifolds: the accelerated forward Euler (AFE) method and the Projected Ralston–Hermite (PRH) method. Both methods build on retractions by using them as a tool for approximating curves on manifolds. The two methods are proven to have local truncation error of order three. Numerical experiments on classical DLRA examples highlight the advantages and shortcomings of these new methods.