Error Analysis Research Articles

System of fractal-fractional differential equations and Bernstein wavelets: a comprehensive study of environmental, epidemiological, and financial applications

Abstract The fractal-fractional derivatives uniquely incorporate memory effects, heterogeneity, and fractal geometry, making them essential for studying dynamic systems compared to integer-order derivatives which cannot capture such intricate dynamics. So, this study aims to analyze the dynamical behaviour of the model of systems of differential equations under Caputo fractal-fractional derivative by designing a numerical framework based on the fractional order Bernstein wavelets. The Caputo fractal-fractional derivative under the power law kernel has been employed to obtain more accurate performances of the considered model as compared to fractional and integer order. A key novelty of this work lies in the implementation of fractional order Bernstein wavelets with collocation grids, which transform the complex differential equations into simpler algebraic equations, ensuring computational efficiency. The validity of the mentioned scheme is demonstrated by examining some famous problems, such as pond pollution problem, SIR epidemic modelling of infectious disease, a model of HIV interactions with CD4+T cells, and a financial model, showcasing its broad applicability across applied science, finance and engineering. For compactness, an error analysis using residual function is performed for fractal-fractional order. The graphs of the solution in integer, fractional and fractal-fractional order show that the achieved solutions are very close to the actual result of the examples and the error progressively decreases as the number of wavelets basis increases. In the case α = β = 1, the obtained approximated wavelet solutions for the suggested model are in good harmony with the exact solutions, the fourth-order Runge-Kutta (RK4) method and existing schemes provided in the literature. This investigation demonstrates that the mentioned scheme is very effective and straightforward for solving such kinds of fractal-fractional models.

A Vieta–Lucas collocation and non-standard finite difference technique for solving space-time fractional-order Fisher equation

The purpose of the article is to analyze an accurate numerical technique to solve a space-time fractional-order Fisher equation in the Caputo sense. For this purpose, the spectral collocation technique is used, which is based on the Vieta–Lucas approximation. By using the properties of Vieta–Lucas polynomials, this technique reduces the nonlinear equations into a system of ordinary differential equations (ODEs). The non-standard finite difference (NSFD) method converts this system of ODEs into algebraic equations which have been solved numerically. Moreover, the error estimate is investigated for the proposed method. To show the accuracy and efficiency of the technique, the obtained numerical results are compared with the analytical results and existing results of the particular forms of the considered fractional order models through error analysis. The important feature of this article is the exhibition of variations of the field variable for various values of spatial and temporal fractional order parameters for different particular cases.

Application of fractional shifted vieta-fibonacci polynomials in nonlinear reaction diffusion equation with variable order time-space fractional derivative

Abstract In this article, an accurate optimization algorithm based on new polynomials namely generalized shifted Vieta-Fibonacci polynomials (GSVFPs) is employed to solve the nonlinear variable order time-space fractional reaction diffusion equation (NVOTSFRDE). The algorithm combines GSVFPs, new variable order fractional operational matrices in the Caputo sense, and the Lagrange multipliers to achieve the optimal solution. First, the solution of the NVOTSFRDE is approximated as a series of GSVFPs with unknown coefficients and parameters. Then, the Lagrange multipliers method is adopted so that the NVOTSFRDE can be transformed into a class of nonlinear algebraic system of equations and we solve these equations using MATLAB and MAPLE software. Solving this system and substituting the coefficients and parameters into the approximation of the guessed functions, the solution of the NVOTSFRDE is obtained. The convergence analysis of the approach are discussed. The accuracy of the algorithm is verified through error analysis and mathematical examples. The accuracy of the new method is higher than that of the exciting method. The reconstruction results demonstrate that the proposed optimization algorithm is efficient for the NVOTSFRDE, and the algorithm is also convergent.

Error analysis and spectral reconstruction of an infrared static miniature interferometric spectrometer.

Scientific-grade spectrometers with high hyperspectral resolution and high spectral accuracy are desirable in miniaturized optical systems to maintain stable and real-time spectral sampling. Fourier transform spectrometers that utilize high-precision moving mirrors generally struggle to enhance their miniaturization and stable real-time performance. A static infrared spectral measurement method is proposed that uses micro/nano-optical devices as the core of static interference and lightweight imaging. The use of micro/nano step mirrors allows for the instantaneous sampling of spectra. By employing an array of micro/nano lenses, interference imaging for each spectral channel can be accomplished. The spectrometer's all-static micro/nano-optical structure results in a reduction in volume and weight of more than half. Enhanced precision in design and fabrication is achieved through optical error analysis via a full-linkage optical field transmission model. An image edge detection-assisted spectral inversion algorithm is proposed, and the sampling stability and reconstruction accuracy are verified. The repeatability accuracy of interference intensity sampling surpasses 2%, and the peak accuracy of the reconstructed spectrum exceeds the resolution.

Error analysis of blade milling considering surface features and deformation

Error analysis of blade milling considering surface features and deformation

Sports Intelligence: Assessing the Sports Understanding Capabilities of Language Models Through Question Answering from Text to Video

Understanding sports presents a fascinating challenge for Natural Language Processing (NLP) due to its intricate and ever-changing nature. Current NLP technologies struggle with the advanced cognitive demands required to reason over complex sports scenarios. To explore the current boundaries of this field, we extensively evaluated mainstream and emerging large models on various sports tasks and addressed the limitations of previous benchmarks. Our study ranges from answering simple queries about basic rules and historical facts to engaging in complex, context-specific reasoning using strategies like few-shot learning and chain-of-thought techniques. Beyond text-based analysis, we also explored the sports reasoning capabilities of mainstream video language models to bridge the gap in benchmarking multimodal sports understanding. Based on a comprehensive overview of main-stream large models on diverse sports understanding tasks, we presented a new benchmark, which highlighted the critical challenges of sports understanding for NLP and the varying capabilities of state-of-the-art large models on sports understanding. We also provided an extensive set of error analyses that pointed to detailed reasoning defects of large model reasoning which model-based error analysis failed to reveal. We hope the benchmark and the error analysis set will help identify future research priorities in this field.

Numerical analysis of a spherical harmonic discontinuous Galerkin method for scaled radiative transfer equations with isotropic scattering

Abstract In highly diffusion regimes when the mean free path $\varepsilon $ tends to zero, the radiative transfer equation has an asymptotic behavior which is governed by a diffusion equation and the corresponding boundary condition. Generally, a numerical scheme for solving this problem has the truncation error containing an $\varepsilon ^{-1}$ contribution that leads to a nonuniform convergence for small $\varepsilon $. Such phenomenons require high resolutions of discretizations, which degrades the performance of the numerical scheme in the diffusion limit. In this paper, we first provide a priori estimates for the scaled spherical harmonic ($P_{N}$) radiative transfer equation. Then we present an error analysis for the spherical harmonic discontinuous Galerkin (DG) method of the scaled radiative transfer equation showing that, under some additional assumptions, its solutions converge uniformly in $\varepsilon $ to the solution of the scaled radiative transfer equation. We further present an optimal convergence result for the DG method with the upwind flux on Cartesian grids. Error estimates of $\left (1+\mathcal{O}(\varepsilon )\right )h^{k+1}$ (where $h$ is the maximum element length) are obtained when tensor product polynomials of degree at most $k$ are used.

A class of discontinuous Galerkin methods for nonlinear variational problems

In the context of discontinuous Galerkin methods, we study approximations of nonlinear variational problems. We propose element-wise nonconforming finite element methods to discretize the continuous minimisation problem. Using Γ \Gamma -convergence arguments we show that for convex energies the discrete minimisers converge to a minimiser of the continuous problem as the mesh parameter tends to zero, under the additional contribution of appropriately defined penalty terms at the level of the discrete energies. In addition, we provide error analysis in the case where the solution of the continuous minimization problem is smooth enough. Under appropriate assumptions we show local existence, uniqueness and corresponding optimal order error estimates for the discrete Euler-Lagrange equations of our scheme. We finally substantiate the feasibility of our methods by numerical examples.

Analysis of errors in junction temperature estimated by temperature-sensitive electrical parameter for parallel-connected SBDs

Abstract Multi-chip topology is widely adopted for large rated current power modules. Thermal management of multi-chip power modules has a significant impact on their long-term reliability. The transient thermal network model used for thermal management relies on the time response of junction temperature, which is estimated using temperature-sensitive electrical parameters (TSEPs) derived from the temperature dependency of I-V characteristics in power devices. However, the parallel connection of power devices complicates the junction temperature estimation because the sensing current is shared unevenly due to differences in the I-V characteristics of individual devices and the temperature gradient within the power module package. Consequently, the temperature estimated using a unified TSEP does not accurately represent the temperature difference among parallel-connected power devices. This paper investigates an error in the junction temperature estimated by using unified static TSEP for parallel-connected silicon carbide (SiC) Schottky barrier diodes (SBDs) as the power device. The results reveal that the unified TSEP underestimates the maximum temperature in multi-chip power modules. However, the semi-physical model of power devices enables us to estimate the junction temperature of each SiC SBD when we measure shared currents for each device.

Theoretical Proof and Implementation of Digital Beam Control and Beamforming Algorithm for Low Earth Orbit Satellite Broadcast Signal Reception Processing Terminal

Compared to analog beamforming, digital beamforming offers better self-calibration and lower sidelobe performance, which has a profound impact on improving low Earth orbit receiver performance. The Digital Beamforming (DBF) module in the low Earth orbit satellite broadcast signal reception terminal can use digital phase shifting to compensate for the phase differences caused by path and spatial distance variations due to inconsistent Radio Frequency (RF) channel delays. This compensation ensures in-phase summation, thereby achieving maximum energy reception in the desired direction. Although DBF has gained widespread attention in the radar field due to its unique functions and advantages, its application is limited by beamforming accuracy and gain. Therefore, with the development of DBF technology, how to improve its accuracy and gain has also attracted extensive attention both domestically and internationally. To address this issue, this paper proposes a beamforming method based on a cap-shaped array for low Earth orbit satellite broadcast signal reception and processing terminals. The method combines prior information and spatial domain search for beam control, and employs a lookup table for beam synthesis. It derives formulas for the Signal-to-Noise Ratio, noise figure, processing flow of the beamforming network, and the determination of beamforming weights for the spherical antenna array. The paper presents a beam control approach that combines prior information with spatial domain search, along with an implementation process for beam synthesis using a lookup table. It also details the corresponding Field-Programmable Gate Array (FPGA) implementation process. Finally, the beamforming algorithm is experimentally validated, and error analysis is conducted. The experimental results show that the measured beamforming sensitivity at all incident angles is below −133 dBm and the G/T values are all greater than −9 dB/K, the beam uniformity at three operating frequencies is less than 3°, and the measured errors in pitch and azimuth angles are both below 2°. The beam pointing error is also below 2°.

Mode Selection for Device to Device Communication in Dynamic Network: A Statistical and Deep Learning Method

A major challenge in device to device (D2D) communications is determining the appropriate communication modes for each potential D2D pair. In dynamic networks, the continuous movement of devices increases the complexity of channel state modeling, which makes it difficult to predict the quality of network service and select appropriate switching thresholds, ultimately affecting the accuracy of D2D mode selection. This paper proposes a novel D2D mode selection method, which integrates deep learning with statistical learning and includes three modules: signal-to-interference-plus-noise-ratio (SINR) prediction, error analysis, and threshold selection. Specifically, the SINR prediction module employs the gated recurrent unit (GRU) method to predict future SINR values; the error analysis module applies a non-parametric method to construct a probability density function of the prediction error. The combination of these two modules provides a significant improvement in prediction value accuracy. Additionally, in the threshold selection module, two constraints are innovatively introduced to mitigate the problem of frequent switching: average reliability (AR) and probably correct reliability (PCR). Simulation results demonstrate that the proposed method achieves higher system throughput, longer D2D mode residence time, and a lower mode switching frequency compared to other methods.

Adsorptive removal of toxic methylene blue and crystal violet dyes from aqueous solutions using Prunus spinosa: isotherm, kinetic, thermodynamic, and error analysis

Abstract In this study, it was aimed to use Prunus spinosa L. fruit pulp as an adsorbent zero-waste and low-cost for the removal of toxic methylene blue (MB) and crystal violet (CV) dyes from aqueous solutions. The adsorbent was characterized utilizing FTIR-ATR, SEM, and pHpzc tests. The pHpzc value of the adsorbent is 4.96. According to optimization experiments, the optimum adsorbent dosage was determined as 0.05 g/50 mL for MB and CV dyes, the optimum pH values were determined as approximately 7 for MB and CV dyes, and the optimum contact time was determined as 45 min for MB and 30 min for CV dyes. The Langmuir model has been used to calculate the maximum adsorption capacities of MB and CV dyes at a temperature of 298 K. The obtained values are 59.59 mg/g for MB and 53.19 mg/g for CV. The experimental data for Prunus spinosa L. for both dyes exhibited a pseudo-second-order kinetic model. According to error analyses, the reproducibility and applicability of isotherm and kinetic models were investigated. From thermodynamic results, the enthalpy values were calculated as − 42.04 kJ/mol for MB and − 24.08 kJ/mol for CV dyes, which indicates that the process is exothermic. Also, the Gibbs free energies of MB and CV dyes were determined as − 34.20 kJ/mol and − 32.33 kJ/mol at 298 K, which indicates the process is spontaneous. Research and comparisons with other adsorbents have demonstrated that Prunus spinosa L. is a cost-effective and appealing choice for removing MB and CV dyes from water solutions. Graphical Abstract

Quantification and Analysis of Human Errors Across the Life Cycle of Electrical Installations

The modern power system has different installations and as they are manmade, their consistency and security are dependent on human intervention. This research study has been carried out to analyse different types of faults in Electrical Installations according to life cycle and human related intervention in the various stages of the life cycle of an electrical installation. Secondary data has been collected to derive the results regarding the electrical faults in Electrical Installations according to life cycle and human intervention in the various stages of the life cycle of an electrical installation. It is found that electrical failures majorly occur in transformers, switchgear and rotating machines. By categorizing electrical faults into its life cycle, the different human errors were realised. The stages of machine life cycle found are design, assembly, inspection, operating and maintenance. By constructing a network and combining it with the life cycle of the installation, different human involvements have been recognized that leads to faults and accidents.

Optimal balanced-norm error estimate of the LDG method for reaction-diffusion problems II: The two-dimensional case with layer-upwind flux

A singularly perturbed reaction-diffusion problem posed on the unit square in R 2 \mathbb {R}^2 is solved numerically by a local discontinuous Galerkin (LDG) finite element method. Typical solutions of this class of 2D problems exhibit boundary layers along the sides of the domain; these layers generally cause difficulties for numerical methods. Our LDG method handles the boundary layers by using a Shishkin mesh and also introducing the new concept of a “layer-upwind flux”—a discrete flux whose values are chosen on the fine mesh (which lies inside the boundary layers) in the direction where the layer weakens. On the coarse mesh, one can use a standard central flux. No penalty terms are needed with these fluxes, unlike many other variants of the LDG method. Our choice of discrete flux makes it feasible to derive an optimal-order error analysis in a balanced norm; this norm is stronger than the usual energy norm and is a more appropriate measure for errors in computed solutions for singularly perturbed reaction-diffusion problems. It will be proved that the LDG method is usually convergent of order O ( ( N − 1 ln ⁡ N ) k + 1 ) O((N^{-1}\ln N)^{k+1}) in the balanced norm, where N N is the number of mesh intervals in each coordinate direction and tensor-product piecewise polynomials of degree k k in each coordinate variable are used in the LDG method. This result is the first of its kind for the LDG method applied to this class of problem and is optimal for convergence on a Shishkin mesh. Its sharpness is confirmed by numerical experiments.

A synthesis of Sphagnum litterbag experiments: initial leaching losses bias decomposition rate estimates

Abstract. Our knowledge of the magnitude and controls of Sphagnum decomposition rates is derived to a large extent from litterbag experiments that do not explicitly consider initial leaching losses. Previous research on vascular plants suggests that decomposition rate (k0) estimates from litterbag experiments are biased when initial leaching losses (l0) are ignored. In contrast, the magnitude and variability of l0 for Sphagnum litterbag experiments are not well known, and it is therefore also not known how much Sphagnum k0 estimates are biased. As Sphagnum is the main peat-forming species in many northern peatlands, and biases in k0 estimates can propagate and amplify in long-term peatland models, minimizing such bias is necessary for accurate predictions of peat accumulation. We present a meta-analysis of 15 Sphagnum litterbag studies to estimate initial leaching losses (l0), to analyze how much Sphagnum k0 estimates are biased when the decomposition model ignores initial leaching losses and to analyze how much the variance in k0 estimates increases due to initial leaching losses even when they are estimated by the decomposition model. Average l0 estimates range between 3 mass-% to 18 mass-%, and average k0 estimates range between 0.01 to 1.16 yr−1. Simulations and models fitted to empirical data indicate that ignoring initial leaching losses leads to an overestimation of k0. An error analysis suggests that k0 and l0 can be estimated only with relatively large errors because of limitations in the design of most available litterbag experiments. Sampling the first litterbags shortly after the start of the experiments allows more accurate estimation of l0 and k0. We also estimated large l0 (>5 mass-%) for only air-dried samples, which could imply that Sphagnum litterbag experiments with dried litter are unrepresentative for natural decomposition processes in which l0 may be smaller according to leaching experiments with fresh litter. We conclude that comparing results of litterbag experiments between experimental treatments and between studies and accurately estimating decomposition rates may only be possible if initial leaching losses are explicitly considered.

Deep learning generalization for diabetic retinopathy staging from fundus images

Objective. Diabetic retinopathy (DR) is a serious diabetes complication that can lead to vision loss, making timely identification crucial. Existing data-driven algorithms for DR staging from digital fundus images (DFIs) often struggle with generalization due to distribution shifts between training and target domains.Approach. To address this, DRStageNet, a deep learning model, was developed using six public and independent datasets with 91 984 DFIs from diverse demographics. Five pretrained self-supervised vision transformers (ViTs) were benchmarked, with the best further trained using a multi-source domain (MSD) fine-tuning strategy.Main results. DINOv2 showed a 27.4% improvement in L-Kappa versus other pretrained ViT. MSD fine-tuning improved performance in four of five target domains. The error analysis revealing 60% of errors due to incorrect labels, 77.5% of which were correctly classified by DRStageNet.Significance. We developed DRStageNet, a DL model for DR, designed to accurately stage the condition while addressing the challenge of generalizing performance across target domains. The model and explainability heatmaps are available atwww.aimlab-technion.com/lirot-ai.

A new error analysis for parabolic Dirichlet boundary control problems

This paper investigates the finite element approximation of a parabolic Dirichlet boundary control problem, presenting a new a priori error estimate. We establish two main convergence results for both semi-discrete and fully discrete optimal control problems, under suitable assumptions. Specifically, we demonstrate convergence orders of $O(k^{\frac{1}{4}})$ and $O(k^{\frac{3}{4}-\varepsilon})$ $(\forall\varepsilon>0)$ for the temporal semi-discretization of control problems on polytopes and smooth domains, respectively. For control problems defined on polyhedra, we achieve a convergence rate of $O(k^{\frac{1}{4}} +h^{\frac{1}{2}})$ in the fully discrete setting. The contributions of this work are twofold. First, we provide an improved temporal convergence rate for parabolic Dirichlet boundary control problems on smooth domains, setting a foundation for further fully discrete error analysis. Second, we refine the existing fully discrete error estimate for boundary control problems on polyhedra by removing the artificial mesh size restriction $k=O(h^2)$. As an intermediate but essential result, we establish both the convergence order and stability of the finite element approximation for parabolic inhomogeneous boundary value problems. Importantly, these results hold under low regularity boundary conditions without imposing mesh size constraints.

Dynamic Simulation of Underground Cable Laying for Digital Three-Dimensional Transmission Lines

In light of the issues associated with the laying process of transmission line cables, including concealed security risks and contact collisions between pulleys and cables, which primarily stem from reliance on drawings, this paper introduces a simulation methodology for the cable laying construction process utilizing Building Information Modeling (BIM) technology. Initially, two-dimensional DWG graphic data are employed to develop a model of the target equipment and construction environment using BIM software (Solid works 2020). Subsequently, the cable is accurately modeled by applying ADAMS virtual prototype technology, the bushing force connection method, and the macro command language. This allows for the construction of a three-dimensional real cable laying system for transmission lines, enabling the simulation of the dynamic cable laying process in the field. Subsequently, an error analysis is conducted to compare the axial tension and laying speed of the cable with theoretical calculation values. The study then proceeds to analyze tension fluctuations during the cable laying process and assess the load-bearing capacity of the pulleys, thus facilitating effective control of the construction process and enhancing safety measures. The findings indicate that the proposed method can accurately and efficiently simulate the on-site cable laying construction process, with numerical errors maintained below 5%, thereby validating the integrity of the model. Furthermore, the traction overload safety protection amplification coefficient is determined to be α = 1.5. It is highlighted that the bearing capacity of the block must exceed 60% of the load carried by the conductor at constant speed. This research provides a theoretical foundation for addressing safety hazards in cable laying engineering and holds certain engineering value.

Generalizable, sequence-invariant deep learning image reconstruction for subspace-constrained quantitative MRI.

To develop a deep subspace learning network that can function across different pulse sequences. A contrast-invariant component-by-component (CBC) network structure was developed and compared against previously reported spatiotemporal multicomponent (MC) structure for reconstructing MR Multitasking images. A total of 130, 167, and 16 subjects were imaged using T1, T1-T2, and T1-T2- -fat fraction (FF) mapping sequences, respectively. We compared CBC and MC networks in matched-sequence experiments (same sequence for training and testing), then examined their cross-sequence performance and generalizability by unmatched-sequence experiments (different sequences for training and testing). A "universal" CBC network was also evaluated using mixed-sequence training (combining data from all three sequences). Evaluation metrics included image normalized root mean squared error and Bland-Altman analyses of end-diastolic maps, both versus iteratively reconstructed references. The proposed CBC showed significantly better normalized root mean squared error than MC in both matched-sequence and unmatched-sequence experiments (p < 0.001), fewer structural details in quantitative error maps, and tighter limits of agreement. CBC was more generalizable than MC (smaller performance loss; p = 0.006 in T1 and p < 0.001 in T1-T2 from matched-sequence testing to unmatched-sequence testing) and additionally allowed training of a single universal network to reconstruct images from any of the three pulse sequences. The mixed-sequence CBC network performed similarly to matched-sequence CBC in T1 (p = 0.178) and T1-T2 (p = 0121), where training data were plentiful, and performed better in T1-T2- -FF (p < 0.001) where training data were scarce. Contrast-invariant learning of spatial features rather than spatiotemporal features improves performance and generalizability, addresses data scarcity, and offers a pathway to universal supervised deep subspace learning.

Scalar Auxiliary Variable (SAV) Stabilization of Implicit‐Explicit (IMEX) Time Integration Schemes for Non‐Linear Structural Dynamics

ABSTRACTImplicit‐explicit (IMEX) time integration schemes are well suited for non‐linear structural dynamics because of their low computational cost and high accuracy. However, the stability of IMEX schemes cannot be guaranteed for general non‐linear problems. In this article, we present a scalar auxiliary variable (SAV) stabilization of high‐order IMEX time integration schemes that leads to unconditional stability. The proposed IMEX‐BDFk‐SAV schemes treat linear terms implicitly using kth‐order backward difference formulas (BDFk) and non‐linear terms explicitly. This eliminates the need for iterations in non‐linear problems and leads to low computational costs. Truncation error analysis of the proposed IMEX‐BDFk‐SAV schemes confirms that up to kth‐order accuracy can be achieved and this is verified through a series of convergence tests. Unlike existing SAV schemes for first‐order ordinary differential equations (ODEs), we introduce a novel SAV for the proposed schemes that allows direct solution of the second‐order ODEs without transforming them into a system of first‐order ODEs. Finally, we demonstrate the performance of the proposed schemes by solving several non‐linear problems in structural dynamics and show that the proposed schemes can achieve high accuracy at a low computational cost while maintaining unconditional stability.