Second-order Error Research Articles

Confidence and second-order errors in cortical circuits.

Minimization of cortical prediction errors has been considered a key computational goal of the cerebral cortex underlying perception, action, and learning. However, it is still unclear how the cortex should form and use information about uncertainty in this process. Here, we formally derive neural dynamics that minimize prediction errors under the assumption that cortical areas must not only predict the activity in other areas and sensory streams but also jointly project their confidence (inverse expected uncertainty) in their predictions. In the resulting neuronal dynamics, the integration of bottom-up and top-down cortical streams is dynamically modulated based on confidence in accordance with the Bayesian principle. Moreover, the theory predicts the existence of cortical second-order errors, comparing confidence and actual performance. These errors are propagated through the cortical hierarchy alongside classical prediction errors and are used to learn the weights of synapses responsible for formulating confidence. We propose a detailed mapping of the theory to cortical circuitry, discuss entailed functional interpretations, and provide potential directions for experimental work.

Second-order error analysis of a corrected average finite difference scheme for time-fractional Cable equations with nonsmooth solutions

In this paper, we first formulate an average finite difference scheme with appropriate correction terms for the time-fractional initial-value problem of y′(t)+0RLDt1−αy(t)=g(t), where 0RLDt1−α denotes the Riemann–Liouville fractional derivative of order 1−α(α∈(0,1)). It is shown that, under some suitable conditions on the data, the numerical solution converges to the exact solution with order O(τ2), where τ is the time step size. The method is then extended to solve the time-fractional Cable equation, combined with a standard discretisation of the spatial derivatives on a uniform mesh. The second-order time convergence rate is proved. Several numerical examples are given to illustrate the good agreement with the theoretical analysis of the presented methods.

Geometrical misalignment-induced nonlinear error in homodyne interferometers

This paper discusses the generation of a novel periodic nonlinear error in homodyne interferometers due to geometric misalignments. These misalignments arise from cumulative assembly errors among the sensor head interior, target mirror, test platform, and detector, leading to dynamic misplacement of the measurement beam on the detector's surface. A physical model was developed to explain this error, focusing on the interference field produced by Gaussian beams under conditions of beam separation and inclined interference. Observations from misaligned setups revealed a strong correlation between the modulation of the interference signal by an envelope curve and the distorted, complex patterns of Lissajous figures, with the movement of the target mirror. Numerical simulations and experimental results demonstrated that segmented elliptical fitting effectively calibrates vortex trajectories and substantially reduces periodic nonlinear errors. However, numerical simulations also uncovered previously undetected picometer-scale second-order nonlinear errors due to the geometric misalignments. This research highlights the physical mechanisms behind new nonlinear errors, emphasizing their importance in advancing homodyne interferometers toward picometer-level accuracy.

A robust second-order low-rank BUG integrator based on the midpoint rule

Dynamical low-rank approximation has become a valuable tool to perform an on-the-fly model order reduction for prohibitively large matrix differential equations. A core ingredient is the construction of integrators that are robust to the presence of small singular values and the resulting large time derivatives of the orthogonal factors in the low-rank matrix representation. Recently, the robust basis-update & Galerkin (BUG) class of integrators has been introduced. These methods require no steps that evolve the solution backward in time, often have favourable structure-preserving properties, and allow for parallel time-updates of the low-rank factors. The BUG framework is flexible enough to allow for adaptations to these and further requirements. However, the BUG methods presented so far have only first-order robust error bounds. This work proposes a second-order BUG integrator for dynamical low-rank approximation based on the midpoint quadrature rule. The integrator first performs a half-step with a first-order BUG integrator, followed by a Galerkin update with a suitably augmented basis. We prove a robust second-order error bound which in addition shows an improved dependence on the normal component of the vector field. These rigorous results are illustrated and complemented by a number of numerical experiments.

Adaptive fuzzy integrated LMS based control strategy for voltage power quality enhancement

In this paper, an adaptive Takagi Sugeno (TS) Fuzzy integrated Mixed Order Normalized Least Mean Square/Fourth Algorithm (MONLMS/F) is proposed for the unresolved voltage power quality issues in distribution system. The proposed control utilizes the cost function of mixed order of errors, which are based on second and fourth order and the feature of cost function overcomes the demerit of traditional basic least mean square method, which is based on second-order error. The implemented control technique has two main objectives. Firstly, the fundamental estimation of correct active and reactive signals from polluted grid voltage for computing the reference load voltages and second is the DC link voltage regulation. A correct weight estimation is achieved by Mixed Order Normalized Least Mean Square/Fourth Algorithm (MONLMS/F) and voltage regulation is performed by Takagi Sugeno Fuzzy (TSF) approach. The proposed integrated control enhances the capability of system against voltage deviation from the reference voltage and successfully eliminates the harmonics from polluted grid voltages. A bio-inspired optimizer like Spotted Hyena (SH) is integrated with TSF for self-tuning of the membership functions (MFs) and fuzzy rule to adapt the system variations and regulate fluctuations in the DC link for dynamic conditions. However, auto-tunned TS-Fuzzy have a fast transient response of DC and AC link voltage based on dynamic indicators like settling time (ts=0.12 s), undershoot (Us=3.4 %) and peak overshoot (Mp=5.6 %) are compared to classical PI controller. Nonetheless, the proposed TS-MONLMS/F control strategy outperforms the other classical control with low oscillations at any power quality scenario with improved dynamic response. The significant MATLAB simulation and experimental results are evaluated, and its performance is demonstrated with the dSPACE (DS1202) processor for experimental realization.

Improvements on the discretisation of boundary conditions to the momentum balance for glacial ice

Abstract The flow of glacial ice is typically approximated as a nonNewtonian viscous fluid, with the momentum balance described by (an approximation to) the Stokes equations, and the nonlinear rheology described by a flow law. The most commonly used rheological law for glacial ice, Glen's flow law, yields infinite viscosity in the case of zero deformation, which can be the case at the ice surface. This poses a problem when solving the momentum balance numerically. We show that two commonly-used discretisation schemes for the boundary conditions at the ice surface and base, which yield proper numerical convergence when applied to simpler problems, produce poor numerical convergence and large errors, when used to solve the momentum balance with Glen's flow law. We show that a discretisation scheme based on the concept of ghost nodes, which substitutes the boundary conditions directly into the momentum balance equations, yields second-order numerical convergence and errors that can be up to four orders of magnitude smaller. These results are robust across different momentum balance approximations. We show that the improved boundary conditions are particularly useful for solving the 3-D higher-order Blatter-Pattyn Approximation (BPA). In general, this work underlines the importance of thoroughly verifying the numerical solvers used in ice-sheet models, before applying them to future projections of ice-sheet mass loss.

Retrospective correction of second-order concomitant fields in 3D axial stack-of-spirals imaging on a high-performance gradient system.

MRI using 3D stack-of-spirals (SoS) readout on a high-performance gradient system is subject to strong second-order, spatially varying concomitant fields, which can lead to signal dropout and blurring artifacts that become more significant at locations farther from the gradient isocenter. A method for compensating for second-order concomitant fields in 3D axial SoS image reconstruction is described. We retrospectively correct for second-order concomitant field-induced phase error in the 3D SoS data by slice-dependent k-space phase compensation based on the nominal spiral readout trajectories. The effectiveness of the method was demonstrated in phantom and healthy volunteer scans in which 3D pseudo-continuous arterial spin labeling imaging was performed with SoS fast spin-echo readout at 3 T. Substantial reduction in blurring was observed with the proposed method. In phantom scans, blurring was reduced by about 53% at 98 mm from the gradient isocenter. In the in vivo 3D pseudo-continuous arterial spin labeling scans, differences of up to 10% were observed at 78 mm from the isocenter, especially around the white-matter and gray-matter interfaces, between the corrected and uncorrected proton density images, perfusion-weighted images, and cerebral blood flow maps. The described retrospective correction method provides a means to correct erroneous phase accruals due to second-order concomitant fields in 3D axial stack-of-spirals imaging.

Temporal High-Order Accurate Numerical Scheme for the Landau–Lifshitz–Gilbert Equation

In this paper, a family of temporal high-order accurate numerical schemes for the Landau–Lifshitz–Gilbert (LLG) equation is proposed. The proposed schemes are developed utilizing the Gauss–Legendre quadrature method, enabling them to achieve arbitrary high-order time discretization. Furthermore, the geometrical properties of the LLG equation, such as the preservation of constant magnetization magnitude and the Lyapunov structure, are investigated based on the proposed discrete schemes. It is demonstrated that the magnetization magnitude remains constant with an error of (2p+3) order in time when utilizing a (2p+2)th-order discrete scheme. Additionally, the preservation of the Lyapunov structure is achieved with a second-order error in the temporal step size. Numerical experiments and simulations effectively verify the performance of our proposed algorithm and validate our theoretical analysis.

Roll Prediction and Parameter Identification of Marine Vessels Under Unknown Ocean Disturbances

Abstract This paper deals with two topics: roll predictions of marine vessels with machine-learning methods and parameter estimation of unknown ocean disturbances when the amplitude, frequency, offset, and phase are difficult to estimate. This paper aims to prevent the risky roll motions of marine vessels exposed to harsh circumstances. First of all, this study demonstrates complex dynamic phenomena by utilising a bifurcation diagram, Lyapunov exponents, and a Poincare section. Without any observers, an adaptive identification applies these four parameters to the globally exponential convergence using linear second-order filters and parameter estimation errors. Then, a backstepping controller is employed to make an exponential convergence of the state variables to zero. Finally, this work presents the prediction of roll motion using reservoir computing (RC). As a result, the RC process shows good performance for chaotic time series prediction in future states. Thus, the poor predictability of Lyapunov exponents may be overcome to a certain extent, with the help of machine learning. Numerical simulations validate the dynamic behaviour and the efficacy of the proposed scheme.

Improving Volume and Biomass Equations for Pinus oocarpa in Nicaragua

We present a new set of equations for tree level volume and aboveground biomass estimation for ocote pine (Pinus oocarpa Schiede ex Schltdl). These equation systems are the first developed for this species in Nicaragua. The first system includes a taper function, a merchantable volume equation, and volume equations for stem, coarse branches, and whole trees. The second system estimates whole tree and individual tree component biomass (stem wood, bark, branches, and needles). Data from 112 sampled trees were used for models’ development. Seemingly Unrelated Iterative Regression and the Generalized Method of Moments were used to simultaneously fit the volume and biomass equations systems, respectively; both methods ensure additivity and compatibility between equations. Weighted regression and a second-order continuous autoregressive error structure were used to correct heteroscedasticity and autocorrelation within the hierarchical dataset. The predictive power of the new proposed equations is higher than the currently used models for P. oocarpa in the country. These equation systems represent a scientific advancement that will enhance forest inventories, optimize timber management of the species, and facilitate accurate monitoring of forest carbon dynamics. Additionally, the new equations will contribute to a more precise accounting of CO2 emissions from the country’s forestry sector.

Research on High-Precision Position Detection Based on a Driven Laser Spot in an Extreme Ultraviolet Light Source

Laser-spot-location detection technology based on photodetectors is widely used in the aerospace, medical, military and communication fields. However, most of the current research focuses on continuous laser detection in the visible and near-infrared bands, and the real-time high-precision position detection of a long-wave infrared pulsed laser is lacking. In this paper, a spot-position detection system based on a four-quadrant detector is designed for a 10.6 μm CO2-driven laser in extreme ultraviolet light source, and a second-order extended error compensation algorithm based on a Gaussian-spot model is proposed. Finally, the algorithm is verified and analyzed experimentally by a spot-position detection system under both focusing and defocusing conditions. The experimental results show that the root-mean-square error, maximum absolute error and average absolute error of the second-order error compensation algorithm are significantly reduced compared with the traditional algorithm, and the detection accuracy of the spot-position is better than 9 μm. The above results show that this spot-position detection system has obvious advantages and high accuracy, which can realize the high-precision real-time detection of a laser’s spot position to obtain accurate spot position information, provide feedback adjustments for subsequent beam pointing control, and provide a theoretical basis for the beam pointing stability of the extreme ultraviolet light source system.

Detection of malicious requests aimed at disrupting the availability of cyber-physical systems

The work is devoted to solving the problem of algorithmization of the security management processes of cyber-physical systems by detecting malicious requests aimed at disrupting the availability of management interfaces. Particular attention is paid to attacks aimed at denial of service of cyber-physical systems by sending HTTP-flood to web management interfaces. This paper proposes algorithmic provision for comprehensive adaptive analysis of incoming requests. The proposed algorithm for the detection of malicious requests analyses the activity of the investigated components of the cyber-physical system's web service at various network levels. The work applies a visual analysis and data processing method based on the representation as a single normalized set. The raw data of the analysed queries is grouped in a special way to detect a particular anomaly as a suspected threat. Examples of data changes and security responses are given. The experimental results confirm that the proposed algorithmic software achieves first- and second-order error reduction compared to the commonly used regression models in modern application-layer firewalls. The results obtained can be applied to the further development of the theory of information security, in particular the information security of cyber-physical systems and systems of processing of especially protected confidential information.

Recursive Constrained Sine Second-Order Error Promoting Adaptive Algorithm

This brief proposes a recursive constrained sine second-order error promoting adaptive (RCSSOEPA) algorithm. Compared with classical recursive method, the RCSSOEPA algorithm can achieve better steady state performance in impulsive-noise. In general, the sine second-order error (SSOE) is constructed to devise a new recursive constrained adaptive-filtering within the least-squares framework for solving linear constrained optimization problems. The mean-square (MS) stability of the RCSSOEPA and its theoretical instantaneous MS deviation under Gaussian and non Gaussian noise are analyzed, numerically investigated and discussed in detail. Simulated results are reported to give a comfirmation of the theoretical analysis, and show that the RCSSOEPA outperforms recent developed constrained adaptive filtering algorithms in the estimation misalignment and when used for system identification under impulsive-noise.

Performance evaluation of dynamic voltage restorer using intelligent control algorithm

Power quality has become an important issue due to increasing applications of modern industrial and commercial sensitive equipment's. Due to high penetration of modern power electronics-based devices leads to voltage disturbances. To address the voltage disturbances the commonly employed series device called Dynamic Voltage Restorer (DVR) is used which requires an effective control technique. A control scheme namely Volterra adaptive expanded filter is developed which incorporate the higher ordered cost function. The conventional filters have like least mean square (LMS) algorithm is restricted to power error of second order. This restricts the computation of fundamental quantity from the disrupted signals and results in degradation under dynamical scenarios. The recommended approach surmount this aforesaid issue with a solution of a mixed learning of step size. This is derived from the error of second and fourth-order. The Volterra based LMS/Fourth (LMS/F) control has the merit of estimating the fundamental quantity under any power quality (PQ) scenario with less delay and low computational burden. The conventional controller like Proportional Integral (PI) has limited gain coefficients to achieve the optimized the voltage regulation. Furthermore, this is overcome by a Fractional Order PID (FOPID) with an additional coefficient of λ and γ obtain more stabilized DC and AC bus voltage by integrating with Kepler optimizer. For achieving optimal gain, an online tuning of FOPID is done with Kepler Optimizer. The Kepler is inspired by planet motion to achieve best gain values. The evaluation of FOPID is made by considering as a Fitness Function (FF) an Integral Time Square Error (ITSE). Owing to the superior performance of Kepler optimizer based FOPID under transitory state and an enhanced DVR performance is observed. The optimized FOPID has improved response in terms of less overshoot (2.9 %), undershoot (5.6 %), and fast settle time (0.104 s). To authenticate the simulating and experimental results dSPACE board is employed to validate the efficacy and accuracy of the proposed Volterra LMS/F technique.

Semidiscrete numerical approximation for dynamic hemivariational inequalities with history-dependent operators

In this paper, we are concerned with a class of second-order hemivariational inequalities involving history-dependent operators. For the problem, we first derive a semidiscrete scheme by implicit Euler formula and prove its unique solvability. The existence and uniqueness of a solution to the inequality problem is given by Rothe method. As the core part of the paper, we propose a two-step semidiscrete approximation for the problem, provide its unique solvability and obtain its second-order error estimates. The two-step scheme is more accurate than the standard implicit Euler scheme. Finally, we apply the results to a dynamic frictionless contact problem with long memory.

IMPLICATIONS OF SECOND-ORDER WAVE GENERATION FOR USE IN WAVESTRUCUTRE RESPONSE EXPERIMENTS

Coastal communities and critical coastal assets are therefore increasingly reliant on engineered protection from wave-induced flooding. Dynamic wave force and wave run-up are among key design parameters of such protection. Present understanding of coastal wave-structure interactions and responses was gained through large databases of experimental data as well as numerical, and field measurements. It is well known that experimental data of wave-structure interaction are contaminated by second-order error waves at sub- and super-harmonic frequencies when first-order wave generation is used. The error waves arise from disparity between linear wave-maker signals and non-linear boundary conditions at the wave generator. Herein, we conduct a novel investigation by experiment of the implications of second-order wave generation for dynamic wave force and run-up on a vertical wall, in shallower depths than previously published (kd = 0.6 - 1.1). The implications of error waves on coastal responses is quantified through comparison of first-order generated (FOG) and secondorder generated (SOG) wave group experiments.

ACCURACY ASSESSMENT OF PRECISE POINT POSITIONING WITH MULTI-CONSTELLATION GNSS DATA UNDER STRONG SOLAR BURST EFECTS

Solar variations modify a layer of the Earth’s upper atmosphere known as the ionosphere. This is of particular concern for the aviation sector because of the way its communications and navigation systems can be affected. At the same time, one of the most complex atmospheric effects is the response of ionospheric regions to geomagnetic storms. The ionospheric response during the same storm can vary in time in different locations, which can introduce significant errors/displacement (meters) in single-frequency relative GNSS positioning (DGNSS technology). The residual effect can be somewhat mitigated by using dual- or multi-frequency GNSS, but dual frequency is not a guarantee against degradation of relative observations results, especially during significant geomagnetic storms. In this regard, PPP absolute positioning technology can be effective. However, another atmospheric effect – ionospheric scintillation can have a significant impact on the accuracy of both GNSS positioning approaches. The main goal of this study was to analyze the effect of second-order ionospheric delay during geomagnetic storms and ionospheric scintillations on GNSS positioning using the PPP method. GNSS data corrected and uncorrected for higher-order ionospheric delay, respectively, were processed by the static PPP-AR method using the PRIDE-PPPAR ver.2.2.6 software for the selected periods of geomagnetic storms. From the analysis of the influence of second-order ionospheric errors, it follows that their values can reach almost 4 cm for first-frequency signals under different states of ionospheric disturbances for the GPS constellation and almost an order of magnitude less for the GNSS quadroconstellation. The appearance of stronger geomagnetic storms increases the second-order ionospheric errors by several millimeters.

Second-order error analysis of the averaged L1 scheme $$\overline {{\rm{L1}}} $$ for time-fractional initial-value and subdiffusion problems

Second-order error analysis of the averaged L1 scheme $$\overline {{\rm{L1}}} $$ for time-fractional initial-value and subdiffusion problems

Second-Order Error Analysis for Fractal Mobile/Immobile Allen–Cahn Equation on Graded Meshes

Second-Order Error Analysis for Fractal Mobile/Immobile Allen–Cahn Equation on Graded Meshes

Least Mean Mixed Norm Square/Fourth Adaptive Algorithm With Optimized FOPID Gains for Voltage Power Quality Mitigation

The conventional adaptive Least Mean Square (LMS) algorithm employs Mean Square Error (MSE) as a cost function based on second-order power error. This limits the extraction of fundamental signals from the noise and suffers from performance degradation in certain dynamic scenarios. The proposed approach addresses this problem with a mixed norm-based step size adaptive algorithm which is devised from the second and fourth-order error optimization. The proposed Mixed-Norm Constraint-based Improved Proportionate Normalized Least Mean Square Fourth (MNC-IPNLMS/F) control strategy extracts the fundamental weight quantity from the polluted grid voltage and generates the reference load voltage. This method exploits the enhancement of Dynamic Voltage Restorer (DVR) performance in terms of convergence and stability. The DC and AC-link voltage is regulated by Fractional Order Proportional Integral Derivative (FOPID) controller. The coefficients of FOPID are self-tuned by several optimizations namely, JAYA algorithm, Accelerated Particle Swarm Optimization (APSO), and Biogeography Based Optimization (BBO). This reduces manual tuning and computational complexity. The major improvements of the proposed approach are less overshoot (6.6%), undershoot (3.3%), and fast settling time (0.14s). The comparative study reveals that the FOPID control based on JAYA Optimization (JO) outperforms the others. The efficacy of the MNC-IPNLMS/F is investigated through the performance results.