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Approximate numerical methods are one of the most used strategies to extract information from many-interacting-agents systems. In particular, numerical approximations are of extended use to deal with epidemic, ecological and biological models, since unbiased methods like the Gillespie algorithm can become unpractical due to high CPU time usage required. However, the use of approximations has been debated and there is no clear consensus about whether unbiased methods or biased approach is the best option. In this work, we derive scaling relations for the errors in approximations based on binomial extractions. This finding allows us to build rules to compute the optimal values of both the discretization time and number of realizations needed to compute averages with the biased method with a target precision and minimum CPU-time usage. Furthermore, we also present another rule to discern whether the unbiased method or biased approach is more efficient. Ultimately, we will show that the choice of the method should depend on the desired precision for the estimation of averages. The binomial method, and similar approximations, are often used for numerical simulations of population models in mathematical epidemiology and ecology. The authors study the binomial method to approximate a stochastic dynamic and compare it with an unbiased Gillespie method for stochastic simulation deriving insights on optimal discretization schemes for the binomial method and provide corresponding rules that would indicate if binomial or unbiased solution methods are more efficient. [ABSTRACT FROM AUTHOR]

In this paper, an adaptive bearing rigid formation control strategy for a class of nonlinear system with unknown dead-zone inputs and external disturbance is proposed. Firstly, the I-Type fuzzy system is used to approximate the unknown nonlinear dynamics of the formation model, and the approximation errors and unknown external disturbance are eliminated by the parameter adaptive estimation. Furthermore, the adaptive dynamic estimation algorithm is utilized to estimate and compensate the unknown dead-zone parameters, effectively suppressing the impact of dead-zone on formation system performance. Finally, the stability of the formation system is proved based on LaSalle's invariance principle, and the effectiveness of the algorithm is verified by simulation results. [ABSTRACT FROM AUTHOR]

This work proposes a hybrid modeling framework based on recurrent neural networks (RNNs) and the finite element (FE) method to approximate model discrepancies in time-dependent, multi-fidelity problems, and use the trained hybrid models to perform bias correction of the low-fidelity models. The hybrid model uses FE basis functions as a spatial basis and RNNs for the approximation of the time dependencies of the FE basis' degrees of freedom. The training data sets consist of sparse, non-uniformly sampled snapshots of the discrepancy function, pre-computed from trajectory data of low- and high-fidelity dynamic FE models. To account for data sparsity and prevent overfitting, data upsampling and local weighting factors are employed, to instigate a trade-off between physically conforming model behavior and neural network regression. The proposed hybrid modeling methodology is showcased in three highly non-trivial engineering test-cases, all featuring transient FE models, namely, heat diffusion out of a heat sink, eddy-currents in a quadrupole magnet, and sound wave propagation in a cavity. The results show that the proposed hybrid model is capable of approximating model discrepancies to a high degree of accuracy and accordingly correct low-fidelity models. • A RNN/FE hybrid model is developed to learn model discrepancies of multi-fidelity FE simulations based on trajectory data. • The RNN captures the dynamics of the discrepancy function while spatial discrepancies are resolved by a FE basis. • Upsampling and Gaussian priors are employed to account for sparsity in the discrepancy data and prevent overfitting. • The model is employed as a correction operator for low-fidelity models in three non-trivial test cases. • The bias corrected models are significantly more accurate compared to the original low-fidelity models. [ABSTRACT FROM AUTHOR]

Large-scale symmetric and asymmetric matrices have emerged in predicting the relationship between genes and diseases. The emergence of large-scale matrices increases the computational complexity of the problem. Therefore, using low-rank matrices instead of original symmetric and asymmetric matrices can greatly reduce computational complexity. In this paper, we propose an approximation conjugate gradient method for solving the low-rank matrix recovery problem, i.e., the low-rank matrix is obtained to replace the original symmetric and asymmetric matrices such that the approximation error is the smallest. The conjugate gradient search direction is given through matrix addition and matrix multiplication. The new conjugate gradient update parameter is given by the F-norm of matrix and the trace inner product of matrices. The conjugate gradient generated by the algorithm avoids SVD decomposition. The backtracking linear search is used so that the approximation conjugate gradient direction is computed only once, which ensures that the objective function decreases monotonically. The global convergence and local superlinear convergence of the algorithm are given. The numerical results are reported and show the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we develop a method for the estimation of airgap spatial variation from the harmonics that appear in the stator's current under static eccentricity conditions. We move beyond the detection of the eccentricity fault and endeavour to quantify the severity of this fault. To achieve this, we formulate the problem as a mathematical inverse problem and attempt to solve it in the Bayesian framework using the so-called Bayesian approximation error approach. The solution takes the form of an algorithm capable of estimating the parameter(s) that characterize the spatial variation of the airgap length from noisy and limited data. Furthermore, it can quantify the uncertainty associated with the results as a byproduct of the Bayesian analysis approach employed in this paper. The results demonstrate that with each iteration of the algorithm, estimation accuracy improves, and the uncertainty of the estimated results reduces. • Bayesian approximation error analysis shows promise in monitoring electrical machines with eccentric rotors. • The paper applies Bayesian inversion for online estimation of airgap length in induction machines with eccentric rotors. • The study estimates static eccentricity ratio from limited data on harmonics of the machine line currents. • This paper gives an analytical method for calculating all inductances of induction machines under static eccentricity. [ABSTRACT FROM AUTHOR]

In this article, we study approximation properties of the variation spaces corresponding to shallow neural networks with a variety of activation functions. We introduce two main tools for estimating the metric entropy, approximation rates, and n-widths of these spaces. First, we introduce the notion of a smoothly parameterized dictionary and give upper bounds on the nonlinear approximation rates, metric entropy, and n-widths of their absolute convex hull. The upper bounds depend upon the order of smoothness of the parameterization. This result is applied to dictionaries of ridge functions corresponding to shallow neural networks, and they improve upon existing results in many cases. Next, we provide a method for lower bounding the metric entropy and n-widths of variation spaces which contain certain classes of ridge functions. This result gives sharp lower bounds on the L 2 -approximation rates, metric entropy, and n-widths for variation spaces corresponding to neural networks with a range of important activation functions, including ReLU k activation functions and sigmoidal activation functions with bounded variation. [ABSTRACT FROM AUTHOR]

Modern electronic devices are supported by multicore processors and they have become an unavoidable part of recent technologies. The Arithmetic Logic Unit (ALU) is one of the indispensable parts of the Central Processing Unit (CPU) which performs most of the operations. Operations like multiplications, and exponentials consume more power than other normal operations. Power management is one of the major challenges in designing processors. To maximize speed and reduce power consumption, a new technique in Multiplier and ALU incorporated with a multicore processor has been proposed. The Proposed Approximate Compatible ALU (ACALU) is designed to perform specified operations that consume more power, using an advanced multiplication technique called the First One First Operand (FOFO) method. The selection process of eight bits is done by the FOFO method followed by error detection and error approximation is performed to get the accurate output. The ACALU performs these operations in the Approximate Computing Mode which utilizes only eight bits for the operations, thus reducing the power consumption. The results of several analyses between multipliers, ALUs, and microprocessors are provided. As we optimize the multiplier and ALU unit, the multicore processor is also enhanced. The proposed multicore processor is compared with the AMD processor and the latest INTEL processor and the comparison results show that the proposed technique is 95.2% and 97.8% more efficient than the existing processors. Thus, the power consumption is minimized and the performance speed is increased in the proposed multicore processor. [ABSTRACT FROM AUTHOR]

Different from the construction process of the traditional finite difference method, we derive a large class of high-accuracy methods for approximating derivatives. Since Taylor expansion is avoided, the requirement for function smoothness in the new methods is greatly reduced. We analyze the approximation errors of the proposed methods and compare their approximation effects. As a typical application, we use the proposed methods to solve Elliptic, Hyperbolic and Parabolic problems numerically. For time-dependent problems, a class of new fully implicit difference schemes and the fourth-order Runge-Kutta method are both used for discretization. When using fully implicit difference schemes, unconditional stability and convergence are proved for hyperbolic and parabolic problems with the periodic boundary conditions. A large number of numerical experiments are provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

To make better decisions on approximation, we may need to increase reliable and useful information on different aspects of approximation. To enhance information about the quality and certainty of approximating the solution of an Apollonius-type quadratic functional equation, we need to measure both the quality and the certainty of the approximation and the maximum errors. To measure the quality of it, we use fuzzy sets, and to achieve its certainty, we use the probability distribution function. To formulate the above problem, we apply the concept of Z-numbers and introduce a special matrix of the form diag (A , B , C) (named the generalized Z-number) where A is a fuzzy time-stamped set, B is the probability distribution function, and C is a degree of reliability of A that is described as a value of A ∗ B . Using generalized Z-numbers, we define a novel control function to investigate H–U–R stability to approximate the solution of an Apollonius-type quadratic functional equation with quality and certainty of the approximation. [ABSTRACT FROM AUTHOR]

Bayesian optimization algorithms are widely used for solving problems with a high computational complexity in terms of objective function evaluation. The efficiency of Bayesian optimization is strongly dependent on the quality of the surrogate models of an objective function, which are built and refined at each iteration. The quality of surrogate models, and hence the performance of an optimization algorithm, can be greatly improved by selecting the appropriate hyperparameter values of the approximation algorithm. The common approach to finding good hyperparameter values for each iteration of Bayesian optimization is to build surrogate models with different hyperparameter values and choose the best one based on some estimation of the approximation error, for example, a cross-validation score. Building multiple surrogate models for each iteration of Bayesian optimization is computationally demanding and significantly increases the time required to solve an optimization problem. This paper suggests a new approach, called exploratory landscape validation, to find good hyperparameter values with less computational effort. Exploratory landscape validation metrics can be used to predict the best hyperparameter values, which can improve both the quality of the solutions found by Bayesian optimization and the time needed to solve problems. [ABSTRACT FROM AUTHOR]

An adaptive neural network energy-based control approach is proposed in this paper to address the issue of stabilizing a partial space elevator during cargo transportation. The method handles the challenges presented by external disturbances and model uncertainties. The dynamic model of the partial space elevator, with a typical multi-input and multi-output underactuated system, is presented. A control scheme using energy-based techniques is proposed to guarantee the stable configuration of the system, and auxiliary kinematic vector variables are designed to incorporate the uncontrollable variables to constitute controllable state subsets. Under these frameworks, adaptive control schemes are formulated using multi-layer neural networks to mitigate model uncertainty. Robust compensators are designed to effectively counteract the unknown boundaries of external disturbances and approximation errors. The stability of the closed control system is rigorously demonstrated through the application of the Lyapunov function and LaSalle's invariance principle. Numerical simulations demonstrate that the proposed control method is highly effective in suppressing oscillations during the transfer period of a partial space elevator. [ABSTRACT FROM AUTHOR]

The paper provides a precise error estimate for an asymptotic expansion of a certain stochastic control problem related to relative entropy minimization. In particular, it is shown that the expansion error depends on the regularity of functionals on path space. An efficient numerical scheme based on a weak approximation with Monte Carlo simulation is employed to implement the asymptotic expansion in multidimensional settings. Throughout numerical experiments, it is confirmed that the approximation error of the proposed scheme is consistent with the theoretical rate of convergence. [ABSTRACT FROM AUTHOR]

The computation of a similarity measure for genomic data is a standard tool in computational genetics. The principal components of such matrices are routinely used to correct for biases due to confounding by population stratification, for instance in linear regressions. However, the calculation of both a similarity matrix and its singular value decomposition (SVD) are computationally intensive. The contribution of this article is threefold. First, we demonstrate that the calculation of three matrices (called the covariance matrix, the weighted Jaccard matrix, and the genomic relationship matrix) can be reformulated in a unified way which allows for the application of a randomized SVD algorithm, which is faster than the traditional computation. The fast SVD algorithm we present is adapted from an existing randomized SVD algorithm and ensures that all computations are carried out in sparse matrix algebra. The algorithm only assumes that row-wise and column-wise subtraction and multiplication of a vector with a sparse matrix is available, an operation that is efficiently implemented in common sparse matrix packages. An exception is the so-called Jaccard matrix, which does not have a structure applicable for the fast SVD algorithm. Second, an approximate Jaccard matrix is introduced to which the fast SVD computation is applicable. Third, we establish guaranteed theoretical bounds on the accuracy (in L 2 norm and angle) between the principal components of the Jaccard matrix and the ones of our proposed approximation, thus putting the proposed Jaccard approximation on a solid mathematical foundation, and derive the theoretical runtime of our algorithm. We illustrate that the approximation error is low in practice and empirically verify the theoretical runtime scalings on both simulated data and data of the 1000 Genome Project. [ABSTRACT FROM AUTHOR]

Estimates of PM2.5 levels are crucial for monitoring air quality and studying the epidemiological impact of air quality on the population. Currently, the most precise measurements of PM2.5 are obtained from ground stations, resulting in limited spatial coverage. In this study, we consider satellite-based PM2.5 retrieval, which involves conversion of high-resolution satellite retrieval of Aerosol Optical Depth (AOD) into high-resolution PM2.5 retrieval. To improve the accuracy of the AOD to PM2.5 conversion, we employ the machine learning based post-process correction to correct the AOD-to-PM conversion ratio derived from Modern-Era Retrospective analysis for Research and Applications, Version 2 (MERRA-2) reanalysis model data. The post-process correction approach utilizes a fusion and downscaling of satellite observation and retrieval data, MERRA-2 reanalysis data, various high resolution geographical indicators, meteorological data and ground station observations for learning a predictor for the approximation error in the AOD to PM2.5 conversion ratio. The corrected conversion ratio is then applied to estimate PM2.5 levels given the high-resolution satellite AOD retrieval data derived from Sentinel-3 observations. Our model produces PM2.5 estimates with a spatial resolution of 100 meters at satellite overpass times. Additionally, we have incorporated an ensemble of neural networks to provide error envelopes for machine learning related uncertainty in the PM2.5 estimates. [ABSTRACT FROM AUTHOR]

This work presents the development of a state-input affine differential neural network that approximates a class of nonlinear systems with uncertain dynamics and perturbations. The feasibility of using differential neural networks is based on the approximation capabilities of artificial neural networks, assuring that the approximation error of the right-hand side of the system with uncertain dynamics is bounded. The dynamics of the free parameters of the differential neural network is obtained using the Lyapunov's second stability method, which provides the convergence of the identification error to an invariant set around the origin. The approximation capabilities of the proposed identifier are compared with a classical differential neural network, aiming to identify the dynamics of a Stewart platform, which is used as testing example. The results show that the cumulative norm of identification's error is smaller using the state-input affine differential neural network, which demonstrates better identification outcomes with the proposed modified identifier. An additional advantage of the state-input affine differential neural network is that it has a quasi-linear structure that could help formulate a controller for several systems, which is a further research interest. • Learning laws for State-input affine differential neural network (SIADNN) identifier. • Design based on the second stability method of Lyapunov. • The novel DNN identifier has more applications than the traditional DNN identifier. • Evaluation of performing the non-parametric identification of a Stewart platform. [ABSTRACT FROM AUTHOR]

This paper provides a convergence and stability analysis of the incremental value iteration algorithm under the influence of various errors. Incremental control is firstly used to linearize the continuous-time nonlinear system, recursive least squares (RLS) identification is then introduced to identify the incremental model online. Based on the incremental model, the value iteration algorithm is used to design an optimal adaptive controller, with an analytical optimal control law. Moreover, the convergence of the developed incremental value iteration algorithm is proved. The stability of the controller is analyzed using Lyapunov stability theory. Finally, a flight control simulation verifies the robustness of the controller to various initial conditions, as well as adaptation to actuator faults. [ABSTRACT FROM AUTHOR]

In this paper, the scope of discrete asymptotic homogenization employing voxel (cartesian) mesh discretization is expanded to estimate high fidelity effective properties of any periodic heterogeneous media with arbitrary Bravais's lattice symmetry, including those with non-orthogonal periodic bases. A framework was developed in Python with a proposed fast–nearest neighbour algorithm to accurately estimate the periodic boundary conditions of the discretized representative volume element of the lattice unit cell. Convergence studies are performed, and numerical errors caused by both voxel meshing and periodic boundary condition approximation processes are discussed in detail. It is found that the numerical error in periodicity approximation is cyclically dependent on the number of divisions performed during the meshing process and, thus, is minimized with a refined voxel mesh. Validation studies are performed by comparing the elastic properties of 2D hexagon lattices with orthogonal and non-orthogonal bases. The developed methodology was also applied to derive the effective properties of several lattice topologies, and variation of their anisotropic macroscopic properties with relative densities is presented as material selection charts. [ABSTRACT FROM AUTHOR]

The success of a multi-kilometre drive by a solar-powered rover at the lunar south pole depends upon careful planning in space and time due to highly dynamic solar illumination conditions. An additional challenge is that the rover may be subject to random faults that can temporarily delay long-range traverses. The majority of existing global spatiotemporal planners assume a deterministic rover-environment model and do not account for random faults. In this paper, we consider a random fault profile with a known, average spatial fault rate. We introduce a methodology to compute recovery policies that maximize the probability of survival of a solar-powered rover from different start states. A recovery policy defines a set of recourse actions to reach a safe location with sufficient battery energy remaining, given the local solar illumination conditions. We solve a stochastic reach-avoid problem using dynamic programming to find an optimal recovery policy. Our focus, in part, is on the implications of state space discretization, which is required in practical implementations. We propose a modified dynamic programming algorithm that conservatively accounts for approximation errors. To demonstrate the benefits of our approach, we compare against existing methods in scenarios where a solar-powered rover seeks to safely exit from permanently shadowed regions in the Cabeus area at the lunar south pole. We also highlight the relevance of our methodology for mission formulation and trade safety analysis by comparing different rover mobility models in simulated recovery drives from the LCROSS impact region. • The safety of a solar-powered rover depends on its location, energy and time of day. • Maximally safe behaviours depend on when and where delays occur during a traverse. • The probability of survival of a rover varies non-monotonically with time. • Min–max dynamic programming can conservatively account for the temporal risk ambiguity. [ABSTRACT FROM AUTHOR]

It has been shown that neural networks are vulnerable to adversarial attacks: adding well-crafted, imperceptible perturbations to their input can modify their output. Adversarial training is one of the most effective approaches to training robust models against such attacks. Unfortunately, this method is much slower than vanilla training of neural networks since it needs to construct adversarial examples for the entire training data at every iteration. By leveraging the theory of coreset selection, we show how selecting a small subset of training data provides a principled approach to reducing the time complexity of robust training. To this end, we first provide convergence guarantees for adversarial coreset selection. In particular, we show that the convergence bound is directly related to how well our coresets can approximate the gradient computed over the entire training data. Motivated by our theoretical analysis, we propose using this gradient approximation error as our adversarial coreset selection objective to reduce the training set size effectively. Once built, we run adversarial training over this subset of the training data. Unlike existing methods, our approach can be adapted to a wide variety of training objectives, including TRADES, ℓ p -PGD, and Perceptual Adversarial Training. We conduct extensive experiments to demonstrate that our approach speeds up adversarial training by 2–3 times while experiencing a slight degradation in the clean and robust accuracy. [ABSTRACT FROM AUTHOR]

Ship operators typically depend on the visual method for draft reading, which may lead to errors or approximations, and it further introduces draft survey calculation errors and cannot provide continuous updates. Ensuring accurate real-time ship draft measurement becomes crucial for enhancing navigational safety, optimizing vessel performance, and achieving precise cargo and consumable measurements. With the current approach towards automation and remote operation of ships, the need to rely on the accuracy of the measurements provided by the sensors has increased. Although different sensor types are gradually being adopted for draft measurement, they encounter challenges in the demanding marine environment which may result in noisy and inaccurate readings. This paper aims to estimate the true draft of a ship in different conditions from noisy sensor measurements using the Kalman filter algorithm. The purpose of the algorithm is to reduce uncertainty in draft measurement that is generated from inaccuracies in the sensor or from the dynamic marine environment. The paper involves designing the Kalman Filter algorithm for draft measurement to work within the different conditions the ship may experience. Simulating different situations and analyzing the result, the application of the filter shows the advantage in real-time draft measurement in both static and dynamic conditions. [ABSTRACT FROM AUTHOR]

In this paper, we provide a new framework for studying continued fractions (CFs) by means of the backwards continued fraction (BCF). We develop an approximation theory for BCFs based on taking expansions of a fixed length, show the correspondence between continued fractions and their BCFs counterpart, and illustrate a rich approximation theory for continued fractions based off the methods of the approximation theory for the backwards case. In particular, we construct explicit functions that are sharp bounds for the BCF or CF error infinitely often over any BCF or CF cylinder set, and work out the details to pass seamlessly between the BCF and CF expansion of any real number. [ABSTRACT FROM AUTHOR]

The detection of elongated structures like lines or edges is an essential component in semantic image analysis. Classical approaches that rely on significant image gradients quickly reach their limits when the structure is context-dependent, amorphous, or not directly visible. This study introduces a principled mathematical description of elongated structures with various origins and shapes. Among others, it serves as an expressive operational description of target functions that can be well approximated by Convolutional Neural Networks. The nominal position of a curve and its positional uncertainty are encoded as a heatmap by convolving the curve distribution with a filter function. We propose a low-error approximation to the expensive numerical integration by evaluating a distance-dependent function, enabling a lightweight implementation with linear time complexity. We analyze the method's numerical approximation error and behavior for different curve types and signal-to-noise levels. Application to surgical 2D and 3D data, semantic boundary detection, skeletonization, and other related tasks demonstrate the method's versatility at low errors. [ABSTRACT FROM AUTHOR]

The choice of the factorization rank of a matrix is critical, e.g., in dimensionality reduction, filtering, clustering, deconvolution, etc., because selecting a rank that is too high amounts to adjusting the noise, while selecting a rank that is too low results in the oversimplification of the signal. Numerous methods for selecting the factorization rank of a non-negative matrix have been proposed. One of them is the cophenetic correlation coefficient ( c c c ), widely used in data science to evaluate the number of clusters in a hierarchical clustering. In previous work, it was shown that c c c performs better than other methods for rank selection in non-negative matrix factorization (NMF) when the underlying structure of the matrix consists of orthogonal clusters. In this article, we show that using the ratio of c c c to the approximation error significantly improves the accuracy of the rank selection. We also propose a new criterion, c o n c o r d a n c e , which, like c c c , benefits from the stochastic nature of NMF; its accuracy is also improved by using its ratio-to-error form. Using real and simulated data, we show that c o n c o r d a n c e , with a CUSUM-based automatic detection algorithm for its original or ratio-to-error forms, significantly outperforms c c c . It is important to note that the new criterion works for a broader class of matrices, where the underlying clusters are not assumed to be orthogonal. [ABSTRACT FROM AUTHOR]

Digital differentiators enable the computation of the derivative of a continuous-time signal at discrete time instances, and they are used in many signal processing applications. This paper derives a unified filter order estimate for digital differentiators that are realized with linear-phase finite-length impulse response filters and designed in the minimax sense. The estimate is useful at the high-level system design when assessing the implementation complexity and it enables fewer designs when finding the minimal filter order required to satisfy a prescribed tolerable approximation error. The proposed unified estimate covers both wideband and lowpass differentiators of integer degrees up to ten. Furthermore, degree-individual filter order estimates are derived which improve and extend previous results. The performance of both the unified and degree-individual order estimates is evaluated through simulation examples and compared with previous estimates. [ABSTRACT FROM AUTHOR]

As we know, feedforward neural networks (FNNs) with sigmoidal activation function are universal approximators. Theoretically, for any continuous function defined on a compact set, there exists an FNN such that the FNN can approximate the function with arbitrary accuracy, which constitutes a theoretical guarantee that FNN can be used as an efficient learning machine. This paper addresses the construction and approximation for FNNs. We construct an FNN with sigmoidal activation function and estimate its approximation error. In particular, an inverse theorem of the approximation is established, which implies the equivalence characterization theorem of the approximation and reveals the relationship between the topological structure of the FNN and its approximation ability. As keys in this study, the concepts of modulus of continuity of function, K-functional, and their relationship are utilized, and two Bernstein-type inequalities are established. [ABSTRACT FROM AUTHOR]

In this paper it is shown, that if a possibly inhomogeneous Markov chain with continuous time and finite state space is weakly ergodic and all the entries of its intensity matrix are locally integrable, then, using available results from the perturbation theory, its time-dependent probability characteristics can be approximately obtained from another Markov chain, having piecewise constant intensities and the same state space. The approximation error (the taxicab distance between the state probability distributions) is provided. It is shown how the Cauchy operator and the state probability distribution for an arbitrary initial condition can be calculated. The findings are illustrated with the numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, a Szasz-Favard-Mirakyan operator is exploited for chaos synchronization of a master-slave system. The universal approximation feature asset lets in Szasz-Favard-Mirakyan operators for approximating uncertainties, including un-modeled dynamics and disturbances. The above subject is considered in detail in this investigation. It is confirmed that the synchronization/approximation errors are uniformly bounded and stable if the Szasz-Favard-Mirakyan operators are applied as the regressors. Additionally, it has been presumed that the synchronization's error rate is not available, and an observer will be designed for its estimation. The Duffing–Holmes oscillator is examined as the computer-generated chaotic framework with the target of examining the functioning of the recommended synchronization observer-based controller. The outcomes are compared with an effective approximation-based control strategy. Unlike Chebyshev Neural Network approximators in which the system's inputs states are needed to define the regressor vector and approximate uncertainties, the suggested Szasz-Favard-Mirakyan operators–based strategy is independent of the system states for constructing the regressor vector. [ABSTRACT FROM AUTHOR]

We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and output of an unknown function defined on an idempotent semifield, the problem is to find a best approximation of the function, by tropical Puiseux polynomial and rational functions. A new solution approach is proposed, which involves the reduction of the problem of polynomial approximation to the best approximate solution of a tropical linear vector equation with an unknown vector on one side (a one-sided equation). We derive a best approximate solution to the one-sided equation, and we evaluate the inherent approximation error in a direct analytical form. Furthermore, we reduce the rational approximation problem to the best approximate solution of an equation with unknown vectors on both sides (a two-sided equation). A best approximate solution to the two-sided equation is obtained in numerical form, by using an iterative alternating algorithm. To illustrate the new technique developed, we solve example approximation problems in terms of a real semifield, where addition is defined as maximum and multiplication as arithmetic addition (max-plus algebra), which corresponds to the best Chebyshev approximation by piecewise linear functions. [ABSTRACT FROM AUTHOR]

The detection of elongated structures like lines or edges is an essential component in semantic image analysis. Classical approaches that rely on significant image gradients quickly reach their limits when the structure is context-dependent, amorphous, or not directly visible. This study introduces a principled mathematical description of elongated structures with various origins and shapes. Among others, it serves as an expressive operational description of target functions that can be well approximated by Convolutional Neural Networks. The nominal position of a curve and its positional uncertainty are encoded as a heatmap by convolving the curve distribution with a filter function. We propose a low-error approximation to the expensive numerical integration by evaluating a distance-dependent function, enabling a lightweight implementation with linear time complexity. We analyze the method's numerical approximation error and behavior for different curve types and signal-to-noise levels. Application to surgical 2D and 3D data, semantic boundary detection, skeletonization, and other related tasks demonstrate the method's versatility at low errors. [ABSTRACT FROM AUTHOR]

The experiences of close relationships-revised (ECR-R) is a widely used 36-item self-report measurement for measuring adult attachment. However, various short versions of the ECR-R have been developed and tested psychometrically. Given the cultural impact, a short version of the Thai ECR-R should be derived from the existing Thai version of the ECR-R. This study aimed to develop a 10-item version of the ECR-R that demonstrates comparable psychometric properties to the previous Thai version and the 18-item ECR-R. This study included four studies with a total of 1,322 participants. In study 1, 434 adults in a nonclinical setting were used for the development of the 10-item Thai ECR-R and tested in an independent sample. Studies 2, 3, and 4 were conducted on 312 adults in the clinical setting, 227 older adults in the nonclinical, and 123 older adults in clinical settings. The Cronbach alphas and corrected correlations between the ECR-R-18 and the ECR-R-10 in each study were calculated. Confirmatory factor analysis of the first-order two-factor solution model with fit statistics was examined with each sample. Correlations of the ECR-R-18 and the ECR-R-10 with other measurements were presented and compared. Known-group validity and measurement invariance test were also examined. The Cronbach alphas of the ECR-R-10 among all samples were acceptable, ranging between.77 and.85 for avoidance subscales and between.82 and.86 for anxiety subscales. The corrected correlation between the ECR-R-18 and ECR-R-10 was between.61 (p <.001) and.82 (p <.001). The values of the comparative fit index and Tucker-Lewis index for the model of ECR-R-10 were between.903 and.985, whereas the root-mean-square error of approximation was between.082 and.036, indicating that the model fits were acceptable. The ECR-R-10 was related to the measurements with a similar construct; however, no difference in the magnitude of correlation was observed between ECR-R-18 and ECR-R-10. Known group validity was established. Measurement invariance was successfully established across different age and gender groups, although it was only partially achieved with respect to clinical status. The ECR-R-10 provided equal or superior psychometric properties to the ECR-R-18 across age groups and settings. As it is a briefer scale, the ECR-R-10 can be practically used in general and clinical samples to reduce the burden of assessment, especially with older adults. Further investigation is needed to test the scale's temporal stability. [ABSTRACT FROM AUTHOR]

Vibration-based health diagnostics technique has shown great potentials to enhance the safety and reliability for many industrial rotary machinery. The emerging sparse representation classification (SRC) paradigm provides a promising tool for intelligent machinery health diagnostics. However, traditional SRC approaches neglect the useful priori information in rotary machinery vibration data, limiting the reconstruction ability and thus restricting their diagnostic accuracy. To address this issue, we present a novel sparsity assisted intelligent recognition (SAIR) methodology for vibration-based machinery health diagnostics. SAIR is constituted by two stages for dictionary design and intelligent recognition. In the dictionary design stage, SAIR exploits the useful priori information of the prediction shift-invariance property, to design class-specific dictionaries via an overlapping segmentation strategy. Additionally, this dictionary design strategy can leverage those local and nonlocal features within data segments. In the intelligent recognition stage, SAIR implements health state recognition by means of the sparsity-based health diagnostic strategy according to minimal sparse approximation errors. Finally, the feasibility and advantage of SAIR have been comprehensively evaluated for planetary gearbox health diagnostics, indicating that SAIR yields an overall diagnostic accuracy of 99.72%. Besides, comparative studies demonstrate that SAIR outperforms the advanced mainstream methods with better diagnostic accuracy and lower computation costs for vibration-based machinery health diagnostics. [ABSTRACT FROM AUTHOR]

In this article, q -Chlodowsky operators are utilized for synchronizing a chaotic master-slave system. The universal approximation property enables q -Chlodowsky operators to approximate uncertainties, which consist of disturbances and unmodeled dynamics. It is verified that uniformly ultimately bounded stability of the tracking/approximation errors can be assured if the q -Chlodowsky operators are utilized as regressors and the polynomial coefficients are tuned by the adaptive laws calculated in the stability analysis. Moreover, it has been assumed that the rate of synchronization error is not at hand and a filter-based strategy has been adopted instead of designing an observer. The Duffing-Holmes oscillator is studied and the results are also compared with three powerful approximation-based and robust control methods. Unlike neural network/fuzzy estimators in which system states are necessary to define the regressor vector, the proposed uncertainty estimator does not require them in the regressor vector. Furthermore, in comparison with radial basis functions neural networks (RBFNNs), we are involved in fewer adaptive coefficients in the regressor vector of the proposed uncertainty estimator. As a result, the tedious and time-consuming procedure of trial and error for tuning the uncertainty estimator is eliminated. [ABSTRACT FROM AUTHOR]

We consider a new approach to approximate stability analysis for a tri-additive functional inequality and to obtain the optimal approximation for permuting tri-derivations and tri-homomorphisms in unital matrix algebras via the vector-valued alternative fixed-point theorem, which is a popular technique of proving the stability of functional equations. We also present a small list of aggregation functions on the classical, well-known special functions to investigate the best approximation error estimates using a different concept of perturbation stability. [ABSTRACT FROM AUTHOR]

Gaussian processes (GPs) are a popular technique for global metamodeling applications. Their objective in such settings is to establish an efficient and globally accurate approximation of the response surface of computationally expensive simulation models. When developing such GPs, the design of (simulation) experiments (DoE) plays an important role in reducing the required number of model runs for obtaining accurate approximations. Sequential (adaptive) selection of experiments can provide significant advantages, especially when the response surface is characterized by localized nonlinearities. Such adaptive DoE strategies for global metamodeling applications typically focus on minimizing the predictive GP variance, representing an exploration strategy, while recent developments have additionally considered the reduction of the GP bias obtained through cross validation, representing an exploitation strategy. While significant focus has been placed on the definition of appropriate adaptive DoE criteria, computational challenges still exist that limit the widespread adoption of adaptive DoE techniques—for example, related to the additional computational demand for identifying the optimal new experiment(s) or the necessity to establish proper schemes to combine exploration and exploitation strategies. To address these specific challenges, this research investigates two new adaptive DoE formulations. The first one focuses on the approximation of the popular integrated mean square error (IMSE) DoE criterion. The computationally demanding GP predictive variance update (after addition of each candidate experiment), required in the original IMSE formulation, is replaced by an approximation based on the current predictive variance and the domain of influence that surrounds each new experiment. The approximation is established through a parametric formulation that leverages the GP kernel to describe the aforementioned domain, with characteristics that are progressively calibrated across the GP training stages, to minimize the discrepancy between the actual and the approximated IMSE. The second formulation establishes a multicriteria search for simultaneously identifying multiple Pareto optimal experiments that balance exploration and exploitation objectives, replacing conventional strategies that establish a weighted combination of these objectives to promote a single DoE selection criterion. [ABSTRACT FROM AUTHOR]

In this paper, we first design a residual-type error estimator for a modified weak Galerkin (MWG) method of 2D H (curl) elliptic problems. Then, we show that the indicator is reliable with respect to the approximation error measured in terms of a natural energy norm, the main ingredient of our approach is to translate the error between weak solution and MWG solution into the conforming part and nonconforming part. We also prove the efficiency of the error estimator by using standard bubble functions. Finally, we provide several experiments to verify the performance of the error estimator within both uniform meshes and adaptive meshes. [ABSTRACT FROM AUTHOR]

The paper presents a posteriori error approximation concept based on residuals in the two-dimensional discontinuous Galerkin (DG) method. The approach is relatively simple and effective in application, and it takes advantage of some unique properties of the DG method. The error function is constructed in an enriched approximation space, utilizing the hierarchical nature of the basis functions. Among many versions of the DG method, the most popular one is based on the interior penalty approach. However, in this paper a DG method with finite difference (DGFD) is utilized, where the continuity of the approximate solution is enforced by finite difference conditions applied on the mesh skeleton. In the DG methods arbitrarily shaped finite elements can be used, so in this paper the meshes with polygonal finite elements are considered, including quadrilateral and triangular elements. Some benchmark examples are presented, in which Poisson's and linear elasticity problems are considered. The examples use various mesh densities and approximation orders to evaluate the errors. The error estimation maps, generated for the discussed tests, indicate a good correlation with the exact errors. In the last example, the error approximation concept is applied for an adaptive hp mesh refinement. [ABSTRACT FROM AUTHOR]

In order to estimate the statistical phase of underwater propagation for coherent acoustic systems, a calculation of background structure functions, based on wavespeeds, provides a wavefront distortion metric that excludes the phase due to focusing. In 1890, Gouy measured a phase change through the focus that differs for spherical vs cylindrical convergence by a factor of 2 from π/2. Statistics for the locations of the probable focus waist can use structure functions based on the measured sound speed profiles. Phase estimates are susceptible to errors in phase estimation unless results consider the Gouy phase while passing through focus waists. These foci may counteract or average with other errors of estimation, including scattering, type of spreading (cylindrical vs spherical foci), and reflections from the surface and bottom. Oceans have a sufficient range for the spatial derivatives of sound speed profiles to contribute persistently to foci that we show have coherence time and cross-sectional spatial (imaging) coherence. As an alternative, specular point theory can estimate deterministic pressure solutions. Convergence of these ray models that violate the stationary phase requires methods such as catastrophe theory to estimate the focal effects. Sommerfeld had an early comprehensive ray theory description that includes the Gouy phase. Using Flatté's data, the foci are clearly time and length coherent. Statistically, the mutual coherence function for the Atlantic, Caribbean, and Gulf of Mexico regions shows that even distorted and out-of-phase convergence has full spatial coherence. [ABSTRACT FROM AUTHOR]

Let F be a real valued generalized biaxisymmetric potential (GBASP) in Lβ on SR, the open sphere of radius R about the origin. In this paper we have obtained the necessary and sufficient conditions on the rate of decrease of a sequence of best harmonic polynomial approximates to F such that F is harmonically continues as an entire function GBASP and determine their (p, q)- order and generalized (p, q)-type with respect to proximate order ρ(r). [ABSTRACT FROM AUTHOR]

In this paper, we apply the well-known aggregation mappings on Mittag-Leffler-type functions to investigating new approximation error estimates of a W -Hilfer fractional differential equation, by a different concept of Ulam-type stability in both bounded and unbounded domains. [ABSTRACT FROM AUTHOR]

We present an open-source system that can optimize compressed trajectory representations for large fleets of vessels. We take into account the type of each vessel in order to choose a suitable configuration that can yield improved trajectory synopses, both in terms of approximation error and compression ratio. We employ a genetic algorithm that converges to a fine-tuned configuration per vessel type without any hyper-parameter tuning. These configurations can provide synopses that retain less than 10% of the original points with less than 20m approximation error in a real world dataset; in another dataset with 90% less samples than the previous one, the synopses retain 20% of the points and achieve less than 80m error. Additionally the level of compression can be chosen by the user, by setting the desired approximation error. Our system also supports incremental optimization by training in data batches, and therefore continuously improves performance. Furthermore, we employ a composite event recognition engine to efficiently detect complex maritime activities, such as ship-to-ship transfer and loitering; thanks to the synopses generated by the genetic algorithm instead of the raw trajectories, we make the recognition process faster while also maintaining the same level of recognition accuracy. Our extensive empirical study demonstrates the effectiveness of our system over large, real-world datasets. [ABSTRACT FROM AUTHOR]

In this paper, the composite control of position and orientation of the quadrotor for tracking purposes are investigated. The quadrotor is an underactuated nonlinear system. So, based on the introduced virtual inputs the quadrotor is transformed into an actuated system. External disturbances and different sources of uncertainty such as input saturation are considered in the dynamic model of the quadrotor. Also, after transform implementation, some complicated terms appear which are considered as other sources of uncertainty. To guarantee the stabilization and tracking goals of the quadrotor in the perturbed model, the composite adaptive fuzzy PID-Like controller is proposed. This controller consists of three parts where the innovation of the paper appears in all terms. The main term is an adaptive fuzzy PID-like controller which able to approximate the coefficients of PID based on the decentralized fuzzy logic. Then, a H∞ compensator based on a modified Riccati-like equation is presented; which can attenuate the fuzzy approximation errors. Also, as the last term of the composite controller, the modified disturbance observer is designed that estimates the uncertainty and disturbance. The experimental result shows that the proposed method leads to resilient stabilization and tracking in comparison with conventional methods. [ABSTRACT FROM AUTHOR]

In this paper, a new criterion for the evaluation of the theoretical efficiency of a greedy algorithm is suggested. Using this criterion, we prove some results on the rate of convergence of greedy algorithms, which provide expansions. We consider both the case of Hilbert spaces and the more general case of Banach spaces. The new component of this paper is that we bound the error of approximation by the product of two norms—the norm of f and the A 1 -norm of f. Typically, only the A 1 -norm of f is used. In particular, we establish that some greedy algorithms (Pure Greedy Algorithm (PGA) and its modifications) are as good as the Orthogonal Greedy Algorithm (OGA) in this new sense of the rate of convergence, while it is known that the PGA is much worse than the OGA in the standard sense. Our new results provide better bounds for the accuracy than known results in the case of small ∥ f ∥ . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the solution to a class of symmetric non-homogeneous two-dimensional fractional integro-differential equations using both analytical and numerical methods. We first show the differences between the Caputo derivative and the symmetric sequential fractional derivative and how they help facilitate the implementation of numerical and analytical approaches. Then, we propose a numerical approach based on the operational matrix method, which involves deriving operational matrices for the differential and integral terms of the equation and combining them to generate a single algebraic system. This method allows for the efficient and accurate approximation of the solution without the need for projection. Our findings demonstrate the effectiveness of the operational matrix method for solving non-homogeneous fractional integro-differential equations. We then provide examples to test our numerical method. The results demonstrate the accuracy and efficiency of the approach, with the graph of exact and approximate solutions showing almost complete overlap, and the approximate solution to the fractional problem converges to the solution of the integer problem as the order of the fractional derivative approaches one. We use various methods to measure the error in the approximation, such as absolute and L 2 errors. Additionally, we explore the effect of the derivative order. The results show that the absolute error is on the order of 10 − 14 , while the L 2 error is on the order of 10 − 13 . Next, we apply the Laplace transform to find an analytical solution to a class of fractional integro-differential equations and extend the approach to the two-dimensional case. We consider all homogeneous cases. Through our examples, we achieve two purposes. First, we show how the obtained results are implemented, especially the exact solution for some 1D and 2D classes. We then demonstrate that the exact fractional solution converges to the exact solution of the ordinary derivative as the order of the fractional derivative approaches one. [ABSTRACT FROM AUTHOR]

This paper focuses on the command-filter control of nonstrict-feedback incommensurate fractional-order systems. We utilized fuzzy systems to approximate nonlinear systems, and designed an adaptive update law to estimate the approximation errors. To overcome the dimension explosion phenomenon in the backstepping process, we designed a fractional-order filter and applied the command filter control technique. The closed-loop system was semiglobally stable, and the tracking error converged to a small neighbourhood of equilibrium points under the proposed control approach. Lastly, the validity of the developed controller is verified with simulation examples. [ABSTRACT FROM AUTHOR]

Deep learning with specific network topologies has been successfully applied in many fields. However, what is primarily called into question by people is its lack of theoretical foundation investigations, especially for structured neural networks. This paper theoretically studies the multichannel deep convolutional neural networks equipped with the downsampling operator, which is frequently used in applications. The results show that the proposed networks have outstanding approximation and generalization ability of functions from ridge class and Sobolev space. Not only does it answer an open and crucial question of why multichannel deep convolutional neural networks are universal in learning theory, but it also reveals the convergence rates. [ABSTRACT FROM AUTHOR]

Geometric error compensation is important in improving the machining accuracy of five-axis machine tools. To improve the efficiency of real-time compensation, this paper proposes a novel compensation method based on a novel representation of the volumetric error. First, 5 error parameters in the workpiece-tool coordinate system are employed to describe the volumetric error, simplifying the actual reverse kinematics from an overdetermined problem to a well-determined problem. Second, the actual inverse kinematics solution, neglecting the position-varying feature of geometric errors, is derived by a simple operation. Finally, the actual inverse kinematics solution considering the position-varying feature of geometric errors is derived based on a linear approximation of geometric errors at the command position nearby. Compared with traditional compensation methods, the proposed method could guarantee higher efficiency since it has a concise analytical expression. Simulation and experiment results are then illustrated to validate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Linear tool path with the inherent shortcoming of tangential and curvature discontinuities leads to poor machining efficiency and surface finish. A lot of local corner smoothing methods which inset a tiny spline curve at the corners have been proposed in literature. However, there exists potential overlap of the two adjacent spline curves while smoothing, and still no optimal solution to fully use the segment length of each corner while eliminating the overlaps. To optimally eliminate the overlaps and smooth the transition curve curvature in real time, a symmetrical non-uniform rational B-spline (NURBS) curve transition based on look ahead optimal method is proposed in this paper, by scanning a set of unconstrained pre-cornering transition curve information, analysing the overlap type and backward determining the optimal transition length with curvature extreme smoothness in consideration, thus fully use the linear segment length of each corner and fully increasing the feedrate at the corners. G 2 continuity, analytical expression of the curvature extreme, approximation error constraint, and real-time performance are considered simultaneously in this paper. Simulation and real machining experiments are made to compare with the existing symmetrical rounding algorithms, the results show that the proposed method has a significant curvature decrease of 17.64% and 4.44% improvement of efficiency, and the curvature distribution of the adjacent curves are balanced, thus improved the machining velocity smoothness. In addition, the real machining experiment results validate the proposed method can work well in 125 μs real-time control period CNC system. [ABSTRACT FROM AUTHOR]

The free-float space manipulator (FFSM) control problem subject to coupling dynamics, uncertainties, and input saturation constraints is investigated in this paper. The screw-based dynamical model is established through the Lagrangian equation and Hamilton's variational principle such that the posture of the rigid body can be described uniquely and globally. The derived dynamics is then reduced to a decoupled one in the light of the conservation of the momentum. On this basis, a novel anti-windup finite-time sliding-mode control strategy is investigated to guarantee the tracking performance in the joint space of FFSM, where a learning-based adaptive neural network is designed to estimate the uncertainties. The approximation error is inhibited by a tanh-type slow-varying adaptive law which is integrated into an auxiliary system to compensate for input saturation nonlinearities. With the contributions above, the stability of the proposed nonsingular terminal sliding-mode (NTSM) controller is comprehensively proved. Comparative simulations based on a 6-DOF FFSM confirm the validity of the proposed control scheme. [ABSTRACT FROM AUTHOR]

B-spline curve approximation is widely used for fitting linear toolpaths to improve machining quality and efficiency in CNC machining. For a high-quality smoothing method, the control of both approximation error and curve curvature needs to be taken into account. To reduce the maximum curve curvature while meeting the precision requirements, a B-spline approximation scheme with tool tip position adjustment is proposed in this paper. Toolpaths are first divided into several subdivisions according to the discrete curvature of each tool tip point. For subdivisions that need adjustment, tool tip positions are adjusted to minimize the maximum discrete curvature. An existing approximation method named energy-term-incorporated progressive and iterative approximation for least square fitting (ELSPIA) is selected and improved to fit the adjusted toolpaths and lower the chord errors. For improving the numerical stability of the ELSPIA method, the way of searching appropriate foot point parameters is discussed in this paper. Deviation expansion factors of tooltip points are also defined to make the ELSPIA method suitable for fitting the adjusted toolpaths. Both simulations and experimental studies are conducted to prove that the proposed method can significantly decrease the maximum curvature of fitted B-spline curves and improve the machining efficiency without exceeding the tolerance. For example, experimental results show that for the tested butterfly and rabbit toolpaths, the proposed method can improve the machining efficiency by 2.61% and 2.37%, compared with the ELSPIA method without tool tip position adjustment. [ABSTRACT FROM AUTHOR]

An efficient numerical algorithm is developed for solving nonlinear functional Volterra integral equations. The core idea is to define an appropriate operator, then combine the Krasnoselskij iterative scheme with collocation at discrete points and the Newton–Cotes quadrature rule. This results in an explicit scheme that does not require solving a nonlinear or linear algebraic system. For the convergence analysis, the discretization error is estimated and proved to converge via a recurrence relation. The discretization error is combined with the Krasnoselskij iteration error to estimate the total approximation error, hence establishing the convergence of the method. Then, numerical experiments are provided, first, to demonstrate the second order convergence of the proposed method, and secondly, to show the better performance of the scheme over the existing nonlinear-based approach. [ABSTRACT FROM AUTHOR]