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An issue for molecular dynamics simulations is that events of interest often involve timescales that are much longer than the simulation time step, which is set by the fastest timescales of the model. Because of this timescale separation, direct simulation of many events is prohibitively computationally costly. This issue can be overcome by aggregating information from many relatively short simulations that sample segments of trajectories involving events of interest. This is the strategy of Markov state models (MSMs) and related approaches, but such methods suffer from approximation error because the variables defining the states generally do not capture the dynamics fully. By contrast, once converged, the weighted ensemble (WE) method aggregates information from trajectory segments so as to yield unbiased estimates of both thermodynamic and kinetic statistics. Unfortunately, errors decay no faster than unbiased simulation in WE as originally formulated and commonly deployed. Here, we introduce a theoretical framework for describing WE that shows that the introduction of an approximate stationary distribution on top of the stratification, as in nonequilibrium umbrella sampling (NEUS), accelerates convergence. Building on ideas from MSMs and related methods, we generalize the NEUS approach in such a way that the approximation error can be reduced systematically. We show that the improved algorithm can decrease the simulation time required to achieve the desired precision by orders of magnitude. [ABSTRACT FROM AUTHOR]

It is shown that the four-component (4C), quasi-four-component (Q4C), and exact two-component (X2C) relativistic Hartree–Fock equations can be implemented in a unified manner by making use of the atomic nature of the small components of molecular 4-spinors. A model density matrix approximation can first be invoked for the small-component charge/current density functions, which gives rise to a static, pre-molecular mean field to be combined with the one-electron term. As a result, only the nonrelativistic-like two-electron term of the 4C/Q4C/X2C Fock matrix needs to be updated during the iterations. A "one-center small-component" approximation can then be invoked in the evaluation of relativistic integrals, that is, all atom-centered small-component basis functions are regarded as extremely localized near the position of the atom to which they belong such that they have vanishing overlaps with all small- or large-component functions centered at other nuclei. Under these approximations, the 4C, Q4C, and X2C mean-field and many-electron Hamiltonians share precisely the same structure and accuracy. Beyond these is the effective quantum electrodynamics Hamiltonian that can be constructed in the same way. Such approximations lead to errors that are orders of magnitude smaller than other sources of errors (e.g., truncation errors in the one- and many-particle bases as well as uncertainties of experimental measurements) and are, hence, safe to use for whatever purposes. The quaternion forms of the 4C, Q4C, and X2C equations are also presented in the most general way, based on which the corresponding Kramers-restricted open-shell variants are formulated for "high-spin" open-shell systems. [ABSTRACT FROM AUTHOR]

In order to ensure the flight safety of small unmanned aerial vehicles (UAVs), a deep neural network-fused mathematical modeling approach is put up for reliable flight control of small UAVs. First, engine torque, thrust eccentricity and initial stop angle are taken into full consideration. A six-degree-of-freedom nonlinear model is formulated for small UAVs, concerning both ground taxiing and air flight status. Then, the model was linearized using the principle of small disturbances. The linearized model expressions for both ground taxiing and air flight were provided. In addition, radial basis function neural networks are used for online approximation to address the nonlinearity and uncertainty caused by changes in aircraft aerodynamic parameters. At the same time, to compensate for the external disturbance and the approximation error of the neural network, the system robustness is improved by selecting reasonable design parameters. This helps the whole flight control system obtain better tracking control performance. At last, some simulation experiments are carried out to evaluate the performance of the proposed mathematical modeling framework. The simulation results show that the proposal has stronger convergence ability, smaller prediction error, and better performance. Thus, proper proactivity can be acknowledged. [ABSTRACT FROM AUTHOR]

This paper presents a numerical comparison of finite-element methods resulting in local mass conservation at the element level for Poisson's problem, namely the primal hybrid and mixed methods. These formulations result in an indefinite system. Alternative formulations yielding a positive-definite system are obtained after hybridizing each method. The choice of approximation spaces yields methods with enhanced accuracy for the pressure variable, and results in systems with identical size and structure after static condensation. A regular pressure precision mixed formulation is also considered based on the classical RT k space. The simulations are accelerated using Open multi-processing (OMP) and Thread-Building Blocks (TBB) multithreading paradigms alongside either a coloring strategy or atomic operations ensuring a thread-safe execution. An additional parallel strategy is developed using C++ threads, which is based on the producer-consumer paradigm, and uses locks and semaphores as synchronization primitives. Numerical tests show the optimal parallel strategy for these finite-element formulations, and the computational performance of the methods are compared in terms of simulation time and approximation errors. Additional results are developed during the process. Numerical solvers often fail to find an accurate solution to the highly indefinite systems arising from finite-element formulations, and this paper documents a matrix ordering strategy to stabilize the resolution. A procedure to enable static condensation based on the introduction of piecewise constant functions that also fulfills Neumann's compatibility condition, and yet computes an average pressure per element is presented. • Numerical comparison of the mixed, primal hybrid, and other locally conservatives finite element methods. • Introduction of a twice hybridized primal hybrid method, yielding a positive definite matrix after static condensation. • Comparison of multi-threaded strategies and thread-safety mechanisms. • A numerical procedure to stabilize the linear system resolution for indefinite matrices is described. • A general procedure that enables static condensation and fulfills Neumann's compatibility condition. [ABSTRACT FROM AUTHOR]

For the first time, we propose an efficient difference spectral approximation for Allen–Cahn equation in a circular domain. Firstly, we introduce the polar coordinate transformation and derive the equivalent form of Allen–Cahn equation under this coordinate system, as well as the corresponding essential polar condition. Then, by using first‐order Euler and second‐order backward difference methods in the temporal direction, we deduce the first‐order and second‐order semi‐implicit schemes, based on which the first‐order and second‐order fully discrete schemes are established by employing Legendre‐Fourier spectral approximation in the spatial direction. In addition, the energy stability and error estimations for the two types of numerical schemes are theoretically proved. Finally, we provide some numerical examples, the results of which demonstrate the stability and convergence of the algorithm. [ABSTRACT FROM AUTHOR]

Objective: This study was to evaluate measurement properties of the Chinese version of the Brief Inventory of Perceived Stress (BIPS-C) and confirm possible solutions for measuring the constructs underlying perceived stress. Methods: A total of 1356 community residents enrolled and were randomly split into two halves. The first half was used to explore the underlying constructs of the BIPS-C by exploratory graph analysis (EGA) and the second half was used to compare and confirm the constructs by confirmatory factor analysis (CFA). Results: The EGA identified a one-factor model of the BIPS-C with an accuracy of 99.3%. One-factor, three-factor, second-order, and bifactor models were compared by CFAs. The bifactor model with one general and three specific factors was found to be the most adequate [comparative fit index (CFI) = 0.990; Tucker-Lewis index (TLI) = 0.979; root mean square error of approximation (RMSEA) = 0.058] and was superior to the other models. The related bifactor indices showed a stronger existence of the general factor. The bifactor model of the BIPS-C also showed adequate internal consistency with McDonald's omega and omega subscales ranging from moderate to strong (0.677–0.869). Conclusion: The BIPS-C demonstrates sufficient measurement properties for assessing general perceived stress. [ABSTRACT FROM AUTHOR]

We consider optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by partial differential equations (PDEs) under model uncertainty. Specifically, we consider inverse problems in which, in addition to the inversion parameters, the governing PDEs include secondary uncertain parameters. We focus on problems with infinite-dimensional inversion and secondary parameters and present a scalable computational framework for optimal design of such problems. The proposed approach enables Bayesian inversion and OED under uncertainty within a unified framework. We build on the Bayesian approximation error (BAE) approach, to incorporate modeling uncertainties in the Bayesian inverse problem, and methods for A-optimal design of infinite-dimensional Bayesian nonlinear inverse problems. Specifically, a Gaussian approximation to the posterior at the maximum a posteriori probability point is used to define an uncertainty aware OED objective that is tractable to evaluate and optimize. In particular, the OED objective can be computed at a cost, in the number of PDE solves, that does not grow with the dimension of the discretized inversion and secondary parameters. The OED problem is formulated as a binary bilevel PDE constrained optimization problem and a greedy algorithm, which provides a pragmatic approach, is used to find optimal designs. We demonstrate the effectiveness of the proposed approach for a model inverse problem governed by an elliptic PDE on a three-dimensional domain. Our computational results also highlight the pitfalls of ignoring modeling uncertainties in the OED and/or inference stages. [ABSTRACT FROM AUTHOR]

This paper proposes an algorithm to obtain the sampling factors to model any frequency partitioning so that it is realized using low complexity rational decimated non-uniform filter banks (RDNUFBs). The proposed algorithm is employed to approximate the Bark scale to find a rational frequency partitioning that can be realized using RDNUFBs with less approximation error. The proposed Bark frequency partitioning is found to reduce the average deviation in centre frequency and bandwidth by 65.04% and 48.50%, respectively, and root-mean-square bandwidth deviation by 54.46% when compared to those of the existing perceptual wavelet approximation of Bark scale. In the first filter bank design approach discussed in the paper, a near-perfect reconstruction partially cosine modulated filter bank is employed to obtain the improved Bark frequency partitioning. In the second design approach, the hardware complexity of the PCM-based RDNUFB is reduced by deriving the channels with different sampling factors from the same prototype filter by the method of channel merging. There is a considerable reduction of 40.83% in the number of multipliers when merging of partially cosine modulated channels is employed. It is also found that both the proposed approaches can reduce the delay by 95.17% when compared to the existing rational tree approximation methods and hence, are suitable for real-time applications. [ABSTRACT FROM AUTHOR]

In this paper, we propose an a posteriori error estimate of the weak Galerkin finite element method (WG‐FEM) solving the Stokes problems with variable coefficients. Its error estimator, based on the property of Stokes' law conservation, Helmholtz decomposition and bubble functions, yields global upper bound and local lower bound for the approximation error of the WG‐FEM. Error analysis is proved to be valid under the mesh assumptions of the WG‐FEM and the way can be extended to other FEMs with the property of Stokes' law conservation, for example, discontinuous Galerkin (DG) FEMs. Finally, we verify the performance of error estimator by performing a few numerical examples. [ABSTRACT FROM AUTHOR]

Background: Distributed statistical analyses provide a promising approach for privacy protection when analyzing data distributed over several databases. Instead of directly operating on data, the analyst receives anonymous summary statistics, which are combined into an aggregated result. Further, in discrimination model (prognosis, diagnosis, etc.) development, it is key to evaluate a trained model w.r.t. to its prognostic or predictive performance on new independent data. For binary classification, quantifying discrimination uses the receiver operating characteristics (ROC) and its area under the curve (AUC) as aggregation measure. We are interested to calculate both as well as basic indicators of calibration-in-the-large for a binary classification task using a distributed and privacy-preserving approach. Methods: We employ DataSHIELD as the technology to carry out distributed analyses, and we use a newly developed algorithm to validate the prediction score by conducting distributed and privacy-preserving ROC analysis. Calibration curves are constructed from mean values over sites. The determination of ROC and its AUC is based on a generalized linear model (GLM) approximation of the true ROC curve, the ROC-GLM, as well as on ideas of differential privacy (DP). DP adds noise (quantified by the ℓ 2 sensitivity Δ 2 (f ^) ) to the data and enables a global handling of placement numbers. The impact of DP parameters was studied by simulations. Results: In our simulation scenario, the true and distributed AUC measures differ by Δ AUC < 0.01 depending heavily on the choice of the differential privacy parameters. It is recommended to check the accuracy of the distributed AUC estimator in specific simulation scenarios along with a reasonable choice of DP parameters. Here, the accuracy of the distributed AUC estimator may be impaired by too much artificial noise added from DP. Conclusions: The applicability of our algorithms depends on the ℓ 2 sensitivity Δ 2 (f ^) of the underlying statistical/predictive model. The simulations carried out have shown that the approximation error is acceptable for the majority of simulated cases. For models with high Δ 2 (f ^) , the privacy parameters must be set accordingly higher to ensure sufficient privacy protection, which affects the approximation error. This work shows that complex measures, as the AUC, are applicable for validation in distributed setups while preserving an individual's privacy. [ABSTRACT FROM AUTHOR]

The primary focus of this research paper is to explore the realm of dynamic learning in sampled‐data strict‐feedback nonlinear systems (SFNSs) by leveraging the capabilities of radial basis function (RBF) neural networks (NNs) under the framework of adaptive control. First, the exact discrete‐time model of the continuous‐time system is expressed as an Euler strict‐feedback model with a sampling approximation error. We provide the consistency condition that establishes the relationship between the exact model and the Euler model with meticulous detail. Meanwhile, a novel lemma is derived to show the stability condition of a digital first‐order filter. To address the non‐causality issues of SFNSs with sampling approximation error and the input data dimension explosion of NNs, the auxiliary digital first‐order filter and backstepping technology are combined to propose an adaptive neural dynamic surface control (ANDSC) scheme. Such a scheme avoids the n$$ n $$‐step time delays associated with the existing NN updating laws derived by the common n$$ n $$‐step predictor technology. A rigorous recursion method is employed to provide a comprehensive verification of the stability, guaranteeing its overall performance and dependability. Following that, the NN weight error systems are systematically decomposed into a sequence of linear time‐varying subsystems, allowing for a more detailed analysis and understanding. In order to ensure the recurrent nature of the input variables, a recursive design is employed, thereby satisfying the partial persistent excitation condition specifically designed for the RBF NNs. Meanwhile, it can verify that the NN estimated weights converge to their ideal values. Compared with the common n$$ n $$‐step predictor technology, there is no need to redesign the learning rules due to the designed NN weight updating laws without time delays. Subsequently, after capturing and storing the convergence weights, a novel neural learning dynamic surface control (NLDSC) scheme is specifically formulated by leveraging the acquired knowledge. The introduced methodology reduces computational complexity and facilitates practical implementation. Finally, empirical evidence obtained from simulation experiments validates the efficacy and viability of the proposed methodology. [ABSTRACT FROM AUTHOR]

In this article, the adaptive event-triggered fault-tolerant control (FTC) issue of uncertain nonlinear systems suffering from sensor and actuator faults as well as external disturbances is studied. A disturbance observer (DO) is constructed to compensate for external disturbances and approximation errors generated by neural networks (NNs). Then, switching event-triggered control and command filtering techniques are introduced to balance the communication frequency and tracking performance of the controlled systems while avoiding “explosion of complexity” issue caused by iterative derivations of virtual controllers. Furthermore, compensation signals are designed to eliminate filtered errors. Finally, it is proven that the developed FTC scheme can esure that all signals are bounded and free from Zeno phenomenon. Simulation results of a spring damping mechanical system verifies the merits of the designed control algorithm. [ABSTRACT FROM AUTHOR]

Forecast error growth as a function of lead time of atmospheric phenomena is caused by initial and model errors. When studying the initial error growth, it may turn out that small-scale phenomena, which contribute little to the forecast product, significantly affect the ability to predict this product. The question under investigation is whether omitting these atmospheric phenomena will improve the predictability of the resulting value. The topic is studied in the extended Lorenz (2005) system. This system shows that omitting small spatiotemporal scales that significantly affect prediction ability will reduce predictability more than modeling it. In other words, a system with model error (omitting phenomena) will not improve predictability. A hypothesis explaining and describing this behavior is developed, with the difference between systems (model error) produced at each time step seen as the error of the initial conditions. The resulting model error is then defined as the sum of the increments of the time evolution of the initial conditions so defined. The hypothesis is compared to the fit parameters that define the model error in certain approximations of the average forecast error growth. Parameters are interpreted in this context, and the approximations are used to estimate the errors described in the hypothesis. A method is proposed to distinguish increments of prediction error growth from small-spatiotemporal-scale phenomena and model errors. Results are presented for the error growth of the ECMWF system, where a 40 % reduction in model error between 1987 and 2011 is calculated based on the developed hypothesis, while over the same period the instability (error growth rate) of the system with respect to initial condition errors has grown. [ABSTRACT FROM AUTHOR]

The relaxation spectrum is a fundamental viscoelastic characteristic from which other material functions used to describe the rheological properties of polymers can be determined. The spectrum is recovered from relaxation stress or oscillatory shear data. Since the problem of the relaxation spectrum identification is ill-posed, in the known methods, different mechanisms are built-in to obtain a smooth enough and noise-robust relaxation spectrum model. The regularization of the original problem and/or limit of the set of admissible solutions are the most commonly used remedies. Here, the problem of determining an optimally smoothed continuous relaxation time spectrum is directly stated and solved for the first time, assuming that discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation experiment are available for identification. The relaxation time spectrum model that reproduces the relaxation modulus measurements and is the best smoothed in the class of continuous square-integrable functions is sought. Based on the Hilbert projection theorem, the best-smoothed relaxation spectrum model is found to be described by a finite sum of specific exponential–hyperbolic basis functions. For noise-corrupted measurements, a quadratic with respect to the Lagrange multipliers term is introduced into the Lagrangian functional of the model's best smoothing problem. As a result, a small model error of the relaxation modulus model is obtained, which increases the model's robustness. The necessary and sufficient optimality conditions are derived whose unique solution yields a direct analytical formula of the best-smoothed relaxation spectrum model. The related model of the relaxation modulus is given. A computational identification algorithm using the singular value decomposition is presented, which can be easily implemented in any computing environment. The approximation error, model smoothness, noise robustness, and identifiability of the polymer real spectrum are studied analytically. It is demonstrated by numerical studies that the algorithm proposed can be successfully applied for the identification of one- and two-mode Gaussian-like relaxation spectra. The applicability of this approach to determining the Baumgaertel, Schausberger, and Winter spectrum is also examined, and it is shown that it is well approximated for higher frequencies and, in particular, in the neighborhood of the local maximum. However, the comparison of the asymptotic properties of the best-smoothed spectrum model and the BSW model a priori excludes a good approximation of the spectrum in the close neighborhood of zero-relaxation time. [ABSTRACT FROM AUTHOR]

Background: DASS-21 is the common and widely used tool for assessing depression, anxiety and stress. However, its validity and Reliability in Ethiopian Amharic language is not assessed. Objective: To translate the DASS-21 and assess its validity and reliability among Ethiopian Defense University college of health science students in Bishoftu, Ethiopia. Method: An institutional based cross-sectional study was conducted using a self-administered questionnaire. A total of 435 students from different departments in accordance with their proportional size were participated in this study. As to the sampling technique, the study units were selected from each department and year of study using simple random sampling proportional to size. Confirmatory factor analyses were employed to assess the factor structure and construct validity of Amharic version of the DASS-21. Cronbach alpha coefficient and corrected item total correlation was calculated to assess the internal consistency of Amharic version of DASS-21. Result: Among 435 undergraduate students who participated in the study, 246(56.6%) were the age of 18–25 year and majority 347(79.8%) were males. Regarding their year of study; 200 (46.0%) were first year students. Confirmatory factor analysis indicates a good model fit of the three correlated factors (Comparative fit index (CFI) = 0.92 with root mean square error of approximation (RMSEA) = 0.059[0.052–0.066] and standardize root mean residual SRMR = 0.045). The internal consistency of overall DASS-21 and each sub scale were in acceptable range (0.91, 0.82, 0.77 and 0.77) respectively. Conclusion: Amharic version of DASS-21 was found to be a valid and reliable instrument to measure the mental problem especially, Depression, Anxiety and Stress among university students. [ABSTRACT FROM AUTHOR]

In this work, the capability to generate the negative group delay (NGD) phenomenon at those frequencies higher than a certain value, that is, the NGD high‐pass (HP) filtering function, of the bilinear double‐order transfer function has been demonstrated. Based on the Type‐I bilinear double‐order filter circuit, our theory has been verified by strong agreements between the formulae and their proof‐of‐concept (POC) circuit‐based simulation results. Both formula‐based and POC circuit‐based simulations give the minimum group delay of −5 ms yet the cut‐off frequency of 10 and 9 rad/s, respectively. Such slight deviation is caused by the approximation error of the bilinear double‐order impedance. By employing two orders, the bilinear double‐order transfer function has been found to be the basis transfer function for the NGD filtering function with the highest degree of freedom. Based on our design equation for the NGD filtering function, it can be seen that the distances between zero and pole and the characteristic frequency of the bilinear double‐order transfer function are governed by such characteristic frequency itself, the cut‐off frequency of the NGD filtering function, and the fractional order of the Laplacian operator. In addition, the effects of the fractional order of the Laplacian operator, the fractional order of the transfer function, and the ratio of the abovementioned distances to the characteristics of the newly found double‐order NGD filtering function have been studied in detail. [ABSTRACT FROM AUTHOR]

In this paper, we determine the error of approximation of functions, conjugate to the functions of two variables ( 2 ⁢ π 2\pi -periodic in both variables) belonging to the Lipschitz class Lip ( ( ξ 1 , ξ 2 ) ; r \operatorname{Lip}((\xi_{1},\xi_{2});r ) ( r ≥ 1 r\geq 1 ), through the double Nörlund means of its conjugate double Fourier series. Additionally, in the form of corollaries, we present some particular cases of the theorem proved here. [ABSTRACT FROM AUTHOR]

We give investigations on a kernel function approximation problem arising from learning theory and show the convergence rate from the view of classical Fourier analysis. First, we provide the general definition for a modulus of smoothness and a K -functional, and show that they are equivalent. In particular, we give explicit representation for some moduli of smoothness. Second, we establish some Jackson-type inequalities for the approximation error associated with some non-radial kernels. Also we apply these results to some concrete classical kernel function spaces and give Jackson-type inequalities for some concrete RKHS approximation problems. Finally, we apply these discussions to learning theory and describe the learning rates with the moduli of smoothness. The tools we used are Fourier analysis and the semigroup operator. The results show that the Jackson-type inequalities of approximation by some radial kernel functions on compact set with nonempty interiors cannot be expressed with the classical moduli of smoothness. [ABSTRACT FROM AUTHOR]

A novel cascade visual control scheme is proposed to tailor for electrically driven robotic manipulators that operate under kinematic and dynamic uncertainties, utilizing an uncalibrated stationary camera. The proposed control approach incorporates adaptive weight radial basis function neural networks (RBFNNs) to learn the behaviors of the uncertain dynamics of the robot and the joint actuators. The controllers are designed to nullify the approximation error and mitigate unknown disturbances through an integrated robust adaptive mechanism. A major advantage of the proposed approach is that prior knowledge of the dynamics of the robotic manipulator and its actuator is no longer required. The controller autonomously assimilates the robot and actuator dynamics online, thereby obviating the need for fussy regression matrix derivation and advance dynamic measurement to establish the adaptive dynamic parameter update algorithm. The proposed scheme ensures closed-loop system stability, bounded system states, and the convergence of tracking errors to zero. Simulation results, employing a PUMA manipulator as a testbed, substantiate the viability of the proposed control policy. [ABSTRACT FROM AUTHOR]

Velocity response spectrum (SV) plays a critical role in the seismic design of structures equipped with velocity-dependent dampers. Often, the SV is approximated using the pseudo-velocity response spectrum (PSV), because of the lack of information on the SV in seismic codes. As indicated in the existing literature, the error of this approximation is significant for short and long periods, and therefore, researchers have started to develop PSV to SV conversion models for obtaining SV more accurately. Recent studies discovered that the relationship between the PSV and SV is affected not only by structural parameters, but also by magnitude, distance, and site class. However, models for converting PSV to SV including magnitude, distance, and site class as input parameters have not been developed. To this end, a PSV to SV conversion model including magnitude, distance, and site class is proposed in this study based on a large number of real ground motion records (16,660 horizontal acceleration time histories) selected from the Japan Strong Motion Network. Furthermore, since the magnitude and distance are not specified in the seismic design, a response-spectrum-shape factor, s, is discussed to reflect the influences of magnitude and distance. Accordingly, an SV/PSV model incorporating s is established. The proposed models show better accuracy than the existing models for cases with different magnitudes, distances, and site classes. [ABSTRACT FROM AUTHOR]

Particle-based Variational Inference (ParVI) methods have been widely adopted in deep Bayesian inference tasks such as Bayesian neural networks or Gaussian Processes, owing to their efficiency in generating high-quality samples given the score of the target distribution. Typically, ParVI methods evolve a weighted-particle system by approximating the first-order Wasserstein gradient flow to reduce the dissimilarity between the particle system's empirical distribution and the target distribution. Recent advancements in ParVI have explored sophisticated gradient flows to obtain refined particle systems with either accelerated position updates or dynamic weight adjustments. In this paper, we introduce the semi-Hamiltonian gradient flow on a novel Information–Fisher–Rao space, known as the SHIFR flow, and propose the first ParVI framework that possesses both accelerated position update and dynamical weight adjustment simultaneously, named the General Accelerated Dynamic-Weight Particle-based Variational Inference (GAD-PVI) framework. GAD-PVI is compatible with different dissimilarities between the empirical distribution and the target distribution, as well as different approximation approaches to gradient flow. Moreover, when the appropriate dissimilarity is selected, GAD-PVI is also suitable for obtaining high-quality samples even when analytical scores cannot be obtained. Experiments conducted under both the score-based tasks and sample-based tasks demonstrate the faster convergence and reduced approximation error of GAD-PVI methods over the state-of-the-art. [ABSTRACT FROM AUTHOR]

The demand for thermal power generation from natural gas has increased globally due to its cleaner burning properties compared to other fossil fuels. Optimizing the gas flow through the network to meet this demand is challenging due to the nonconvex Weymouth equation constraining gas flow and nodal pressures in pipelines. Traditional methods for addressing this nonconvexity lead to significant approximation errors or high operational costs. This study poses the Weymouth constraint as a Mathematical Programming with Complementarity Constraints (MPCC) for an optimal gas flow problem. The complementarity constraints reformulate the discontinuous sign function using binary-behaving continuous variables. This MPCC-based approach avoids solving mixed-integer programming problems while enhancing the accuracy of conventional linear and second-order approximations. Testing the approach on various interconnected systems, including Colombia's national gas transportation grid, demonstrated significant reductions in Weymouth approximation errors, thereby supporting effective optimization for interconnected networks. [ABSTRACT FROM AUTHOR]

One‐hidden‐layer feedforward neural networks are described as functions having many real‐valued parameters. Approximation properties of neural networks are established (universal approximation property), and the approximation error is related to the number of parameters in the network. The essentially optimal order of approximation error bounds was already derived in 1996. We focused on the numerical experiment that indicates the neural networks whose parameters contain stochastic perturbations gain better performance than ordinary neural networks and explored the approximation property of neural networks with stochastic perturbations. In this paper, we derived the quantitative order of variance of stochastic perturbations to achieve the essentially optimal approximation order and verified the justifiability of our theory by numerical experiments. [ABSTRACT FROM AUTHOR]

Directional interpolation is a fast and efficient compression technique for high‐frequency Helmholtz boundary integral equations, but requires a very large amount of storage in its original form. Algebraic recompression can significantly reduce the storage requirements and speed up the solution process accordingly. During the recompression process, weight matrices are required to correctly measure the influence of different basis vectors on the final result, and for highly accurate approximations, these weight matrices require more storage than the final compressed matrix. We present a compression method for the weight matrices and demonstrate that it introduces only a controllable error to the overall approximation. Numerical experiments show that the new method leads to a significant reduction in storage requirements. [ABSTRACT FROM AUTHOR]

Let X,Y$X, Y$ be two Banach spaces, F:X→Y$F:X\rightarrow Y$ be an ε$\varepsilon$‐norm‐additive map for some ε⩾0$\varepsilon \geqslant 0$, that is, |∥F(x)+F(y)∥−∥x+y∥|⩽ε,for allx,y∈X.$$\begin{equation*} \big\vert \Vert F(x)+F(y)\Vert -\Vert x+y\Vert \big\vert \leqslant \varepsilon, {\text{\,for all\,}} x,y\in X. \end{equation*}$$In this paper, we prove that if F$F$ is surjective with F(0)=0$F(0)=0$, then there exists a linear surjective isometry U:X→Y$U:X\rightarrow Y$ such that ∥F(x)−U(x)∥⩽32ε,for allx∈X.$$\begin{equation*} \Vert F(x)-U(x)\Vert \leqslant \frac{3}{2}\varepsilon, {\text{\,for all\,}} x\in X. \end{equation*}$$The estimate 32ε$\frac{3}{2}\varepsilon$ is sharp. We also approximate standard surjective ε$\varepsilon$‐norm‐additive maps between the positive cones of continuous function spaces by linear isometries within a sharp approximation error 32ε$\frac{3}{2}\varepsilon$. [ABSTRACT FROM AUTHOR]

The predefined performance tracking control issue of a vertical take-off and landing (VTOL) aircraft system is discussed in this work. Employing the presented control performance function, the constrained error signal of the aircraft is switched into an equivalent unconstrained signal. Moreover, fuzzy logic systems are introduced to address unknown nonlinear dynamics problems in aircraft system. Furthermore, in order to address the approximation errors and the external disturbances of aircraft system, parameter-adaptive update laws are proposed in the analysis process. Subsequently, adaptive fuzzy sliding-mode control laws are designed by combining fuzzy control method with sliding-mode control technique. By applying the redesigned sliding-mode functions and the Lyapunov stability method, the output of the VTOL aircraft system can track the desired trajectory, and the tracking errors are constrained in a predefined range. Finally, the availability of the proposed control laws is testified through simulation examples. [ABSTRACT FROM AUTHOR]

To strike a balance between modeling accuracy and computational efficiency for simulations of ultrasound waves in soft tissues, we derive a pseudodifferential factorization of the wave operator with fractional attenuation. This factorization allows us to approximately solve the Helmholtz equation via one-way (transmission) or two-way (transmission and reflection) sweeping schemes tailored to high-frequency wave fields. We provide explicitly the three highest order terms of the pseudodifferential expansion to incorporate the well-known square-root first order symbol for wave propagation, the zeroth order symbol for amplitude modulation due to changes in wave speed and damping, and the next symbol to model fractional attenuation. We also propose wide-angle Padé approximations for the pseudodifferential operators corresponding to these three highest order symbols. Our analysis provides insights regarding the role played by the frequency and the Padé approximations in the estimation of error bounds. We also provide a proof-of-concept numerical implementation of the proposed method and test the error estimates numerically. [ABSTRACT FROM AUTHOR]

Building on the successes of local kernel methods for approximating the solutions to partial differential equations (PDEs) and the evaluation of definite integrals (quadrature/cubature), a local estimate of the error in such approximations is developed. This estimate is useful for determining locations in the solution domain where increased node density (equivalently, reduction in the spacing between nodes) can decrease the error in the solution. An adaptive procedure for adding nodes to the domain for both the approximation of derivatives and the approximate evaluation of definite integrals is described. This method efficiently computes the error estimate at a set of prescribed points and adds new nodes for approximation where the error is too large. Computational experiments demonstrate close agreement between the error estimate and actual absolute error in the approximation. Such methods are necessary or desirable when approximating solutions to PDEs (or in the case of quadrature/cubature), where the initial data and subsequent solution (or integrand) exhibit localized features that require significant refinement to resolve and where uniform increases in the density of nodes across the entire computational domain is not possible or too burdensome. [ABSTRACT FROM AUTHOR]

In this work is considered an elliptic problem, referred to as the Ventcel problem, involving a second-order term on the domain boundary (the Laplace--Beltrami operator). A variational formulation of the Ventcel problem is studied, leading to a finite element discretization. The focus is on the construction of high-order curved meshes for the discretization of the physical domain and on the definition of the lift operator, which is aimed at transforming a function defined on the mesh domain into a function defined on the physical one. This lift is defined in such a way as to satisfy adapted properties on the boundary relative to the trace operator. The Ventcel problem approximation is investigated both in terms of geometrical error and of finite element approximation error. Error estimates are obtained both in terms of the mesh order r > 1 and to the finite element degree k > 1, whereas such estimates usually have been considered in the isoparametric case so far, involving a single parameter k = r. The numerical experiments we led in both 2 and 3 dimensions allow us to validate the results obtained and proved on the a priori error estimates depending on the 2 parameters k and r. A numerical comparison is made between the errors using the former lift definition and the lift defined in this work establishing an improvement in the convergence rate of the error in the latter case. [ABSTRACT FROM AUTHOR]

In approximation of functions based on point values, least-squares methods provide more stability than interpolation, at the expense of increasing the sampling budget. We show that near-optimal approximation error can nevertheless be achieved, in an expected L2 sense, as soon as the sample size m is larger than the dimension n of the approximation space by a constant multiplicative ratio. On the other hand, for m = n, we obtain an interpolation strategy with a stability factor of order n. The proposed sampling algorithms are greedy procedures based on [Batson, Spielman, and Srivastava, Twice-Ramanujan sparsifiers, in Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, 2009, pp. 255--262] and [Lee and Sun, SIAM J. Comput., 47 (2018), pp. 2315--2336], with polynomial computational complexity. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fast and accurate method for approximating the Minkowski sum of two rotational ellipsoids with a superellipsoid. The Minkowski sum is used in a variety of applications such as robot motion planning and particle flow simulation requiring collision detection. Many of them are computed based on Minkowski sum, whose accuracy and processing time depend on how the mesh is created. We approximate Minkowski sum with a superellipsoid function. The superellipsoid has various shapes with the value of the exponents, and the computation of the parameters including the exponents is an algebraic computation with the time complexity O (1) . As a result, approximation error is small and computation is fast in all combinations of rotational ellipsoids. In addition, collision is quickly detected by solving the inequality of superellipsoid. [ABSTRACT FROM AUTHOR]

The increasingly popular GW method is becoming a convenient tool to determine vertical ionization energies in molecular systems. However, depending on the formalism used and the range of orbitals investigated, it may be hampered by a steep computational scaling. To alleviate this issue, correlated natural virtual orbitals (NVOs) based on second-order Møller–Plesset (MP2) and direct MP2 correlation energies are implemented, and the resulting correlated NVOs are tested on GW quasiparticle energies. Test cases include the popular GW variants G0W0 and evGW0 as well as more elaborate vertex corrections. We find that for increasingly larger molecular systems and basis sets, NVOs considerably improve efficiency. Furthermore, we test the performance of the truncated (frozen) NVO ansatz on the GW100 test set. For the latter, it is demonstrated that, using a carefully chosen truncation threshold, NVOs lead to a negligible loss in accuracy while providing speedups of one order of magnitude. Furthermore, we compare the resulting quasiparticle energies to very accurate vertical ionization energies obtained from coupled-cluster theory with singles, doubles, and noniterative triples [CCSD(T)], confirming that the loss in accuracy introduced by truncating the NVOs is negligible compared to the methodical errors in the GW approximation. It is also demonstrated that the choice of basis set impacts results far more than using a suitably truncated NVO space. Therefore, at the same computational expense, more accurate results can be obtained using NVOs. Finally, we provide improved reference CCSD(T) values for the GW100 test set, which have been obtained using the def2-QZVPP basis set. [ABSTRACT FROM AUTHOR]

We derive rigorous bounds on the error resulting from the approximation of the solution of parametric hyperbolic scalar conservation laws with ReLU neural networks. We show that the approximation error can be made as small as desired with ReLU neural networks that overcome the curse of dimensionality. In addition, we provide an explicit upper bound on the generalization error in terms of the training error, number of training samples and the neural network size. The theoretical results are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

We study the approximation and quadrature error of points that satisfy Marcinkiewicz-Zygmund inequalities. First, we investigate the use of Marcinkiewicz-Zygmund inequalities in an abstract Hilbert space for an abstract approximation and quadrature rule. The setting is then specified to Sobolev spaces induced by Freud weights e^{-2\sigma |x|^\alpha } with \alpha >1 and \sigma >0, and we derive specific bounds for the approximation and quadrature error. For the Gaussian weight e^{-2\pi x^2}, we verify that the Sobolev spaces essentially coincide with a specific class of modulation spaces that are well known in (time-frequency) analysis. [ABSTRACT FROM AUTHOR]

In this paper, a numerical strategy is introduced for solving a class of linear fractional-order Darboux problem that includes the Caputo derivative. The presented strategy is grounded on the Newton-Cotes quadrature rule and is established by implementing Lagrange interpolation polynomial. A formula is developed to evaluate the two-dimensional fractional integrals by using the multiplicative integration strategy. Then, this formula is employed to derive a scheme for solving a class of fractional-order Darboux problem. The accuracy and efficiency of all the presented schemes are illustrated through numerical experiments. Furthermore, the estimation of the upper bounds of error in the numerical approximation for the proposed multiplicative integration strategy is established. [ABSTRACT FROM AUTHOR]

We obtain an expression for the error in the approximation of f(A)b$$ f(A)\boldsymbol{b} $$ and bTf(A)b$$ {\boldsymbol{b}}^Tf(A)\boldsymbol{b} $$ with rational Krylov methods, where A$$ A $$ is a symmetric matrix, b$$ \boldsymbol{b} $$ is a vector and the function f$$ f $$ admits an integral representation. The error expression is obtained by linking the matrix function error with the error in the approximate solution of shifted linear systems using the same rational Krylov subspace, and it can be exploited to derive both a priori and a posteriori error bounds. The error bounds are a generalization of the ones given in Chen et al. for the Lanczos method for matrix functions. A technique that we employ in the rational Krylov context can also be applied to refine the bounds for the Lanczos case. [ABSTRACT FROM AUTHOR]

This paper proposes a Generalized Multilevel B-spline Approximation (GMBA) method, which addresses scattered data interpolation problems in image processing. Mathematically, the GMBA provides a better solution for the B-spline control lattice by superimposing identical level B-splines compared with traditional Multilevel B-spline Approximation (MBA). Specifically, the GMBA allows the spacing of next control lattice to be subdivided arbitrarily or remained unchanged, which is determined by a predefined spacing set or the current error level. These improvements bring higher approximation accuracy and more flexibility for algorithm design to avoid over-fitting. In this paper, basic GMBA algorithm and its refined algorithm are compiled for image processing. Finally, six relevant cases are involved to test the GMBA, including surface approximation, image enlargement, image completion, and Salt-and-Pepper (SAP) noise removal. The experimental results show that the GMBA has better performance than the MBA in surface approximation and image processing, performs comparatively fast with the best performance on more than half of the standard test images compared with traditional algorithms, and has partially better performance even than deep learning algorithms. The GMBA can effectively recover meaningful details in images contaminated with even extremely high SAP noise level (up to 99%). • We give a generalized multilevel B-spline approximation (GMBA) method. • GMBA allows the control spacing to be subdivided arbitrarily or remained unchanged. • GMBA offers higher approximation accuracy and greater flexibility than traditional MBA. • GMBA provides superior performance and less run time than many of the state-of-the-art methods for SAP noise removal. • GMBA can recovery meaningful detail at noise levels as high as 99% for SAP noise removal. [ABSTRACT FROM AUTHOR]

Corporate social responsibility (CSR) is a constantly evolving concept that reflects changes in society and the expectations of stakeholders in a process that leads companies to adapt to respond to new challenges and demands. Similarly, the theory and practice of CSR have moved from regulatory compliance to a more holistic approach that requires more sophisticated models that allow for a deeper understanding. Following the fact that companies now have access to a greater amount of information related to their social and environmental performance, as well as comparative data from other organizations, which is why it is necessary to have a management tool that allows collecting, process, and generating more accurate and meaningful metrics that more effectively reflect a company's impact on society. Consequently, the present work aims to identify a system of variables and dimensions representative of the management of organizations that allows the validation of a generic model of corporate social responsibility determined through factorial and structural modeling capable of reflecting more accurate and up-to-date dimensions and metrics of model variables and dimensions. To carry out the study, a sample of 667 middle and senior managers from medium and large companies in Guayas, Ecuador, were randomly contacted in their respective business contexts by trained interviewers. The data processing was carried out in stages. First, the exploratory factorial analysis method of the items was applied to form relevant factors. Then, the dimensions' structural modeling was formulated and ratified through the pertinent goodness-of-fit indices. The results determined a system of correlated factors whose items present estimators of commonality and high and significant factorial loads, excluding variables that did not exceed the sensitization criteria applied. Finally, a model made up of 15 factors and 66 variables ratified using the comparative adjustment indices, the chi-square likelihood ratio, and the mean square error of approximation is introduced, confirming the proposed corporate social responsibility model. [ABSTRACT FROM AUTHOR]

Let ω be a positive, Lebesgue integrable and exponential decay function defined on an infinite interval I and let L1,ω(I) be the space of weighted Lebesgue integrable functions on I. In this paper, we give the relations of the best one-sided approximation and the optimal Hermite–Fejér interpolation by the set of algebraic polynomials of degree not exceeding a given number for the smooth function classes BC2r,+(I), r ∈ ℕ+, in the metric of the space L1,ω(I), and prove that the Hermite–Fejér interpolation based on the set of the zeros of some orthogonal polynomials is optimal in L1,ω(I). In addition, we show that the approximation error of the optimal Hermite–Fejér interpolation and quadrature errors of the weighted Gaussian quadrature formula are equal, and give the exact constant of the errors. [ABSTRACT FROM AUTHOR]

In this paper, the optimal approximation algorithm is proposed to simplify non-linear functions and/or discrete data as piecewise polynomials by using the constrained least squares. In time-sensitive applications or in embedded systems with limited resources, the runtime of the approximate function is as crucial as its accuracy. The proposed algorithm searches for the optimal piecewise polynomial (OPP) with the minimum computational cost while ensuring that the error is below a specified threshold. This was accomplished by using smooth piecewise polynomials with optimal order and numbers of intervals. The computational cost only depended on polynomial complexity, i.e., the order and the number of intervals at runtime function call. In previous studies, the user had to decide one or all of the orders and the number of intervals. In contrast, the OPP approximation algorithm determines both of them. For the optimal approximation, computational costs for all the possible combinations of piecewise polynomials were calculated and tabulated in ascending order for the specific target CPU off-line. Each combination was optimized through constrained least squares and the random selection method for the given sample points. Afterward, whether the approximation error was below the predetermined value was examined. When the error was permissible, the combination was selected as the optimal approximation, or the next combination was examined. To verify the performance, several representative functions were examined and analyzed. [ABSTRACT FROM AUTHOR]

A randomized quaternion singular value decomposition algorithm based on block Krylov iteration (RQSVD-BKI) is presented to solve the low-rank quaternion matrix approximation problem. The upper bounds of deterministic approximation error and expected approximation error for the RQSVD-BKI algorithm are also given. It is shown by numerical experiments that the running time of the RQSVD-BKI algorithm is smaller than that of the quaternion singular value decomposition, and the relative errors of the RQSVD-BKI algorithm are smaller than those of the randomized quaternion singular value decomposition algorithm in Liu et al. (SIAM J. Sci. Comput., 44(2): A870-A900 (2022)) in some cases. In order to further illustrate the feasibility and effectiveness of the RQSVD-BKI algorithm, we use it to deal with the problem of color image inpainting. [ABSTRACT FROM AUTHOR]

Privacy-preserving neural networks offer a promising solution to train and predict without user privacy leakage, and fully homomorphic encryption (FHE) stands out as one of the key technologies, as it enables homomorphic operations over encrypted data. However, only addition and multiplication homomorphisms are supported by FHE, and thus, it faces huge challenges when implementing non-linear functions with ciphertext inputs. Among the non-linear functions in neural networks, one may refer to the activation function, the argmax function, and maximum pooling. Inspired by using a composition of low-degree minimax polynomials to approximate sign and argmax functions, this study focused on optimizing the homomorphic argmax approximation, where argmax is a mathematical operation that identifies the index of the maximum value within a given set of values. For the method that uses compositions of low-degree minimax polynomials to approximate argmax, in order to further reduce approximation errors and improve computational efficiency, we propose an improved homomorphic argmax approximation algorithm that includes rotation accumulation, tree-structured comparison, normalization, and finalization phases. And then, the proposed homomorphic argmax algorithm was integrated into a neural network structure. Comparative experiments indicate that the network with our proposed argmax algorithm achieved a slight increase in accuracy while significantly reducing the inference latency by 58 % , as the homomorphic sign and rotation operations were rapidly reduced. [ABSTRACT FROM AUTHOR]

Aiming to provide a faster and convenient truncated SVD algorithm for large sparse matrices from real applications (i.e., for computing a few of the largest singular values and the corresponding singular vectors), a dynamically shifted power iteration technique is applied to improve the accuracy of the randomized SVD method. This results in a dynamic shifts-based randomized SVD (dashSVD) algorithm, which also collaborates with the skills for handling sparse matrices. An accuracy-control mechanism is included in the dashSVD algorithm to approximately monitor the per vector error bound of computed singular vectors with negligible overhead. Experiments on real-world data validate that the dashSVD algorithm largely improves the accuracy of a randomized SVD algorithm or attains the same accuracy with fewer passes over the matrix, and provides an efficient accuracy-control mechanism to the randomized SVD computation, while demonstrating the advantages on runtime and parallel efficiency. A bound of the approximation error of the randomized SVD with the shifted power iteration is also proved. [ABSTRACT FROM AUTHOR]

In this paper, we construct an approximation to the Pitman–Yor process by truncating its two‐parameter Poisson–Dirichlet representation. The truncation is based on a decreasing sequence of random weights, thus having a lower approximation error compared to the popular truncated stick‐breaking process. We develop an exact simulation algorithm to sample from the approximation process and provide an alternative MCMC algorithm for the parameter regime where the exact simulation algorithm becomes slow. The effectiveness of the simulation algorithms is demonstrated by the estimation of the functionals of a Pitman–Yor process. Then we adapt the approximation process into a Pitman–Yor process mixture model and devise a blocked Gibbs sampler for posterior inference. [ABSTRACT FROM AUTHOR]

We consider the design of fast and reliable neural network‐based approximations of traditional stabilizing controllers for linear systems with polytopic uncertainty, including control laws with variable structure and those based on a (minimal) selection policy. Building upon recent approaches for the design of reliable control surrogates with guaranteed structural properties, we develop a systematic procedure to certify the closed‐loop stability and performance of a linear uncertain system when a trained rectified linear unit (ReLU)‐based approximation replaces such traditional controllers. First, we provide a sufficient condition, which involves the worst‐case approximation error between ReLU‐based and traditional controller‐based state‐to‐input mappings, ensuring that the system is ultimately bounded within a set with adjustable size and convergence rate. Then, we develop an offline, mixed‐integer optimization‐based method that allows us to compute that quantity exactly. [ABSTRACT FROM AUTHOR]

New necessary and sufficient stability tests for neutral‐type linear time delay systems are presented. They are based on functionals with prescribed derivative depending on the delay Lyapunov matrix and recent stability/instability theorems. A piecewise linear approximation of the initial function argument in the functional is employed. As a result, the functional is approximated via a quadratic form whose matrix provides stability criteria. This matrix involves the integrals based on the delay Lyapunov matrix which are handled via an efficient computational procedure. The main feature of these tests is that they are verified in a finite number of operations, which is achieved through rigorous bounding of the approximation error on a specific class of functions. Some examples which validate the obtained results are presented and discussed. [ABSTRACT FROM AUTHOR]

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]

An adaptive state-feedback control system is proposed for a class of linear time-varying systems represented in the controller canonical form. The adaptation problem is reduced to the one of Taylor series-based first approximations of the ideal controller parameters. The exponential convergence of identification and tracking errors of such an approximation to an arbitrarily small and adjustable neighbourhood of the equilibrium point is ensured if the condition of the regressor persistent excitation with a sufficiently small time period is satisfied. The obtained theoretical results are validated via numerical experiments. [ABSTRACT FROM AUTHOR]