Search

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]

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 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]

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]

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]

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]

Every sufficiently big matrix with small spectral norm has a nearby low-rank matrix if the distance is measured in the maximum norm (Udell and Townsend, 2019 [7]). We use the Hanson–Wright inequality to improve the estimate of the distance for matrices with incoherent column and row spaces. In numerical experiments with several classes of matrices we study how well the theoretical upper bound describes the approximation errors achieved with the method of alternating projections. [ABSTRACT FROM AUTHOR]

In this paper, an online event-triggered self-learning scheme based on adaptive dynamic programming (ADP) is developed to address tracking control design for nonlinear systems with constrained input and uncertain disturbance. Firstly, the value function with non-quadratic function is defined for the augmented nominal system, and the constrained robust tracking problem is equivalent to the optimal control for solving the tracking event-triggered Hamilton–Jacobi–Bellman (ETHJB) equation. Then, a single-critic network is developed to obtain the value function and control law related to the solution of the tracking ETHJB equation, greatly reducing approximation errors and computational costs. To alleviate the requirement for the entire state sampling, we propose a triggering rule that ensures system stability while limiting control updates. Theoretical proof demonstrates that the tracking state of the closed-loop system and the weight approximation error of the neural network are uniformly ultimately bounded (UUB). Finally, two examples are provided to validate the availability of the proposed scheme. [ABSTRACT FROM AUTHOR]

This paper expands the cyclic block proximal gradient method for block separable composite minimization by allowing for inexactly computed gradients and pre-conditioned proximal maps. The resultant algorithm, the inexact cyclic block proximal gradient (I-CBPG) method, shares the same convergence rate as its exactly computed analogue provided the allowable errors decrease sufficiently quickly or are pre-selected to be sufficiently small. We provide numerical experiments that showcase the practical computational advantage of I-CBPG for certain fixed tolerances of approximation error and for a dynamically decreasing error tolerance regime in particular. Our experimental results indicate that cyclic methods with dynamically decreasing error tolerance regimes can actually outpace their randomized siblings with fixed error tolerance regimes. We establish a tight relationship between inexact pre-conditioned proximal map evaluations and δ -subgradients in our (δ , B) -Second Prox theorem. This theorem forms the foundation of our convergence analysis and enables us to show that inexact gradient computations can be subsumed within a single unifying framework. [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]

Summary: This paper presents neural networks (NNs) adaptive controller for an uncertain fractional‐order nonlinear system in strict‐feedback form, subject to input saturation, unavailable states for measurement, and external disturbances. The fractional‐order adaptive laws are derived based only on the output tracking error thanks to the implementation of the strictly positive real (SPR) property, differently from the existing results in the literature of fractional‐order strict‐feedback systems where all system states are used in the adaptive laws. The proposed design approach addresses the nonaffine nature of the control input due to saturation nonlinearity by using the mean value theorem and follows a nonrecursive design by using a state transformation where the advantage is twofold. First, it eliminates the explosion complexity found in back‐stepping control‐based approaches design, and second, it reduces the approximators units and parameters in controller implementation. Furthermore, an observer is introduced to estimate the unavailable newly defined states, and then an output adaptive feedback control design is ensured. An NN is used to approximate the unknown ideal control law, and an auxiliary control term is appended to deal with saturation effect, unknown disturbance, and approximation errors. The tracking error is proved to converge asymptotically to a bounded set using the Lyapunov theory. Simulation results on three examples show the effectiveness of the proposed approach. [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]

The autocovariance and cross‐covariance functions naturally appear in many time series procedures (e.g. autoregression or prediction). Under assumptions, empirical versions of the autocovariance and cross‐covariance are asymptotically normal with covariance structure depending on the second‐ and fourth‐order spectra. Under non‐restrictive assumptions, we derive a bound for the Wasserstein distance of the finite‐sample distribution of the estimator of the autocovariance and cross‐covariance to the Gaussian limit. An error of approximation to the second‐order moments of the estimator and an m‐dependent approximation are the key ingredients to obtain the bound. As a worked example, we discuss how to compute the bound for causal autoregressive processes of order 1 with different distributions for the innovations. To assess our result, we compare our bound to Wasserstein distances obtained via simulation. [ABSTRACT FROM AUTHOR]

This paper investigates the spectral norm version of the column subset selection problem. Given a matrix |$\textbf{A}\in \mathbb{R}^{n\times d}$| and a positive integer |$k\leq \textrm{rank}(\textbf{A})$|⁠ , the objective is to select exactly |$k$| columns of |$\textbf{A}$| that minimize the spectral norm of the residual matrix after projecting |$\textbf{A}$| onto the space spanned by the selected columns. We use the method of interlacing polynomials introduced by Marcus–Spielman–Srivastava to derive a new upper bound on the minimal approximation error. This new bound is asymptotically sharp when the matrix |$\textbf{A}\in \mathbb{R}^{n\times d}$| obeys a spectral power-law decay. The relevant expected characteristic polynomial is a variation of the expected polynomial for the restricted invertibility problem, incorporating two extra variable substitution operators. Finally, we propose a deterministic polynomial-time algorithm that achieves this error bound up to a computational error. [ABSTRACT FROM AUTHOR]

In this paper, a control-error-based decentralized neural network (NN) direct adaptive controller is presented for uncertain interconnected nonlinear systems, in strict-feedback form, subject to input saturation and external disturbances with unavailable states for measurement. Different from the existing results in the literature, the proposed approach is based on the control error instead of the tracking error resulting in a separation-like principle. Furthermore, the explosion of complexity due to back-stepping recursive design is completely avoided along with discarding all restrictive assumptions imposed on the unmatched interconnections. Actually, NNs are used to approximate the unknown ideal control laws, and auxiliary control terms are appended to deal with approximation errors and enhance the stability of the closed-loop system. Besides, fuzzy inference systems are introduced to estimate the unknown control errors, leading to simplified derivation of adaptive laws. Thanks to the strictly positive real (SPR) property, the tracking errors are proved to converge asymptotically to zero using Lyapunov theory, which is superior to bounded stability results usually found in the literature. Simulation results show the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

This article proposes a novel saturated PID‐type formation controller for a class of nonholonomic wheeled mobile robots (WMRs) of type (m, s) with mobility m and steerability s. Since the path curvature causes a severe deviation of the followers from the leaders' trajectory in the corners, a new definition of output equations is suggested to construct the coordinates of a virtual reference point in the front of each follower based on the concept of the extended look‐ahead control that utilizes a corrective angle to compensate the trajectory curvature efficiently. A nonlinear transformation and the robot's kinematic and dynamic equations are heuristically combined to develop a novel second‐order Euler–Lagrange formulation of unconstrained errors from constrained errors. Then, a novel saturated PID‐type controller is proposed for WMRs to build a desirable formation with a prescribed performance. An innovative nonlinear observer is presented to estimate followers' velocities. An efficient mixture of a radial basis function neural network (RBFNN) and an adaptive mechanism is proposed to compensate for model uncertainties, external disturbances, and network approximation errors. Lyapunov's direct method verifies semi‐global uniform ultimate boundedness stability by estimating a gain‐dependent region of attraction. Numerical simulations finally show the efficiency of the proposed controller. [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]

Summary: For a class of multiple‐input multiple‐output large‐scale nonlinear systems in strict‐feedback form with input saturation, external disturbances and immeasurable states, an adaptive decentralized neural network (NN) control strategy on the basis of event triggered mechanism is investigated in this article. In contrast to the literature, the proposed method is centered on the control‐error as a replacement to the tracking‐error that leads to a simplified derivation approach of adaptive laws. Furthermore, the control gains for this class of systems are considered unknown nonlinear functions and not assumed as simple unity or known gains as always done in the literature. Moreover, the challenge of losing controllability that typically arises in state transformation‐based methods, as reported in the literature, is entirely resolved in our approach. Last and not least, all restrictions imposed on unmatched interconnections are eliminated along with avoiding the complexity explosion caused by recursive back‐stepping designs. For this end, the unknown ideal control laws are approximated using NNs, while additional control terms are added to handle saturation effects, unknown interactions, and approximation errors. Additionally, fuzzy inference systems are employed to estimate unknown control errors. Due to the strictly positive real property, the tracking errors are proved to belong to a small compact set using Lyapunov theory. Simulation results demonstrate the effectiveness of the proposed approach. [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]

Estimating the quadratic variation (QV) using high‐frequency financial data is studied in this article, and this work makes two major contributions: first, the fundamental Itô isometry E∫baf(t,ω)dBt2=E∫baf2(t,ω)dt is generalized to some extent, and as an application example, the widely known convergence property of realized volatility (RV) estimator of QV is analyzed by alternatively utilizing this generalized Itô isometry; second, we intuitively establish two types of new estimators of QV which permit volatility varying with time. To construct such estimators, the RV combined with realized bipower variation and RV with realized quarticity are employed, respectively. By applying the generalized Itô isometry, we further show that each of the proposed estimators can converge to QV, at a rate of O(n−1/2), almost surely and in mean square sense, both of which are stronger than the existing convergence in probability. Moreover, the error of approximation is also provided for each estimation. In addition, the obtained convergence property for both types of estimators is demonstrated by empirical investigations based on high‐frequency data of IBM stock. [ABSTRACT FROM AUTHOR]

AbstractThis article investigates the optimal excess-loss reinsurance, investment, and the rationality of using a diffusion model to approximate a jump model. We assume that the instantaneous rate of return of the risky asset is modelled by an Ornstein-Uhlenbeck (O-U) process, and the insurance claims are modeled by a compound Poisson (CP) process and a diffusion approximation (DA) model. By using the stochastic dynamic programming method, the closed-form expressions for the optimal reinsurance and investment strategies and the corresponding value functions are derived. We find that the insurer with the CP claim model always has higher reinsurance demand than the insurer with the DA claim model. Moreover, the error of using the DA model to approximate the CP model is very much dependent on the reinsurance price. Numerical analysis shows that the insurer adjusts the investment strategy as the state of the financial market changes. [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]

This paper proposes a novel learning‐based model predictive control (LMPC) scheme for discrete‐time nonlinear systems. It overcomes the challenge of manually designing the terminal conditions for traditional MPC and enhances the control performance. The scheme employs the value iteration (VI) in reinforcement learning (RL), and autonomously designs the terminal cost by iteratively performing value function learning and policy update under known dynamics and constraints. In contrast to the existing schemes that combine RL with MPC, the proposed scheme explicitly considers the approximation errors in each iteration. Further, a rigorous theoretical analysis is provided, including the convergence of VI, the stability and performance of the closed‐loop system. In addition, the influences of the prediction horizon and the initial terminal cost on performance are also investigated. Simulation results of a linear system verify the theoretical properties of the LMPC and show that it achieves (near‐)optimal performance. Moreover, its unique superiority over traditional MPC is fully demonstrated in a nonholonomic vehicle regulation example. [ABSTRACT FROM AUTHOR]

In this paper, we present the definitions of nonfragile high‐gain observers and design method for lower‐triangular nonlinear systems with output uncertainty. Radial basis function neural networks (RBFNNs) are used to approximate the output uncertainty. By inserting an output filter and an input‐output filter, a new augmented adaptive observable canonical form is derived. Then, a corresponding observer with gain perturbations is designed to estimate the states and the coefficients of the RBFNNs, and a disturbance observer is designed to estimate the approximation error. The maximum allowable gain perturbation is also given. Then, the obtained results are extended to nonlinear systems in adaptive observer form with output uncertainty. Finally, some numerical simulations are offered to corroborate the theoretical results. [ABSTRACT FROM AUTHOR]

In this work we consider the two dimensional instationary Navier–Stokes equations with homogeneous Dirichlet/no-slip boundary conditions. We show error estimates for the fully discrete problem, where a discontinuous Galerkin method in time and inf-sup stable finite elements in space are used. Recently, best approximation type error estimates for the Stokes problem in the L∞(I; L2(Ω)), L2(I; H1(Ω)) and L2(I; L2(Ω)) norms have been shown. The main result of the present work extends the error estimate in the L∞(I; L2(Ω)) norm to the Navier–Stokes equations, by pursuing an error splitting approach and an appropriate duality argument. In order to discuss the stability of solutions to the discrete primal and dual equations, a specially tailored discrete Gronwall lemma is presented. The techniques developed towards showing the L∞(I; L2(Ω)) error estimate, also allow us to show best approximation type error estimates in the L2(I; H1(Ω)) and L2(I; L2(Ω)) norms, which complement this work. [ABSTRACT FROM AUTHOR]

In this paper we establish best approximation type estimates for the fully discrete Galerkin solutions of transient Stokes problem in L 2 (I ; L 2 (Ω) d) and L 2 (I ; H 1 (Ω) d) norms. These estimates fill the gap in the error analysis of the transient Stokes problems and have a number of applications. The analysis naturally extends to inhomogeneous parabolic problems. The best type L 2 (I ; H 1 (Ω)) error estimate are new even for scalar parabolic problems. [ABSTRACT FROM AUTHOR]

We propose the use of low bit-depth Sigma-Delta and distributed noise-shaping methods for quantizing the random Fourier features (RFFs) associated with shift-invariant kernels. We prove that our quantized RFFs—even in the case of |$1$| -bit quantization—allow a high-accuracy approximation of the underlying kernels, and the approximation error decays at least polynomially fast as the dimension of the RFFs increases. We also show that the quantized RFFs can be further compressed, yielding an excellent trade-off between memory use and accuracy. Namely, the approximation error now decays exponentially as a function of the bits used. The quantization algorithms we propose are intended for digitizing RFFs without explicit knowledge of the application for which they will be used. Nevertheless, as we empirically show by testing the performance of our methods on several machine learning tasks, our method compares favourably with other state-of-the-art quantization methods. [ABSTRACT FROM AUTHOR]

Over the last several decades, improvements in the fields of analytic combinatorics and computer algebra have made determining the asymptotic behaviour of sequences satisfying linear recurrence relations with polynomial coefficients largely a matter of routine, under assumptions that hold often in practice. The algorithms involved typically take a sequence, encoded by a recurrence relation and initial terms, and return the leading terms in an asymptotic expansion up to a big-O error term. Less studied, however, are effective techniques giving an explicit bound on asymptotic error terms. Among other things, such explicit bounds typically allow the user to automatically prove sequence positivity (an active area of enumerative and algebraic combinatorics) by exhibiting an index when positive leading asymptotic behaviour dominates any error terms. In this article, we present a practical algorithm for computing such asymptotic approximations with rigorous error bounds, under the assumption that the generating series of the sequence is a solution of a differential equation with regular (Fuchsian) dominant singularities. Our algorithm approximately follows the singularity analysis method of Flajolet and Odlyzko [SIAM J. Discrete Math. 3 (1990), pp. 216–240], except that all big-O terms involved in the derivation of the asymptotic expansion are replaced by explicit error terms. The computation of the error terms combines analytic bounds from the literature with effective techniques from rigorous numerics and computer algebra. We implement our algorithm in the SageMath computer algebra system and exhibit its use on a variety of applications (including our original motivating example, solution uniqueness in the Canham model for the shape of genus one biomembranes). [ABSTRACT FROM AUTHOR]

The Tucker tensor decomposition is a natural extension of the singular value decomposition (SVD) to multiway data. We propose to accelerate Tucker tensor decomposition algorithms by using randomization and parallelization. We present two algorithms that scale to large data and many processors, significantly reduce both computation and communication cost compared to previous deterministic and randomized approaches, and obtain nearly the same approximation errors. The key idea in our algorithms is to perform randomized sketches with Kronecker-structured random matrices, which reduces computation compared to unstructured matrices and can be implemented using a fundamental tensor computational kernel. We provide probabilistic error analysis of our algorithms and implement a new parallel algorithm for the structured randomized sketch. Our experimental results demonstrate that our combination of randomization and parallelization achieves accurate Tucker decompositions much faster than alternative approaches. We observe up to a 16X speedup over the fastest deterministic parallel implementation on 3D simulation data. [ABSTRACT FROM AUTHOR]

We present a new restricted SVD-based CUR (RSVD-CUR) factorization for matrix triplets (A, B, G) that aims to extract meaningful information by providing a low-rank approximation of the three matrices using a subset of their rows and columns. The proposed method utilizes the discrete empirical interpolation method (DEIM) to select the subset of rows and columns from the orthogonal and nonsingular matrices obtained through an RSVD of the matrix triplet. We explore the relationships between a DEIM type RSVD-CUR factorization, a DEIM type CUR factorization, and a DEIM type generalized CUR decomposition and provide an error analysis that establishes the accuracy of the RSVD-CUR decomposition within a factor of the approximation error of the RSVD of the given matrices. The RSVD-CUR factorization can be used in applications that require approximating one data matrix relative to two other given matrices. We discuss two such applications, namely multiview dimension reduction and data perturbation problems where a correlated noise matrix is added to the input data matrix. Our numerical experiments demonstrate the advantages of the proposed method over the standard CUR approximation in these scenarios. [ABSTRACT FROM AUTHOR]

This paper investigates the impact of subcooling degree and exhaust pressure on the performance of a high-pressure piston pump for pressurizing cryogenic liquid. We fabricate a lab-scale cryogenic liquid pump, conduct experiments with liquid nitrogen, and observe the significant effect of subcooling changes on the pump performance. The lab-scale pump achieves up to 30 bar of exhaust pressure and up to 45.4 g exhaust mass per cycle. This paper introduces the concept of duty cycle for the exhaust mass as a measure of pump performance. The results show that as the subcooling degree and the exhaust pressure increase, the duty cycle and the exhaust mass also increase, improving the pump performance. However, the increase in performance does not happen proportionally, and an optimal subcooling degree must be selected. This paper establishes a generalized computational model to predict the pump performance, achieving the mean absolute errors of 1.8 bar and 0.3 g/s in pressure and mass flow rate, respectively. The model can predict the exhaust mass during one stroke cycle of the pump with an approximation error of 10 %. In addition, the model predicts that pressurizing liquid hydrogen to 900 bar requires subcooling degree and its duty cycle cannot reach 100 % by reduction of specific volume. The findings suggest that subcooling can be used to enhance cryogenic liquid pump performance, and the optimal subcooling degree can be determined during the design of piston pumps to achieve high pressure. [Display omitted] • A piston-type tension-assisted cryogenic liquid pump was developed. • We identified the performance indicator of the pump as duty cycle. • This paper investigated the effect of the parameters on the duty cycle. • Subcooling degree effect on liquid hydrogen is predicted. [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]

This paper presents a sliding mode control based on particle swarm optimization neural network and adaptive reaching law, and the proposed control method solves the problem of chattering and tracking performance degradation of a multi-joint manipulator caused by uncertainties such as external disturbances and modeling error. First, to address the problem that the precise dynamic system of the manipulator is difficult to establish, the radial basis function neural network (RBFNN) is used to approximate the uncertainty of the manipulator model, and the parameters of the neural network are optimized through the adaptive natural selection particle swarm optimization algorithm (ASelPSO) to improve the approximation ability and reduce the approximation error. Second, to eliminate chattering, adaptive reaching law is selected to improve the dynamic quality of approaching motion. Finally, a comparative simulation experiment is carried out with a 3-DOF manipulator as the research object. The results show that the control method has obvious improvements in eliminating chattering, improving tracking accuracy, and increasing convergence speed, which verifies the feasibility and superiority of the control scheme. [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]

This article is concerned with the problem of approximating a not necessarily bounded spectrahedral shadow, a certain convex set, by polyhedra. By identifying the set with its homogenization, the problem is reduced to the approximation of a closed convex cone. We introduce the notion of homogeneous δ -approximation of a convex set and show that it defines a meaningful concept in the sense that approximations converge to the original set if the approximation error δ diminishes. Moreover, we show that a homogeneous δ -approximation of the polar of a convex set is immediately available from an approximation of the set itself under mild conditions. Finally, we present an algorithm for the computation of homogeneous δ -approximations of spectrahedral shadows and demonstrate it on examples. [ABSTRACT FROM AUTHOR]

Approximating Bézier curve by polygon is a traditional technique in computer-aided geometric design, and there are several methods for constructing such polygons. In this paper, we propose to approximate Bézier curve by the least square polygon (LSP). LSP was derived by least square fitting which minimizes the integral of approximating errors. We provide a closed formula for the vertices of LSP. Because of its optimization method, LSP has lower approximation errors than existing approximating polygons. This observation is verified by many numerical examples we tested and several typical examples are included in this paper. [ABSTRACT FROM AUTHOR]

This paper proposes Brenke-type polynomials for uncertainty approximation in adaptive control of cooperative electrically driven manipulators. The contribution of this paper is reducing the number of signals needed in the construction of the regressor vector for uncertainty approximation due to the simplicity of Brenke-type polynomials in comparison with other alternatives. It is verified that uniformly bounded stability of the tracking/approximation errors can be assured if the Brenke-type polynomials are utilised as regressors. Moreover, it has been assumed that velocity signals are not available for feedback. Different trajectories, including simple and highly complicated and fast paths, have been tested in simulation results. The results are also compared with four state-of-the-art powerful approximation methods to show the accuracy and efficiency of the proposed controller structure. [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]

Currently, social networks serve as communication tools, but the problem of excessive usage leading to addiction is on the rise. Therefore, the purpose of this study is to validate the Social Network addiction (SNA) scale in university students. The study is developed from a quantitative approach, of an exploratory and confirmatory type. The type of sampling is non-probabilistic by self-determination, and the participation of 200 students was achieved, to whom the reason for the survey was explained, who agreed to participate voluntarily. For data analysis, the instrument was validated according to Cronbach's alpha criteria and after, for the analysis, the SEM methodology was used. The most important findings show only the conformation of a two-factor model, which includes the seven most representative items, and which are feasible to analyze in this study group. The goodness-of-fit index (GFI) value is acceptable since it exceeds the theoretical threshold greater than 0.9 and the root mean square error of approximation (RMSEA) is less than 0.05. The Tucker-Lewis index (TLI) and the norm adjustment index (NFI) are greater than 0.9. Based on the results, it can be concluded that students exhibit a strong desire for connectivity. Furthermore, they invest a significant amount of time in their online networks and experience uncertainty about how to occupy their when they are not connected. [ABSTRACT FROM AUTHOR]

It has been suggested that equivalence testing (otherwise known as negligible effect testing) should be used to evaluate model fit within structural equation modelling (SEM). In this study, we propose novel variations of equivalence tests based on the popular root mean squared error of approximation and comparative fit index fit indices. Using Monte Carlo simulations, we compare the performance of these novel tests to other existing equivalence testing‐based fit indices in SEM, as well as to other methods commonly used to evaluate model fit. Results indicate that equivalence tests in SEM have good Type I error control and display considerable power for detecting well‐fitting models in medium to large sample sizes. At small sample sizes, relative to traditional fit indices, equivalence tests limit the chance of supporting a poorly fitting model. We also present an illustrative example to demonstrate how equivalence tests may be incorporated in model fit reporting. Equivalence tests in SEM also have unique interpretational advantages compared to other methods of model fit evaluation. We recommend that equivalence tests be utilized in conjunction with descriptive fit indices to provide more evidence when evaluating model fit. [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]

The adaptive predefined settling time performance tracking control for a class of uncertain nonlinear network systems with fast time-varying or even abrupt reference signals is studied. In this paper, an improved performance function with time-varying boundary (following the change of reference signal) is designed to overcome the problem that the traditional prescribed performance control (PPC) can only converge to a constant in steady state. Compared with the existing adaptive finite time prescribed performance congestion control for tracking fixed value reference signals, an adaptive predefined settling time prescribed performance congestion control strategy is proposed by means of some new recursive construction (introducing nonautonomous differential equations (NDE)) and analysis innovation. In addition, the network system considered in this paper is more general. Under the framework of backstepping, NDE is used iteratively to control the approximation error with boundary inequality. It is proved theoretically that all signals of the closed-loop system are predefined settling time bounded. Finally, the effectiveness of the proposed control strategy is verified by simulations. [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]