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We develop approximate estimation methods for exponential random graph models (ERGMs), whose likelihood is proportional to an intractable normalizing constant. The usual approach approximates this constant with Monte Carlo simulations; however, convergence may be exponentially slow. We propose a deterministic method, based on a variational mean-field approximation of the ERGM's normalizing constant. We compute lower and upper bounds for the approximation error for any network size, adapting nonlinear large deviation results. This translates into bounds on the distance between true likelihood and mean-field likelihood. Monte Carlo simulations suggest that in practice, our deterministic method performs better than our conservative theoretical approximation bounds imply, for a large class of models. [ABSTRACT FROM AUTHOR]

This paper is aimed to establish a new result on error of approximation of function in Lipschitz Lip (p (t), r)(r ≥ 1)-class via almost (N,pk,qk)-means using the concept of head bounded variation sequences (HBVS) and rest bounded variation sequences (RBVS) of Fourier series. [ABSTRACT FROM AUTHOR]

A major gap between few-shot and many-shot learning is the data distribution empirically oserved by the model during training. In few-shot learning, the learned model can easily become over-fitted based on the biased distribution formed by only a few training examples, while the ground-truth data distribution is more accurately uncovered in many-shot learning to learn a well-generalized model. In this paper, we propose to calibrate the distribution of these few-sample classes to be more unbiased to alleviate such an over-fitting problem. The distribution calibration is achieved by transferring statistics from the classes with sufficient examples to those few-sample classes. After calibration, an adequate number of examples can be sampled from the calibrated distribution to expand the inputs to the classifier. Specifically, we assume every dimension in the feature representation from the same class follows a Gaussian distribution so that the mean and the variance of the distribution can borrow from that of similar classes whose statistics are better estimated with an adequate number of samples. Extensive experiments on three datasets, miniImageNet, tieredImageNet, and CUB, show that a simple linear classifier trained using the features sampled from our calibrated distribution can outperform the state-of-the-art accuracy by a large margin. Besides the favorable performance, the proposed method also exhibits high flexibility by showing consistent accuracy improvement when it is built on top of any off-the-shelf pretrained feature extractors and classification models without extra learnable parameters. The visualization of these generated features demonstrates that our calibrated distribution is an accurate estimation thus the generalization ability gain is convincing. We also establish a generalization error bound for the proposed distribution-calibration-based few-shot learning, which consists of the distribution assumption error, the distribution approximation error, and the estimation error. This generalization error bound theoretically justifies the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a quasi-quadratic online adaptive dynamic programming (QOADP) algorithm is proposed to realize optimal economic dispatch for smart buildings. Load demand of high volatility is considered, which is modeled by an uncontrollable state. To reduce residual errors of the approximation structure, a quasi-quadratic-form parametric structure was designed elaborately with a bias term to counteract effects of uncertainties. Based on action-dependent heuristic dynamic programming (ADHDP), an implementation of the QOADP algorithm is presented that involved obtaining optimal economic dispatch for smart buildings. Finally, hardware-in-loop (HIL) experiments were conducted, and the performance of the proposed QOADP algorithm is superior to that of two other typical algorithms. [ABSTRACT FROM AUTHOR]

Researchers in statistical shape analysis often analyze outlines of objects. Even though these contours are infinite-dimensional in theory, they must be discretized in practice. When discretizing, it is important to reduce the number of sampling points considerably to reduce computational costs, but to not use too few points so as to result in too much approximation error. Unfortunately, determining the minimum number of points needed to achieve sufficiently approximate the contours is computationally expensive. In this paper, we fit regression models to predict these lower bounds using characteristics of the contours that are computationally cheap as predictor variables. However, least squares regression is inadequate for this task because it treats overestimation and underestimation equally, but underestimation of lower bounds is far more serious. Instead, to fit the models, we use the LINEX loss function, which allows us to penalize underestimation at an exponential rate while penalizing overestimation only linearly. We present a novel approach to select the shape parameter of the loss function and tools for analyzing how well the model fits the data. Through validation methods, we show that the LINEX models work well for reducing the underestimation for the lower bounds. [ABSTRACT FROM AUTHOR]

This article presents an effective method to address the tracking control problem arising in uncertain switched high-order nonlinear system in strict-feedback form. The system under consideration contains unknown functions, which causally make the asymptotic tracking performance difficult to be achieved. By adopting the adding a power integrator approach in the framework of backstepping, a novel tracking controller is developed to guarantee an asymptotic tracking performance in the presence of the approximation error cased by neural networks (NNs) under arbitrary switching. The main contributions lie in: 1) the article for the first time embeds the backstepping technique in designing a kind of discontinuous controller with proportional integral (PI) compensation and 2) with the help of Filippov’s theory, a new defined system described by differential inclusions can be first obtained by taking some transformations, and then a novel nonsmooth Lyapunov function approach along with its upper right Dini derivative technique is applied to complete the construction of the discontinuous controller. Finally, two simulation examples are exhibited to verify the validity of the proposed design techniques. [ABSTRACT FROM AUTHOR]

In this article, we consider an iterative adaptive dynamic programming (ADP) algorithm within the Hamiltonian-driven framework to solve the Hamilton–Jacobi–Bellman (HJB) equation for the infinite-horizon optimal control problem in continuous time for nonlinear systems. First, a novel function, “min-Hamiltonian,” is defined to capture the fundamental properties of the classical Hamiltonian. It is shown that both the HJB equation and the policy iteration (PI) algorithm can be formulated in terms of the min-Hamiltonian within the Hamiltonian-driven framework. Moreover, we develop an iterative ADP algorithm that takes into consideration the approximation errors during the policy evaluation step. We then derive a sufficient condition on the iterative value gradient to guarantee closed-loop stability of the equilibrium point as well as convergence to the optimal value. A model-free extension based on an off-policy reinforcement learning (RL) technique is also provided. Finally, numerical results illustrate the efficacy of the proposed framework. [ABSTRACT FROM AUTHOR]

In biological and medical research, the cost and collateral damage caused during the collection and measurement of a sample are the reasons behind a compromise on the inference with a fixed and accepted approximation error. The ranked set sampling (RSS) performs better in such scenarios, and the use of auxiliary information even enhances the performance of the estimators. In this study, two generalized classes of estimators are proposed to estimate the population variance using RSS and information of auxiliary variable. The bias and mean square errors of the proposed classes of estimators are derived up to first order of approximation. Some special cases of one of the proposed class of estimators are also considered in the presence of available population parameters. A simulation study was conducted to see the performance of the members of the proposed family by using various sample sizes. The real-life data application is done to estimate the variance of gestational age of fetuses with supplementary information. The results showed that RSS design is a more accurate method than simple random sampling, to determine the population variance of hard-to-measure or destructive sampling units. [ABSTRACT FROM AUTHOR]

The existing magnetic target localization methods are greatly affected by the geomagnetic field and exist approximation errors. In this paper, a two-point magnetic gradient tensor localization model is established by using the spatial relation between the magnetic target and the observation points derived from magnetic gradient tensor and tensor invariants. Based on the model, the equations relating to the position vector of magnetic target are constructed. Solving the equations, a new magnetic target localization method using only a two-point magnetic gradient tensor and no approximation errors is achieved. To accurately evaluate the localization accuracy of the method, a circular trajectory that varies in all three directions is proposed. Simulation results show that the proposed method is almost error-free in the absence of noise. After adding noise, the maximum relative error percentage is reduced by 28.4% and 2.21% compared with the single-point method and the other two-point method, respectively. Furthermore, the proposed method is not affected by the variation in the distance between two observation points. At a detection distance of 20 m, the maximum localization error is 1.86 m. In addition, the experiments also verify that the new method can avoid the influence of the geomagnetic field and the variation in the distance, and achieve high localization accuracy. The average relative error percentage in the y-direction is as low as 3.78%. [ABSTRACT FROM AUTHOR]

In this paper, we adopt the choice theory of the shape parameters contained in the smooth radial basis functions to solve Poisson equations. Luh's choice theory, based on harmonic analysis, is mathematically complicated and applies only to function interpolation. Here, we aim at presenting an easily accessible approach to solving differential equations with the choice theory which proves to be very successful, not only by its easy accessibility but also by its striking accuracy and efficiency. Our emphases are on the highly reliable prediction of the optimal value of the shape parameter and the extremely small approximation errors of the numerical solutions to the differential equations. We hope that our approach can be accepted by both mathematicians and non-mathematicians. [ABSTRACT FROM AUTHOR]

Given several nonnegative matrices with a single pattern of allocation among their zero/nonzero elements, the average matrix should have the same pattern as well. This is the first tenet of the pattern-multiplicative average (PMA) concept, while the second one suggests the multiplicative nature of averaging. The concept of PMA was motivated in a number of application fields, of which we consider the matrix population models and illustrate solving the PMA problem with several sets of model matrices calibrated in particular botanic case studies. The patterns of those matrices are typically nontrivial (they contain both zero and nonzero elements), the PMA problem thus having no exact solution for a fundamental reason (an overdetermined system of algebraic equations). Therefore, searching for the approximate solution reduces to a constrained minimization problem for the approximation error, the loss function in optimization terms. We consider two alternative types of the loss function and present a general algorithm of searching the optimal solution: basin-hopping global search, then local descents by the method of conjugate gradients or that of penalty functions. Theoretical disadvantages and practical limitations of both loss functions are discussed and illustrated with a number of practical examples. [ABSTRACT FROM AUTHOR]

In this work we extend some ideas about greedy algorithms, which are well-established tools for, e.g., kernel bases, and exponential-polynomial splines whose main drawback consists in possible overfitting and consequent oscillations of the approximant. To partially overcome this issue, we develop some results on theoretically optimal interpolation points. Moreover, we introduce two algorithms which perform an adaptive selection of the spline interpolation points based on the minimization either of the sample residuals (f-greedy), or of an upper bound for the approximation error based on the spline Lebesgue function (λ -greedy). Both methods allow us to obtain an adaptive selection of the sampling points, i.e., the spline nodes. While the f-greedy selection is tailored to one specific target function, the λ -greedy algorithm enables us to define target-data-independent interpolation nodes. [ABSTRACT FROM AUTHOR]

The use of nondifferentiable priors in Bayesian statistics has become increasingly popular, in particular in Bayesian imaging analysis. Current state-of-the-art methods are approximate in the sense that they replace the posterior with a smooth approximation via Moreau-Yosida envelopes, and apply gradient-based discretized diffusions to sample from the resulting distribution. We characterize the error of the Moreau-Yosida approximation and propose a novel implementation using underdamped Langevin dynamics. In misson-critical cases, however, replacing the posterior with an approximation may not be a viable option. Instead, we show that piecewise-deterministic Markov processes (PDMP) can be used for exact posterior inference from distributions satisfying almost everywhere differentiability. Furthermore, in contrast with diffusion-based methods, the suggested PDMP-based samplers place no assumptions on the prior shape, nor require access to a computationally cheap proximal operator, and consequently have a much broader scope of application. Through detailed numerical examples, including a nondifferentiable circular distribution and a nonconvex genomics model, we elucidate the relative strengths of these sampling methods on problems of moderate to high dimensions, underlining the benefits of PDMP-based methods when accurate sampling is decisive. for this article are available online. [ABSTRACT FROM AUTHOR]

In this article we derive the asymptotic distribution of estimated best linear unbiased predictors (EBLUPs) of the random effects in a nested error regression model. Under very mild conditions which do not require the assumption of normality, we show that asymptotically the distribution of the EBLUPs as both the number of clusters and the cluster sizes diverge to infinity is the convolution of the true distribution of the random effects and a normal distribution. This result yields very simple asymptotic approximations to and estimators of the prediction mean squared error of EBLUPs, and then asymptotic prediction intervals for the unobserved random effects. We also derive a higher order approximation to the asymptotic mean squared error and provide a detailed theoretical and empirical comparison with the well-known analytical prediction mean squared error approximations and estimators proposed by Kackar and Harville and Prasad and Rao. We show that our simple estimator of the predictor mean squared errors of EBLUPs works very well in practice when both the number of clusters and the cluster sizes are sufficiently large. Finally, we illustrate the use of the asymptotic prediction intervals with data on radon measurements of houses in Massachusetts and Arizona. [ABSTRACT FROM AUTHOR]

This paper considers the positivity-preserving model reduction for discrete-time positive systems. Given a stable high-order positive system, we aim to find a reduced-order model such that the approximation error is minimized within a prescribed H ∞ performance and positivity is preserved. Regarding the bounded real lemma, the sufficient and necessary condition for the existence of a reduced-order model is established in terms of bilinear matrix inequality and convex semi-definite constraint, which ensures that the reduced-order system is positive and the resulted error system is stable and has an H ∞ performance level. Based on the inner-approximation strategy, we approximate the bilinear constraints with convex ones, under which an iterative procedure is provided to calculate the desired reduced-order model. Finally, an example is provided to demonstrate the effectiveness and potential benefits of the presented results. [ABSTRACT FROM AUTHOR]

The curse of dimensionality problem refers to a set of troubles arising when dealing with huge amount of data as happens, e.g., applying standard numerical methods to solve partial differential equations related to financial modeling. To overcome the latter issue, we propose a Deep Learning approach to efficiently approximate nonlinear functions characterizing financial models in a high dimension. In particular, we consider solving the Black–Scholes–Barenblatt non-linear stochastic differential equation via a forward-backward neural network, also calibrating the related stochastic volatility model when dealing with European options. The obtained results exhibit accurate approximations of the implied volatility surface. Specifically, our method seems to significantly reduce the neural network's training time and the approximation error on the test set. [ABSTRACT FROM AUTHOR]

We present a technique for producing valid dual bounds for nonconvex quadratic optimization problems. The approach leverages an elegant piecewise linear approximation for univariate quadratic functions due to Yarotsky (Neural Netw 94:103–114, 2017), formulating this (simple) approximation using mixed-integer programming (MIP). Notably, the number of constraints, binary variables, and auxiliary continuous variables used in this formulation grows logarithmically in the approximation error. Combining this with a diagonal perturbation technique to convert a nonseparable quadratic function into a separable one, we present a mixed-integer convex quadratic relaxation for nonconvex quadratic optimization problems. We study the strength (or sharpness) of our formulation and the tightness of its approximation. Further, we show that our formulation represents feasible points via a Gray code. We close with computational results on problems with quadratic objectives and/or constraints, showing that our proposed method (i) across the board outperforms existing MIP relaxations from the literature, and (ii) on hard instances produces better bounds than exact solvers within a fixed time budget. [ABSTRACT FROM AUTHOR]

We consider the spectral approximation of flow optimal control problems with state constraint. The main contribution of this work is to derive an a posteriori error estimator, and show the upper and lower bounds for the approximation error. Numerical experiment shows the efficiency of the error indicator, which will be used to construct reliable adaptive hp spectral element approximation for flow control in the future work. [ABSTRACT FROM AUTHOR]

Deep learning—in particular, deep neural networks (DNNs)—as a mesh-free and self-adapting method has demonstrated its great potential in the field of scientific computation. In this work, inspired by the Deep Ritz method proposed by Weinan E et al. to solve a class of variational problems that generally stem from partial differential equations, we present a coupled deep neural network (CDNN) to solve the fourth-order biharmonic equation by splitting it into two well-posed Poisson's problems, and then design a hybrid loss function for this method that can make efficiently the optimization of DNN easier and reduce the computer resources. In addition, a new activation function based on Fourier theory is introduced for our CDNN method. This activation function can reduce significantly the approximation error of the DNN. Finally, some numerical experiments are carried out to demonstrate the feasibility and efficiency of the CDNN method for the biharmonic equation in various cases. [ABSTRACT FROM AUTHOR]

Chirps are familiar in nature, have a built-in resistance to noise and interference, and are connected to a wide range of highly oscillatory processes. Detecting chirp oscillating patterns by traditional Fourier series is challenging because the chirp frequencies constantly change over time. Estimating such types of functions considering the partial sums of a Fourier series in Fourier analysis does not permit an approximate solution, which entails more Fourier coefficients required for signal reconstruction. The standard Fourier series, therefore, has a poor convergence rate and is an inadequate approximation. In this study, we use a parameterized orthonormal basis with an adjustable parameter to match the oscillating behavior of the chirp to approximate linear chirps using the partial sums of a generalized Fourier series known as fractional Fourier series, which gives the best approximation with only a small number of fractional Fourier coefficients. We used the fractional Fourier transform to compute the fractional Fourier coefficients at sample points. Additionally, we discover that the fractional parameter has the best value at which fractional Fourier coefficients of zero degrees have the most considerable magnitude, leading to the rapid decline of fractional Fourier coefficients of high degrees. Furthermore, fractional Fourier series approximation with optimal fractional parameters provides the minimum mean square error over the fractional Fourier parameter domain. [ABSTRACT FROM AUTHOR]

Spiking neural networks (SNNs) have shown clear advantages over traditional artificial neural networks (ANNs) for low latency and high computational efficiency, due to their event-driven nature and sparse communication. However, the training of deep SNNs is not straightforward. In this paper, we propose a novel ANN-to-SNN conversion and layer-wise learning framework for rapid and efficient pattern recognition, which is referred to as progressive tandem learning. By studying the equivalence between ANNs and SNNs in the discrete representation space, a primitive network conversion method is introduced that takes full advantage of spike count to approximate the activation value of ANN neurons. To compensate for the approximation errors arising from the primitive network conversion, we further introduce a layer-wise learning method with an adaptive training scheduler to fine-tune the network weights. The progressive tandem learning framework also allows hardware constraints, such as limited weight precision and fan-in connections, to be progressively imposed during training. The SNNs thus trained have demonstrated remarkable classification and regression capabilities on large-scale object recognition, image reconstruction, and speech separation tasks, while requiring at least an order of magnitude reduced inference time and synaptic operations than other state-of-the-art SNN implementations. It, therefore, opens up a myriad of opportunities for pervasive mobile and embedded devices with a limited power budget. [ABSTRACT FROM AUTHOR]

In this paper, we solve a system of mixed variational inclusions involving a generalized Cayley operator and the generalized Yosida approximation operator. An iterative algorithm is suggested to discuss the convergence analysis. We have shown that our system admits a unique solution by using the properties of q-uniformly smooth Banach space, and we discuss the convergence criteria for sequences generated by iterative algorithm. Two examples are constructed, and an application is provided. [ABSTRACT FROM AUTHOR]

Homomorphic encryption (HE) is the ultimate tool for performing secure computations even in untrusted environments. Application of HE for deep learning (DL) inference is an active area of research, given the fact that DL models are often deployed in untrusted environments (e.g., third-party servers) yet inferring on private data. However, existing HE libraries [somewhat (SWHE), leveled (LHE) or fully homomorphic (FHE)] suffer from extensive computational and memory overhead. Few performance optimized high-speed homomorphic libraries are either suffering from certain approximation issues leading to decryption errors or proven to be insecure according to recent published attacks. In this article, we propose architectural tricks to achieve performance speedup for encrypted DL inference developed with exact HE schemes without any approximation or decryption error in homomorphic computations. The main idea is to apply quantization and suitable data packing in the form of bitslicing to reduce the costly noise handling operation, Bootstrapping while achieving a functionally correct and highly parallel DL pipeline with a moderate memory footprint. Experimental evaluation on the MNIST dataset shows a significant ($37\times$) speedup over the nonbitsliced versions of the same architecture. Low memory bandwidths (700 MB) of our design pipelines further highlight their promise toward scaling over larger gamut of Edge-AI analytics use cases. [ABSTRACT FROM AUTHOR]

This paper presents an online recorded data-based composite neural finite-time control scheme for underactuated marine surface vessels (MSVs) subject to uncertain dynamics and time-varying external disturbances. The underactuation problem of the MSVs was solved by introducing the line-of-sight (LOS)method. The uncertain dynamics of MSVs are approximated by the composite neural networks (NNs). A modified prediction error signal is designed by virtue of online recorded data. The weight updating law of NN is driven by both tracking error and prediction error, introducing additional correction information to the weights of NN, thus improving the learning ability of the NN. Furthermore, disturbance observers can be devised to estimate the compound disturbances consisting of the approximation errors of NNs and external disturbances. Moreover, the smooth function is inserted into the design of the control scheme, and the finite-time composite neural trajectory tracking control of MSVs is achieved. The stability of the MSVs trajectory tracking closed-loop control system is guaranteed rigorously by the Lyapunov approach, and the tracking error will converge to the set of residuals around zero within a finite time. The simulation tests on anMSV verify the effectiveness of the proposed control scheme. [ABSTRACT FROM AUTHOR]

Geometric approximation errors of periodically distributed surface facets of cable-network reflector antennas can result in grating lobes undesired in some space missions. As the surface is shaped by the tensioned flexible cable-network structure, the electromagnetic (EM) and structural parameters are highly correlated, and the well-designed surface obtained only concerning the EM performance may not be shaped by the tension-only cable network with specified cable-tension constraints. To address the problem, this study proposes an integrated structural-EM optimization method to enable low grating lobe cable-network antenna design. In this method, force densities of the cables, which can determine both the cable tensions and surface node positions that decide the boundary conditions of the EM field, are selected as the design variables. An adaptive objective/constraint function is proposed, which enables the grating lobe level (GLL) to be minimized with constrained cable tensions, antenna gain, and first sidelobe level (SLL) in a simultaneous, feasible and relatively fast way. To demonstrate the feasibility and effectiveness of the proposed method, it was implemented on a 12 m-diameter cable-network antenna. The results indicate that the grating lobes are greatly degraded and the achieved surface can be formed by well tensioned cables. [ABSTRACT FROM AUTHOR]

The sum of lognormal variables has been a topic of interest in several fields of research such as engineering, biology and finance, among others. For example, in the field of telecommunications, the aggregate interference of radio frequency signals is modeled as a sum of lognormal variables. To date, there is no closed expression for the probability distribution function (PDF) of this sum. Several authors have proposed approximations for this PDF, with which they calculate the mean and variance. However, each method has limitations in its range of parameters for mean, variance and number of random variables to be added. In other cases, long approximations as power series are used, which makes the analytical treatment impractical and reduces the computational performance of numerical operations. This paper shows an alternative method for calculating the mean and variance of the sum of lognormal random variables from a computational performance approach. Our method has been evaluated extensively by Monte Carlo simulations. As a result, this method is computationally efficient and yields a low approximation error computation for a wide range of mean values, variances and number of random variables. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the approximation of a diffusion model problem with contrasted diffusivity for various nonconforming approximation methods. The essential difficulty is that the Sobolev smoothness index of the exact solution may be just barely larger than 1. The lack of smoothness is handled by giving a weak meaning to the normal derivative of the exact solution at the mesh faces. We derive robust and quasi-optimal error estimates. Quasi-optimality means that the approximation error is bounded, up to a generic constant, by the best approximation error in the discrete trial space, and robustness means that the generic constant is independent of the diffusivity contrast. The error estimates use a mesh-dependent norm that is equivalent, at the discrete level, to the energy norm and that remains bounded as long as the exact solution has a Sobolev index strictly larger than 1. Finally, we briefly show how the analysis can be extended to the Maxwell's equations. [ABSTRACT FROM AUTHOR]

The adaptive cubic regularization method solves an unconstrained optimization model by using a three-order regularization term to approximate the objective function at each iteration. Similar to the trust-region method, the calculation of the sub-problem highly affects the computing efficiency. The Lanczos method is an useful tool for simplifying the objective function in the sub-problem. In this paper, we implement the adaptive cubic regularization method with the aid of the Lanczos method, and analyze the error of Lanczos approximation. We show that both the error between the Lanczos objective function and the original cubic term, and the error between the solution of the Lanczos approximation and the solution of the original cubic sub-problem are bounded up by the condition number of the optimal Hessian matrix. Furthermore, we compare the numerical performances of the adaptive cubic regularization algorithm when using the Lanczos approximation method and the adaptive cubic regularization algorithm without using the Lanczos approximation for unconstrained optimization problems. Numerical experiments show that the Lanczos method improves the computation efficiency of the adaptive cubic method remarkably. [ABSTRACT FROM AUTHOR]

In this paper, we totally discard the traditional trial-and-error algorithms of choosing the acceptable shape parameter c in the multiquadrics − c 2 + ∥ x ∥ 2 when dealing with differential equations, for example, the Poisson equation, with the RBF collocation method. Instead, we choose c directly by the MN-curve theory and hence avoid the time-consuming steps of solving a linear system required by each trial of the c value in the traditional methods. The quality of the c value thus obtained is supported by the newly born choice theory of the shape parameter. Experiments demonstrate that the approximation error of the approximate solution to the differential equation is very close to the best approximation error among all possible choices of c. [ABSTRACT FROM AUTHOR]

Modeling complex hydrodynamic processes in coastal systems is an important problem of mathematical modeling that cannot be solved analytically. The approximation of convective terms is difficult from the point of view of error reduction. This paper proposes a difference scheme based on a linear combination of the Upwind Leapfrog scheme with 2/3 weight coefficient, and the Standard Leapfrog scheme with 1/3 weight coefficient. The weight coefficients are obtained as a result of solving the problem of minimizing the approximation error. Numerical experiments show the advantage of the developed scheme in comparison with other modifications of the Upwind Leapfrog scheme in the case when the convective transport prevails over the diffusion one. The proposed difference scheme solves transport problems more effectively than classical difference schemes in the case when the Péclet number falls in the range from 2 to 20. It follows that the considered difference scheme allows hydrodynamic problems to be solved in regions of complex shape effectively. [ABSTRACT FROM AUTHOR]

The sigmoid function is a widely used nonlinear activation function in neural networks. In this article, we present a modular approximation methodology for efficient fixed-point hardware implementation of the sigmoid function. Our design consists of three modules: piecewise linear (PWL) approximation as the initial solution, Taylor series approximation of the exponential function, and Newton–Raphson method-based approximation as the final solution. Its modularity enables the designer to flexibly choose the most appropriate approximation method for each module separately. Performance evaluation results indicate that our work strikes an appropriate balance among the objectives of approximation accuracy, hardware resource utilization, and performance. [ABSTRACT FROM AUTHOR]

The article proposes an adaptive robust controller with synchronization compensation for the Maglev wind turbine yaw system to achieve stable nacelle suspension and pitching suppression. First, two-degree-of-freedom suspension model is constructed with consideration of magnetic coupling and pitching interference, and then deduces a two-terminal suspension control model based on the inverse system. Second, a nominal suspension controller is designed based on the composite error; and then an adaptive synchronization controller is proposed to adjust coupling parameters based on reconstructed synchronous function; an adaptive tracking controller is used to online update dominant control input parameters. The robust controller is to compensate for the influence of the approximation errors during the parameters convergence process. It is proved that the proposed strategy can guarantee the globally uniformly ultimately bounded of all signals, and suspension tracking error and synchronization error can converge to the desired smaller neighborhood of zero. Finally, simulation and experimental results show that the system with the proposed strategy has the faster dynamic response, smaller steady error and synchronization error, and stronger robustness than other two strategies. [ABSTRACT FROM AUTHOR]

This article investigates the finite-time distributed approximate optimization problem of higher order multiagent systems, where the local cost functions are considered to be quadratic functions. This problem is solved via penalty-function-based method. First, by the penalty-function method, a global approximate cost function is constructed. Second, nonlinear distributed optimization algorithms are proposed for higher order multiagent systems by the tool of adding a power integrator technique. In the optimization algorithms design, the gradients of the approximate cost function are utilized. Under the proposed optimization algorithms, the agents approach the approximate optimal solution in finite time. Although there exist errors (they may be called approximation errors) between the approximate minimizers and the global accurate minimizer, the approximation errors can be regulated by penalty parameter and the relationship between the bound of the approximation errors and the penalty parameter is given explicitly. Furthermore, the proposed distributed approximate optimization algorithms are applied to the optimal rendezvous problem of wheeled multimobile robots, making the mobile robots achieve approximate optimization rendezvous in finite time. The effectiveness of the proposed distributed optimization algorithms and their applications to optimal rendezvous problem are validated by simulations. [ABSTRACT FROM AUTHOR]

Most of the existing classification systems assume that the data used is high-quality labeled. However, the labeling process in real-world may inevitably introduce corruptions into labels which can confuse the performances of classifiers. In this paper, based on Broad Learning System (BLS), we propose a novel label noise tolerant method to classify the pattern with corrupted labels. The standard BLS has shown promising efficiency and accuracy in general classification, but its learning process is prone to be affected by the noisy labels. Here, by detailed probabilistic analysis, we first give the reason for lacks of robustness in standard BLS. Then a maximum likelihood estimation-based objective function is derived for robust classification. In addition, a manifold regularization term is integrated to preserve the local geometry of data, which makes the model to be more robust and flexible to learn the output weights. Given some basic assumptions on the approximation errors, the obtained model can be transformed to a graph regularized reweighted BLS problem. The negative effects of noisy labels in data can be inhibited adaptively by assigning reasonable weights. Theoretical analysis and extensive experiments are provided to demonstrate the robustness and effectiveness of the proposed robust BLS model, especially for the case of large amounts of noisy labels. [ABSTRACT FROM AUTHOR]

The recent success of generative adversarial networks and variational learning suggests that training a classification network may work well in addressing the classical two-sample problem, which asks to differentiate two densities given finite samples from each one. Network-based methods have the computational advantage that the algorithm scales to large datasets. This paper considers using the classification logit function, which is provided by a trained classification neural network and evaluated on the testing set split of the two datasets, to compute a two-sample statistic. To analyze the approximation and estimation error of the logit function to differentiate near-manifold densities, we introduce a new result of near-manifold integral approximation by neural networks. We then show that the logit function provably differentiates two sub-exponential densities given that the network is sufficiently parametrized, and for on or near manifold densities, the needed network complexity is reduced to only scale with the intrinsic dimensionality. In experiments, the network logit test demonstrates better performance than previous network-based tests using classification accuracy, and also compares favorably to certain kernel maximum mean discrepancy tests on synthetic datasets and hand-written digit datasets. [ABSTRACT FROM AUTHOR]

The paper presents a model of structured objects in a grayscale or color image, described by means of optimal piecewise constant image approximations, which are characterized by the minimum possible approximation errors for a given number of pixel clusters, where the approximation error means the total squared error. An ambiguous image is described as a non-hierarchical structure but is represented as an ordered superposition of object hierarchies, each containing at least one optimal approximation in g0 = 1, 2,..., etc., colors. For the selected hierarchy of pixel clusters, the objects-of-interest are detected as the pixel clusters of optimal approximations, or as their parts, or unions. The paper develops the known idea in cluster analysis of the joint application of Ward's and K-means methods. At the same time, it is proposed to modernize each of these methods and supplement them with a third method of splitting/merging pixel clusters. This is useful for cluster analysis of big data described by a convex dependence of the optimal approximation error on the cluster number and also for adjustable object detection in digital image processing, using the optimal hierarchical pixel clustering, which is treated as an alternative to the modern informally defined "semantic" segmentation. [ABSTRACT FROM AUTHOR]

In this work we test the numerical behaviour of matrix-valued fields approximated by finite element subspaces of [ H 1 ] 3 × 3 , [ H (curl) ] 3 and H (sym Curl) for a linear abstract variational problem connected to the relaxed micromorphic model. The formulation of the corresponding finite elements is introduced, followed by numerical benchmarks and our conclusions. The relaxed micromorphic continuum model reduces the continuity assumptions of the classical micromorphic model by replacing the full gradient of the microdistortion in the free energy functional with the Curl. This results in a larger solution space for the microdistortion, namely [ H (curl) ] 3 in place of the classical [ H 1 ] 3 × 3 . The continuity conditions on the microdistortion can be further weakened by taking only the symmetric part of the Curl. As shown in recent works, the new appropriate space for the microdistortion is then H (sym Curl) . The newly introduced space gives rise to a new differential complex for the relaxed micromorphic continuum theory. [ABSTRACT FROM AUTHOR]

Holographic optical tweezers have unique non-physical contact and can manipulate and control single or multiple cells in a non-invasive way. In this paper, the dynamics model of the cells captured by the optical trap is analyzed, and a control system based on a novel self-organizing fuzzy cerebellar model neural network (NSOFCMNN) is proposed and applied to the cell manipulation control of holographic optical tweezers. This control system consists of a main controller using the NSOFCMNN with a new self-organization mechanism, a robust compensation controller, and a higher order sliding mode. It can accurately move the captured cells to the expected position through the optical trap generated by the holographic optical tweezers system. Both the layers and blocks of the proposed NSOFCMNN can be adjusted online according to the new self-organization mechanism. The compensation controller is used to eliminate the approximation errors. The higher order sliding surface can enhance the performance of controllers. The distances between cells are considered in order to further realize multi-cell cooperative control. In addition, the stability and convergence of the proposed NSOFCMNN are proved by the Lyapunov function, and the learning law is updated online by the gradient descent method. The simulation results show that the control system based on the proposed NSOFCMNN can effectively complete the cell manipulation task of optical tweezers and has better control performance than other neural network controllers. [ABSTRACT FROM AUTHOR]

This paper presents a new nonlinear projection based model reduction using convolutional Variational AutoEncoders (VAEs). This framework is applied on transient incompressible flows. The accuracy is obtained thanks to the expression of the velocity and pressure fields in a nonlinear manifold maximising the likelihood on pre-computed data in the offline stage. A confidence interval is obtained for each time instant thanks to the definition of the reduced dynamic coefficients as independent random variables for which the posterior probability given the offline data is known. The parameters of the nonlinear manifold are optimized as the ones of the decoder layers of an autoencoder. The parameters of the conditional posterior probability of the reduced coefficients are the ones of the encoder layers of the same autoencoder. The optimization of both sets of the encoder and the decoder parameters is obtained thanks to the application of a variational Bayesian method, leading to variational autoencoders. This Reduced Order Model (ROM) is not a regression model over the offline pre-computed data. The numerical resolution of the ROM is based on the Chorin projection method. We apply this new nonlinear projection-based Reduced Order Modeling (ROM) for a 2D Karman Vortex street flow and a 3D incompressible and unsteady flow in an aeronautical injection system. [ABSTRACT FROM AUTHOR]

In this paper a novel low-power approximate floating-point multiplier is presented. Since the mantissa computation is responsible for the largest part of the power consumption, we apply a novel approximation technique to mantissa multiplication, based on static segmentation. In our approach, the inputs of the mantissa multiplier are properly segmented so that a small inner multiplier can be used to calculate the output, with beneficial impact on power and area. To further improve performance, we introduce a novel segmentation-and-truncation approach which allows us to eliminate the shifter normally present at the output of the segmented multiplier. In addition, a simple compensation term for reducing approximation error is employed. The accuracy of the circuit can be tailored at the design time, by acting on a single parameter. The proposed approximate floating-point multiplier is compared with the state-of-the-art, showing good performance in terms of both precision and hardware saving. For single-precision floating-point format, the obtained NMED is in the range 10−5–7 × 10−7, while MRED is in the range 3 × 10−3–1.7 × 10−4. Synthesis results in 28 nm CMOS show area and power saving of up to 82% and 85%, respectively, compared to the exact floating-point multiplier. Image processing applications confirm the expectations, with results very close to the exact case. [ABSTRACT FROM AUTHOR]

The low-order approximation emerges as a promising method for capturing the dynamic behavior of Printed Circuit Heat Exchangers (PCHEs) in sCO 2 Recompression Closed Brayton Cycles (RCBCs). However, two primary challenges hinder widespread adoption: concerns regarding feasibility and the determination of key parameters, such as the time constant. To address these challenges, this study first affirms the feasibility of approximating PCHE's dynamic behaviors with a low-order system. In specific, the validity of first-order approximation on the dynamics of high-temperature recuperator (HTR) with mild non-linear property variations is initially assumed. However, a discrepancy is noticed under various operating conditions by employing the same time constant after being compared with the reference model (SCOPE). Then, a novel concept of the optimal time constant, τ op , is introduced to underscore the influence of operating conditions on parameter accuracy. Through extensive parameter sweeping, the impact of operating conditions on approximation error is evaluated from two perspectives. First, utilizing τ op effectively reduces approximation errors to acceptable levels. The mean squared error remains below 2 ° C 2 for mass flow rate fluctuations below 32.5 kg/s, and below 4 ° C 2 for temperature variations below 40 ° C. Second, the optimal time constant is significantly influenced by mass flow rate, notably affected by geometry configuration, and minimally influenced by hot inlet temperature. Additionally, this study provides insights for design optimization and control strategy selection, as the findings highlight the trade-off between response speed and thermal performance, and the relatively rapid response of sCO 2 -PCHEs to temperature control, compared with mass flow rate control. [ABSTRACT FROM AUTHOR]

This paper proposes a fault-tolerant control scheme for tracking and controlling hypersonic vehicles with unknown dynamics, actuator failures, and unmeasurable states. The approach involves using a radial-based neural network to approximate the unknown dynamics and reconstruct the entire system model. Additionally, a neural network state observer is proposed to estimate the unmeasurable state of the system. To address the impact of actuator faults, a nonlinear observer is designed to estimate and compensate for the approximation error of the neural network system and fault values. Furthermore, a prescribed performance function is introduced to ensure both transient and steady-state performance of the system. The bounded stability of the closed-loop system is demonstrated through Lyapunov stability analysis. • Unmeasurable states, actuator faults, and state constraints in hypersonic flight vehicle control are investigated. • The neural network state observer realizes the estimation of the unmeasurable state of the system under fault conditions. • The nonlinear observer can estimate perturbations, and the prescribed performance function guarantees transient and steady-state performance. [ABSTRACT FROM AUTHOR]

Signed distance fields (SDFs) have emerged as an alternative shape representation for real-time collision detection and lighting effects. Computing these for complex models can be expensive, so one popular approach is to prepare an approximation via sampling and interpolation. Then, these may be rendered using sphere marching, which gets close to the surface quickly, but needs several iterations to converge to it. In this paper, we propose an alternative that computes the intersection of a given ray and the surface analytically at a narrow band. This may be combined with other enhancements like having variable error for the approximation depending on the distance to the surface and skipping regions that do not contain the surface to accelerate the outer band ray traversal while reducing the required memory. To achieve smoother representations with minimal computational cost, we propose a method for computing surface intersections and normals from separate interpolants. We evaluate all these to find the optimal combination improving the rendering performance and memory consumption of these SDF approximations. [Display omitted] • Enhancements for rendering SDF piecewise approximations. • Solving narrow band intersections using analytical methods. • Reducing SDF approximations for improving required memory and outer band ray traversal. [ABSTRACT FROM AUTHOR]

Recently, Cheval-Garabédian et al. (2023) published a paper in Ore Geology Reviews entitled "The La Lucette Sb-Au-(W) vein-deposit (Armorican Massif, France): new time and genetic constraints to decipher the 310–295 Ma Sb-Au metallogenic peak in the Variscan Belt." Studies on Sb mineralisation are highly welcomed, especially in France, where there is a lack of updated data hindering a comprehensive understanding of this mineral system. The paper provides new mineralogical, fluid inclusion and geochronological data from historical collection samples of La Lucette mine, a mining site no more accessible since at least 40 years (Serment, 1978). These data, together with structural data from literature, lead the authors to build a metallogenic model related to dextral shear zone during the Carboniferous-Permian transition. The authors further argue for a single and unique 310–295 Ma Sb-Au metallogenic peak in the European Variscan Belt, a late Variscan hydrothermal event coeval with post-thickening tectonics (Munoz et al., 1992; Bouchot et al., 1997, 2005). This assertion is not correct. In this comment, we highlight that (i) several errors and approximations question the validity of their model, and (ii) the authors omit certain information that contradicts their metallogenic model. In our view, their data do not fully support the existence of a generalised and unique Late Carboniferous Sb-Au metallogenic peak in the French Variscan Belt. Furthermore, the idea defended by the authors that ore deposits occurred during dextral strike-slip tectonics that affected the area may better support a possible long-lived history during Palaeozoic times. [ABSTRACT FROM AUTHOR]

Hybrid quantum mechanics/molecular mechanics (QM/MM) models play a pivotal role in molecular simulations. These models provide a balance between accuracy, surpassing pure MM models, and computational efficiency, offering advantages over pure QM models. Adaptive approaches have been developed to further improve this balance by allowing on-the-fly selection of the QM and MM subsystems as necessary. We propose a novel and robust adaptive QM/MM method for practical material defect simulations. To ensure mathematical consistency with the QM reference model, we employ machine-learning interatomic potentials (MLIPs) as the MM models (Chen et al., 2022 and Grigorev et al., 2023). Our adaptive QM/MM method utilizes a residual-based error estimator that provides both upper and lower bounds for the approximation error, thus indicating its reliability and efficiency. Furthermore, we introduce a novel adaptive algorithm capable of anisotropically updating the QM/MM partitions. This update is based on the proposed residual-based error estimator and involves solving a free interface motion problem, which is efficiently achieved using the fast marching method. We demonstrate the robustness of our approach via numerical tests on a range of crystalline defects comprising edge dislocations, cracks and di-interstitials. [ABSTRACT FROM AUTHOR]

Reconstructed temperature field will provide critical input for monitoring the aerodynamic heating on the thermal protection system (TPS). In this paper, a novel data-driven method is proposed that allows real-time reconstruction of the 3D transient non-uniform temperature field in a TPS ceramic tile. The measured temperature is obtained by embedding sparse sensors at different layers of the tile. The spatial feature bases and time projection coefficients of the non-uniform temperature fields are extracted using the multi-parameter proper orthogonal decomposition (POD) method. The Fully-connected Long Short-Term Memory (FC-LSTM) networks are used to establish the prediction relationship between the measurements and the full-field temperatures. The problem to estimate the projection coefficients of the spatial feature bases is solved by optimizing the loss function using Adam algorithm. The numerical examples show an approximation error less than 5% and the reconstruction time-consuming less than 1 s in 3D temperature field reconstruction (TFR). The proposed reconstruction method is robust at different measurement noise levels and tested point arrangements. [ABSTRACT FROM AUTHOR]

To address ambiguity arising from linguistic decision judgments, Guo and Zhao introduced a model of the well-known Multi-Criteria Decision-Making (MCDM) method, Best-Worst Method (BWM), using fuzzy sets, known as Fuzzy Best–Worst Method (FBWM). Despite its versatility, FBWM has limitations, such as criteria-weights being independent of fuzzy comparison values' shape and reliance on approximate fuzzy arithmetic operations. In response to these limitations, we introduce a novel model of Fuzzy Best–Worst Method based on α -cut intervals. In this model, the calculation of optimal weights involves formulating an optimization problem that utilizes exact fuzzy arithmetic operations defined in terms of α -cut intervals. This approach aims to optimize the entire shape of fuzzy comparison values simultaneously. However, it turns out that, although we have proven the existence of optimal solution for this problem, solving it proves to be challenging. To resolve this issue, we approximate optimal weight set(s) using triangular fuzzy weight sets and assess the Error of Approximation (EA). The approximate weight set with the desired EA is then defuzzified to obtain a crisp weight set. In situations where there are multiple approximate weight sets with the same EA, interval-weights are calculated. Following that, we establish the necessary conditions for a Fuzzy Pairwise Comparison System (FPCS) to be consistent. Using these conditions, we calculate a lower bound of the Consistency Index (CI), providing a measure of the accuracy of weights. Finally, we discuss numerical examples and a real-world scenario, ranking of risk factors in supply chain 4.0, to validate the effectiveness of the proposed model. • Introducing an α -cut interval based model of the best–worst method. • Approximation of optimal weights and estimation of error of approximation. • Addressing non-uniqueness of optimal weights by calculating interval-weights. • Introduction of a consistency ratio to assess the reliability of weights. • The proposed model outperforms existing models in a thorough comparison. [ABSTRACT FROM AUTHOR]

An adaptive fuzzy sliding control (AFSMC) approach is adopted in this paper to address the problem of angular position control and vibration suppression of rotary flexible joint systems. AFSMC consists of fuzzy-based singleton control action and switching control law. By adjusting fuzzy parameters with the self-learning ability to discard system nonlinearities and uncertainties, singleton control based on fuzzy approximation theory can estimate the perfect control law of feedback linearization. To enhance robustness, an additional switching control law is incorporated to reduce the approximation error between the derived singleton control action and the perfect control law of feedback linearization. AFSMC's closed-loop stability will be demonstrated via sliding surface and Lyapunov function analysis of error function. In order to demonstrate the effectiveness of the AFSMC in tracking performance as well as its ability to respond to model uncertainties and external perturbations, simulations are carried out using Simulink and Matlab in order to demonstrate how well it adapts to these situations. Based on these results, it can be concluded that the AFSMC performs satisfactorily in terms of tracking. [ABSTRACT FROM AUTHOR]

Virtual element methods (VEM) provide great flexibility in solving numerical problems defined on arbitrarily shaped polygonal or polyhedral discretizations. In this paper, we develop a framework for two dimensional elastic problems defined on complex topology models using high-order virtual element methods from an engineering perspective. The VEM discrete formulations are detailedly derived following the rule used in standard FEM. An arbitrarily complex model is first embedded into a rectangular domain which is then discretized into a structured grid. The elements intersecting with the boundaries are further adaptively refined through a quad-tree refinement strategy controlled by a subdivision level or an approximation error. An optimization method is proposed to avoid the generation of tiny elements and two averaged schemes for stress recovery in post-processing are discussed. The behavior of the proposed VEM is thoroughly studied and the results are compared with analytical solutions and that obtained from FEM. The heavy burden placed on meshing complex CAD geometries is greatly alleviated and the convergence studies confirm the accuracy and convergence of the method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the concept of (s,P)-function and establish Hermite-Hadamard's inequalities for this type of functions. Also, we obtain some new Hermite-Hadamard type inequalities for functions whose first derivative in absolute value is (s,P)- function by using Hölder and power-mean integral inequalities. We also extend our initial results to functions of several variables. Next, we point out some applications of our results to give estimates for the approximation error of the integral function in the trapezoidal formula and for some inequalities related to special means of real numbers. [ABSTRACT FROM AUTHOR]