Nonlinear Approximation Method Research Articles

FINITE TEMPERATURE EFFECTS WITHIN SCALAR FIELD DARK MATTER MODEL

The distribution of dark matter in four low surface brightness spiral galaxies is studied using two models within the scalar field theory of dark matter, an alternative to the cold dark matter paradigm. The first model is a Bose-Einstein condensate, in which bosons occupy the ground state at zero temperature. The second model includes finite temperature corrections to the scalar field potential, which allows the introduction of excited states. A nonlinear least squares approximation method is used to determine the free parameters of the models, including scale radius, characteristic (central) density and total mass, based on observational data of rotation curves. Quantitative analysis shows the importance of considering finite temperatures at the galactic level. In addition, the two models are compared with results from widely used and accepted phenomenological dark matter profiles such as the isothermal sphere, Navarro-Frank-White and Burkert profiles. The reliability of each model was assessed based on the Bayesian information criterion of completeness. Statistical analysis provides meaningful interpretation of the choice of a particular profile. Ultimately, this study contributes to a better understanding of the distribution of dark matter in low surface brightness spiral galaxies by shedding light on the performance of scalar field models compared to traditional phenomenological profiles.

Space Extended Target Tracking Using Poisson Multi-Bernoulli Mixture Filtering with Nonlinear Measurements

The number of space targets in the near-Earth orbit has greatly increased, and space-based radar has the advantage of high resolution and high accuracy tracking to enhance the tracking efficiency of space extended targets (SET). We propose a gamma Gaussian inverse Wishart based on a matched linearization-Poisson multi-Bernoulli mixture (GGIWML-PMBM) filter to estimate the motion state and shape size of the SET. For the random matrix that can only describe the distribution under which the measurements are linear, the measurements of the spatial target are linearized using matched linearization, and the extended information is preserved in the second-order central moments. The transfer density and likelihood function of GGIW are nonlinear for Poisson measurements with nonlinear Gaussian spatial distributions, and the single-target densities and normalizing constants of the PMBM filter are not closed form. The prediction and update of PMBM include Gaussian-weighted integral calculation, for which different nonlinear approximation methods are used to calculate the weighted integral and derive the closed form of GGIWML-PMBM. Finally, the simulation scenarios of low-orbit single-radar sensor tracking near SET and group SET are established, and the results show that the tracking accuracy can reach 2.6 m for near SET and 6.6 m in group SET.

Augmenting Basis Sets by Normalizing Flows

AbstractApproximating functions by a linear span of truncated basis sets is a standard procedure for the numerical solution of differential and integral equations. Commonly used concepts of approximation methods are well‐posed and convergent, by provable approximation orders. On the down side, however, these methods often suffer from the curse of dimensionality, which limits their approximation behavior, especially in situations of highly oscillatory target functions. Nonlinear approximation methods, such as neural networks, were shown to be very efficient in approximating high‐dimensional functions. We investigate nonlinear approximation methods that are constructed by composing standard basis sets with normalizing flows. Such models yield richer approximation spaces while maintaining the density properties of the initial basis set, as we show. Simulations to approximate eigenfunctions of a perturbed quantum harmonic oscillator indicate convergence with respect to the size of the basis set.

Error estimate of the cell-centered nonlinear positivity-preserving two-point flux approximation schemes

We present the error estimate of nonlinear positivity-preserving finite volume schemes for diffusion problems. These schemes are also named by nonlinear two-point flux approximation (NTPFA) methods. Due to their conservation and monotonicity, such schemes have attracted wide attention. Under suitable regularity assumptions on the exact solution, we prove a discrete H1 error estimate through the analysis of consistent errors. Some numerical examples are presented to demonstrate the theoretical results.

Laser-modified nanocrystalline NiMoO4 as an electrode material in hybrid supercapacitors

The nanocrystalline NiMoO4 obtained as a result of hydrothermal synthesis was exposed to laser radiation with a pulse energy of 70 mJ/cm2 for 5 minutes. The phase composition and size of crystallites of the triclinic structure of NiMoO4 were determined by X-ray analysis. The average crystallite size was 18 nm for laser-irradiated nickel molybdate. Impedance analysis was used to analyze the temperature dependence of the electrical conductivity of laser-modified NiMoO4. The frequency index of the power law, determined by the nonlinear approximation method, was 0.5-0.67, which corresponds to the hopping mechanism of charge carriers. The electrochemical behavior of NiMoO4 was studied using cyclic voltammetry and galvanostatic charge/discharge testing. The laser-irradiated NiMoO4 reaches a specific capacitance of 553 F/g at a scan rate of 1 mV/s. The hybrid electrochemical system based on electrodes of modified NiMoO4 and carbon material provides high Coulombic efficiency (95%) for a significant number of charge/discharge cycles.

An Optimized Method for Nonlinear Function Approximation Based on Multiplierless Piecewise Linear Approximation

In this paper, we propose an optimized method for nonlinear function approximation based on multiplierless piecewise linear approximation computation (ML-PLAC), which we call OML-PLAC. OML-PLAC finds the minimum number of segments with the predefined fractional bit width of input/output, maximum number of shift-and-add operations, user-defined widths of intermediate data, and maximum absolute error (MAE). In addition, OML-PLAC minimizes the actual MAE as much as possible by iterating. As a result, under the condition of satisfying the maximum number of segments, the MAE can be minimized. Tree-cascaded 2-input and 3-input multiplexers are used to replace multi-input multiplexers in hardware architecture as well, reducing the depth of the critical path. The optimized method is applied to logarithmic, antilogarithmic, hyperbolic tangent, sigmoid and softsign functions. The results of the implementation prove that OML-PLAC has better performance than the current state-of-the-art method.

Reward-Reinforced Generative Adversarial Networks for Multi-Agent Systems

Multi-agent systems deliver highly resilient and adaptable solutions for common problems in telecommunications, aerospace, and industrial robotics. However, achieving an optimal global goal remains a persistent obstacle for collaborative multi-agent systems, where learning affects the behaviour of more than one agent. A number of nonlinear function approximation methods have been proposed for solving the Bellman equation, which describe a recursive format of an optimal policy. However, how to leverage the value distribution based on reinforcement learning, and how to improve the efficiency and efficacy of such systems remain a challenge. In this work, we developed a reward-reinforced generative adversarial network to represent the distribution of the value function, replacing the approximation of Bellman updates. We demonstrated our method is resilient and outperforms other conventional reinforcement learning methods. This method is also applied to a practical case study: maximising the number of user connections to autonomous airborne base stations in a mobile communication network. Our method maximises the data likelihood using a cost function under which agents have optimal learned behaviours. This reward-reinforced generative adversarial network can be used as a generic framework for multi-agent learning at the system level.

Kernel least logarithm absolute difference algorithm based on P-norm

The kernel adaptive filtering is an efficient and nonlinear approximation method which is developed in reproducing kernel Hilbert space (RKHS). Kernel function is used to map input data from original space to RKHS space, thus solving nonlinear problems is efficient.Impulse noise and non-Gaussian noise exist in the real application environment, and the probability density distribution of these noise characteristics shows a relatively heavy trailing phenomenon in the statistical sense. α stable distribution can be used to model this kind of non-Gaussian noise well. The kernel least mean square(KLMS) algorithms usually perform well in Gaussian noise, but the mean square error criterion only captures the second-order statistics of the error signal, this type of algorithm is very sensitive to outliers, in other words, it lacks robustness in α stable distribution noise. The kernel least logarithm absolute difference(KLLAD) algorithm can deal with outliers well, but it has the problem of slow convergence.In order to further improve the convergence speed of nonlinear adaptive filtering algorithm in α stable distributed noise background, a new kernel least logarithm absolute difference algorithm based on p-norm (P-KLLAD) is presented in this paper. The algorithm combining least logarithm absolute difference algorithm and p norm, on the one hand, the least logarithm difference criteria is ensure the algorithm to have good robustness in α stable distribution noise environment, and on the other hand, add p norm on the absolute value of error.The steepness of the cost function is controlled by p norm and a posititive constant ɑ to improve the convergence speed of the algorithm.The computer simulation results of Mackey-Glass chaotic time series prediction and nonlinear system identification show that this algorithm improves the convergence speed with good robustness,and the convergence speed and robustness better than the kernel least mean square algorithm,the kernel fractional lower power algorithm, the kernel least logarithm absolute difference algorithm and the kernel least mean p-norm algorithm.

A New Nonlinear Two-Point Flux Approximation Method for Solving the Anisotropic Diffusion Equation with Reduced Violations of the Discrete Maximum/Minimum Principle

Summary A class of monotone cell-centered nonlinear finite-volume methods has been proposed in the past decade to solve the anisotropic diffusion equation. The nonlinear two-point flux approximation (TPFA) (NTPFA) method preserves the nonnegativity of the solution values but can violate the discrete maximum/minimum principle (DMP). To enforce DMP, the nonlinear multipoint flux approximation (NMPFA) method ought to be used. In this work, we propose a novel NTPFA method that can reduce the severity of DMP violations significantly compared with the standard NTPFA method. The new formulation uses conormal decomposition for the construction of the one-sided fluxes. To define the unique flux through a connection between two cells, we choose a convex combination of the two one-sided fluxes such that the absolute differences of the magnitudes of the two transmissibility terms associated with the two neighboring cells are minimized, thus bringing the discrete coefficient matrix closer to having the zero row-sum property. Numerical experiments are conducted to test the performance of the new NTPFA method. The results demonstrate that the new scheme has comparable convergence order for both the solution and the flux compared with the standard NTPFA method or the classical multi-point flux approximation (MPFA-O) method. Moreover, the new NTPFA formulation shows marked improvements over the standard NTPFA in terms of reducing DMP violations. However, depending on the specific problem configuration, our new NTPFA formulation can lead to a system of nonlinear equations that is more difficult to solve.

Fault tolerant robust control with transients for over-actuated nonlinear systems

For the problem of robust fault tolerant control of over-actuated nonlinear systems, a new robust synthesis method is investigated. The nonlinear system is modeled via a T–S fuzzy system approach with actuators layout. Based on a designed non-parallel distributed compensation form controller, the closed loop system is obtained. Then a more generalized performance index is introduced by involving the system transients and actuator faults. With a newly proposed nonlinear approximation method, a series of linear matrix inequalities which can make the system robust stable are listed. Finally, based on a dynamic positioning vessel control task, the validity of the proposed method is verified.

Randomized linear algebra for model reduction—part II: minimal residual methods and dictionary-based approximation

A methodology for using random sketching in the context of model order reduction for high-dimensional parameter-dependent systems of equations was introduced in [Balabanov and Nouy 2019, Part I]. Following this framework, we here construct a reduced model from a small, efficiently computable random object called a sketch of a reduced model, using minimal residual methods. We introduce a sketched version of the minimal residual based projection as well as a novel nonlinear approximation method, where for each parameter value, the solution is approximated by minimal residual projection onto a subspace spanned by several vectors picked (online) from a dictionary of candidate basis vectors. It is shown that random sketching technique can improve not only efficiency but also numerical stability. A rigorous analysis of the conditions on the random sketch required to obtain a given accuracy is presented. These conditions may be ensured a priori with high probability by considering for the sketching matrix an oblivious embedding of sufficiently large size. Furthermore, a simple and reliable procedure for a posteriori verification of the quality of the sketch is provided. This approach can be used for certification of the approximation as well as for adaptive selection of the size of the random sketching matrix.

Distributed adaptive output feedback containment control for time-delay nonlinear multiagent systems

This paper addresses the distributed adaptive containment control problem for uncertain nonlinear multiagent systems (MASs) with time delays and unmodeled dynamics under fixed directed graph. With outputs information of neighbors, the local reference generator of each agent is constructed to generate corresponding virtual tracking trajectory belonging to the convex hull spanned by dynamic leaders. Based on the reference generator, we utilize the nonlinear function approximation method and dynamic gain scaling technique to design the linear-like distributed adaptive output feedback controller. The proposed scheme can significantly relax conditions on nonlinearities with time delays, and greatly simplify the controller design procedure. Using a novel lemma, it is proved that the containment errors can converge to an arbitrarily small compact set and the resulting closed-loop system is semi-globally stable. In addition, if uncertain nonlinearities can be modeled as the product of unknown constant parameters and known functions, the global containment control can be achieved. Finally, simulation results are given to illustrate the effectiveness of proposed method.

Non-Canonic Representation of the Random Process in Tasks of Simulating Seismic Impacts for Calculating Buildings and Structures

The seismic impact model is analyzed as a combination of simple functions with random parameters. A random process simulating seismic impact is defined by a simple non-canonical representation. A method for generating artificial accelerograms of local earthquakes is proposed, based on the representation of the correlation function as the sum of cosine-exponential terms. The approximation parameters of the correlation function were determined earlier based on the least squares method using the MATLAB package using nonlinear approximation methods. As a quality criterion, the initial and generated spectral curves coincide. Samples of random process realizations from 20 to 10,000 realizations (artificial accelerograms) are used. Algorithms are simply implemented in modern computer mathematics systems MATLAB, SCILAB. The mathematical model of the accelerogram of the earthquake in El Centro, 1940 is also considered. The area of application of the algorithms is calculations using Eurocode 8 and ISO 2394, ISO 8930 standards, including using probabilistic methods, determining the reliability of buildings and structures based on the statistical test method.

ATA: Attentional Non-Linear Activation Function Approximation for VLSI-Based Neural Networks

In this letter, we present an attentional non-linear activation function approximation method called ATA for VLSI-based neural networks. Unlike other approximation methods that pursue the low hardware resources with a high recognition accuracy loss, the ATA utilizes the pixel attention to focus on the important features to keep the recognition accuracy and reduce resource cost. Specifically, attention applied in the activation function is realized by the approximated activation functions with different fitting errors for VLSI-based neural networks. The important features are highlighted by the piecewise linear function and improved look-up table with low fitting error, while the trivial features are ignored with the large fitting error. Experimental results demonstrate that the ATA outperforms other state-of-the-art approximation methods in recognition accuracy, power and area.

Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces

We consider the problem of model reduction of parametrized PDEs where the goal is to approximate any function belonging to the set of solutions at a reduced computational cost. For this, the bottom line of most strategies has so far been based on the approximation of the solution set by linear spaces on Hilbert or Banach spaces. This approach can be expected to be successful only when the Kolmogorov width of the set decays fast. While this is the case on certain parabolic or elliptic problems, most transport-dominated problems are expected to present a slow decaying width and require to study nonlinear approximation methods. In this work, we propose to address the reduction problem from the perspective of general metric spaces with a suitably defined notion of distance. We develop and compare two different approaches, one based on barycenters and another one using tangent spaces when the metric space has an additional Riemannian structure. Since the notion of linear vectorial spaces does not exist in general metric spaces, both approaches result in nonlinear approximation methods. We give theoretical and numerical evidence of their efficiency to reduce complexity for one-dimensional conservative PDEs where the underlying metric space can be chosen to be theL2-Wasserstein space.

BSDEs driven by G-Brownian motion with time-varying Lipschitz condition

The present paper is devoted to the study of backward stochastic differential equations driven by G-Brownian motion (G-BSDEs) with time-varying Lipschitz condition. With the help of nonlinear stochastic analysis technique and approximation method, we prove that the G-BSDEs admit a unique solution on finite or infinite time horizon. Furthermore, we obtain the corresponding comparison theorem.

Hybrid consensus‐based distributed pseudomeasurement information filter for small UAVs tracking in wireless sensor network

In this study, a novel state estimation method is proposed for small unmanned aerial vehicles (UAVs) tracking in passive sensors (PSs) wireless sensor network (WSN), where the measurement equations are highly non-linear. Different from many non-linear function approximation methods, the non-linear measurement functions are recast into a linear form with respect to UAV states. The centralised 3D pseudomeasurement information filter (PMIF) is derived by embedding the linear form measurements into conventional information filtering schemes. For distributed state estimation in a not fully connected WSN, a hybrid consensus-based distributed PMF (DHCPMIF) is developed in which each PS independently calculates local contribution by using its own pseudomeasurement. To guarantee bounded estimation error as close as possible to that obtained by the optimal centralised estimator, the hybrid consensus method is developed by combining consensus on measurement and consensus on information method, which retains the advantages of both approaches, thus providing estimation accuracy improvement when the number of consensus iterations is moderate. The effectiveness of the proposed distributed filter is verified via two simulation examples involving tracking a UAV using four and eight PSs, respectively.

A success history-based adaptive differential evolution optimized support vector regression for estimating plastic viscosity of fresh concrete

Plastic viscosity is an important parameter of fresh concrete mixes. This research investigates a machine learning-based method for constructing a functional mapping between concrete mix properties and the plastic viscosity. The investigated machine learning method relies on the support vector regression (SVR) which is a robust method for nonlinear and multivariate function approximation. Moreover, the history-based adaptive differential evolution with linear population size reduction (L-SHADE) is employed to optimize the SVR model construction phase. Thus, the proposed method, named L-SHADE-SVR, is an integration of machine learning and metaheuristic optimization. To train and verify the L-SHADE-SVR model, a dataset consisting of 142 experimental tests was collected. Experimental results with repetitive phases of model training and testing reveal that the newly constructed model is capable of delivering highly accurate estimation of the plastic viscosity with mean absolute percentage error of 12% and coefficient of determination of 0.82. These outcomes are superior compared to the employed benchmark methods including artificial neural network, multivariate adaptive regression spline, and sequential piecewise multiple linear regression. Therefore, the L-SHADE-SVR model is a promising tool to assist construction engineers in estimating the plastic viscosity of fresh concrete mixes.

The study of the dependence of spring crops yield on the abiotic environmental factors using nonlinear interpolation

The article discusses the effect of abiotic environmental factors on crop yields. Abiotic factors cannot be changed by human activity; therefore, considering their impact on yield is very important for predicting crop yields. The article analyzes the dependence of spring wheat yield on such abiotic environmental factors as average ten-day air temperatures during the growing season, ten-day amounts of precipitation during the growing season and the Selyaninov hydrothermal coefficient during the growing season. To identify dependencies, the authors applied a non-linear approximation method using a third-degree polynomial dependence with one variable. The study was conducted during the most important growing season of spring wheat, in May and June. The authors show the equations of the third-degree polynomial dependences and graphs, as well as their graphic interpretation. Based on the obtained graphs, the authors have analyzed favorable intervals of abiotic factors for the formation of high yields of spring wheat. The authors indicate the most optimal temperature and humidity conditions for each ten days of May and June. These dependences are suitable not only for spring wheat but also for other spring grain crops, such as oats and barley.

The Framing of a Problem through Gaming (Mathematics-STEM) Subject Content Areas

According to Demeter (2015) over the past fourty years, scientists, engineers, philosophers, and psychologists have been concerned with the infinite number of social, economic, and physical issues that exist and that cannot be dealt with in isolation. As agruged by Wong, Luo, and Fong (2019) it has been found through experiments and piloting techniques that the existing traditional methods and knowledge is not suited into coping and addressing the unique and sophisticated situations. In the process of searching and coming up with techniques such the use of Bernoulli’s principle for aerodynamics and aviation, perturbation techniques, and the nonlinear approximation methods, it was discovered that physics, chemistry, and mathematics brings out a limited area of interest and therefore can only offer solutions to problems in narrow perspectives. For example, the aforementioned disciplines majorly concentrated on a particular subject of interest and use limited principles towards solving the identified problem, that is to say, a significant number of assumptions are brought into play in order to help in arriving at the intended destinations. Notably, the techniques applied in solving scientific problems do not regard competing frameworks and this contributed to their limited nature in solving social and physical problems. As suggested by Leaver,Willson, and Torres (2016) it is against this backdrop that, framing of a problem through gaming techniques in subject content areas is being encouraged since it involves competing frameworks and consequently works to reduce the number of assumptions plunged in the process of solving mathematical problem. Gaming techniques entails the combination of various techniques that can deal with adversities, uncertainty, unique, and sophisticated situations in the field of science. Fundamentally, framing of a problem through gaming tricks provides an advantage of combining the social-human with economic, technological, and physical domains. The outcome associated with the gaming technique is scientific improvements and increased applications of the available knowledge and information (Wong, Luo, and Fong, 2019).