Search Results (32)

This article presents an alternative variant of motion planning techniques for autonomous vehicles (AVs) centered on an inverse approach that concurrently optimizes both trajectory and speed. This method emphasizes searching for a trajectory and distributing its speed within a single road segment, regarded as a final element. The references for the road lanes are represented by splines that interpolate the path length, derivative, and curvature using Cartesian coordinates. This approach enables the determination of parameters at the final node of the road segment while varying the reference length. Instead of directly modeling the trajectory and velocity, the second derivatives of curvature and speed are modeled to ensure the continuity of all kinematic parameters, including jerk, at the nodes. A specialized inverse numerical integration procedure based on Gaussian quadrature has been adapted to reproduce the trajectory, speed, and other key parameters, which can be referenced during the motion tracking phase. The method emphasizes incorporating kinematic, dynamic, and physical restrictions into a set of nonlinear constraints that are part of the optimization procedure based on sequential quadratic optimization. The objective function allows for variation in multiple parameters, such as speed, longitudinal and lateral jerks, final time, final angular position, final lateral offset, and distances to obstacles. Additionally, several motion planning variants are calculated simultaneously based on the current vehicle position and the number of lanes available. Graphs depicting trajectories, speeds, accelerations, jerks, and other relevant parameters are presented based on the simulation results. Finally, this article evaluates the efficiency, speed, and quality of the predictions generated by the proposed method. The main quantitative assessment of the results may be associated with computing performance, which corresponds to time costs of 0.5–2.4 s for an average power notebook, depending on optimization settings, desired accuracy, and initial conditions. Full article

In this paper, we investigate efficient numerical methods for highly oscillatory integrals with Bessel function kernels over finite and infinite domains. Initially, we decompose the two types of integrals into the sum of two integrals. For one of these integrals, we reformulate the Bessel function $J_{\nu }(z)$ as a linear combination of the modified Bessel function of the second kind $K_{\nu }(z)$, subsequently transforming it into a line integral over an infinite interval on the complex plane. This transformation allows for efficient approximation using the Cauchy residue theorem and appropriate Gaussian quadrature rules. For the other integral, we achieve efficient computation by integrating special functions with Gaussian quadrature rules. Furthermore, we conduct an error analysis of the proposed methods and validate their effectiveness through numerical experiments. The proposed methods are applicable for any real number $\nu$ and require only the first $\lfloor \nu \rfloor$ derivatives of f at 0, rendering them more efficient than existing methods that typically necessitate higher-order derivatives. Full article

This paper considers a numerical method for solving the stochastic semilinear subdiffusion equation which is driven by integrated fractional Gaussian noise and the Hurst parameter $H\in (1/2,1)$. The finite element method is employed for spatial discretization, while the L1 scheme and Lubich’s first-order convolution quadrature formula are used to approximate the Caputo time-fractional derivative of order $\alpha \in (0,1)$ and the Riemann–Liouville time-fractional integral of order $\gamma \in (0,1)$, respectively. Using the semigroup approach, we establish the temporal and spatial regularity of the mild solution to the problem. The fully discrete solution is expressed as a convolution of a piecewise constant function with the inverse Laplace transform of a resolvent-related function. Based on the Laplace transform method and resolvent estimates, we prove that the proposed numerical scheme has the optimal convergence order $O({\tau }^{min\{H+\alpha +\gamma -1-\epsilon ,\alpha \}}),\epsilon >0$. Numerical experiments are presented to validate these theoretical convergence orders and demonstrate the effectiveness of this method. Full article

We investigate the underwater propagation of multi-Gaussian correlated asymmetric Bessel beam with partial coherence in the condition of quadrature amplitude modulation. The oceanic turbulence optical power spectrum is used to characterize turbulence effects under variable temperature and salinity. Based on the derivation of orbital angular momentum mode distribution, the theoretical model of bit error rate (BER) is constructed. Numerical analyses show that the low-temperature oceanic channel is more beneficial to BER reduction than the low-salinity channel. Due to the better resistance to turbulence, low-order modulation is superior in BER performance. As for beam optimization, the increments in wavelength and source coherence width, or the decrements of topological charge and asymmetry factor, help to obtain a lower BER. The research is instructive for the construction of underwater transmission links based on vortex beams. Full article

We analyze the error estimates of a fully discrete scheme for solving a semilinear stochastic subdiffusion problem driven by integrated fractional Gaussian noise with a Hurst parameter $H\in (0,1)$. The covariance operator Q of the stochastic fractional Wiener process satisfies $\parallel A^{-\rho }{Q}^{1/2}{\parallel }_{HS}<\infty$ for some $\rho \in [0,1)$, where ${\parallel ·\parallel }_{HS}$ denotes the Hilbert–Schmidt norm. The Caputo fractional derivative and Riemann–Liouville fractional integral are approximated using Lubich’s convolution quadrature formulas, while the noise is discretized via the Euler method. For the spatial derivative, we use the spectral Galerkin method. The approximate solution of the fully discrete scheme is represented as a convolution between a piecewise constant function and the inverse Laplace transform of a resolvent-related function. By using this convolution-based representation and applying the Burkholder–Davis–Gundy inequality for fractional Gaussian noise, we derive the optimal convergence rates for the proposed fully discrete scheme. Numerical experiments confirm that the computed results are consistent with the theoretical findings. Full article

As the era of sixth-generation (6G) communications approaches, there will be an unprecedented increase in the number of wireless internet-connected devices and a sharp rise in mobile data traffic. Faced with the scarcity of spectrum resources in traditional communication networks and challenges such as rapidly establishing communications after disasters, this study leverages unmanned aerial vehicles (UAVs) to promote an integrated multi-hop communication system combining free-space optical (FSO) communication, terahertz (THz) technology, and intelligent reflecting surface (IRS). This innovative amalgamation capitalizes on the flexibility of UAVs, the deployability of IRS, and the complementary strengths of FSO and THz communications. We have developed a comprehensive channel model that includes the effects of atmospheric turbulence, attenuation, pointing errors, and angle-of-arrival (AOA) fluctuations. Furthermore, we have derived probability density functions (PDFs) and cumulative distribution functions (CDFs) for various switching techniques. Employing advanced methods such as Gaussian–Laguerre quadrature and the central limit theorem (CLT), we have calculated key performance indicators including the average outage probability, bit error rate (BER), and channel capacity. The numerical results demonstrate that IRS significantly enhances the performance of the UAV-based hybrid FSO/THz system. The research indicates that optimizing the number of IRS elements can substantially increase throughput and reliability while minimizing switching costs. Additionally, the multi-hop approach specifically addresses the line-of-sight (LoS) dependency limitations inherent in FSO and THz systems by utilizing UAVs as dynamic relay points. This strategy effectively bridges longer distances, overcoming physical and atmospheric obstacles, and ensures stable communication links even under adverse conditions. This study underscores that the enhanced multi-hop FSO/THz link is highly effective for emergency communications after disasters, addressing the challenge of scarce spectrum resources. By strategically deploying UAVs as relay points in a multi-hop configuration, the system achieves greater flexibility and resilience, making it highly suitable for critical communication scenarios where traditional networks might fail. Full article

Hamiltonian Neural Networks (HNNs) provide structure-preserving learning of Hamiltonian systems. In this paper, we extend HNNs to structure-preserving inversion of stochastic Hamiltonian systems (SHSs) from observational data. We propose the quadrature-based models according to the integral form of the SHSs’ solutions, where we denoise the loss-by-moment calculations of the solutions. The integral pattern of the models transforms the source of the essential learning error from the discrepancy between the modified Hamiltonian and the true Hamiltonian in the classical HNN models into that between the integrals and their quadrature approximations. This transforms the challenging task of deriving the relation between the modified and the true Hamiltonians from the (stochastic) Hamilton–Jacobi PDEs, into the one that only requires invoking results from the numerical quadrature theory. Meanwhile, denoising via moments calculations gives a simpler data fitting method than, e.g., via probability density fitting, which may imply better generalization ability in certain circumstances. Numerical experiments validate the proposed learning strategy on several concrete Hamiltonian systems. The experimental results show that both the learned Hamiltonian function and the predicted solution of our quadrature-based model are more accurate than that of the corrected symplectic HNN method on a harmonic oscillator, and the three-point Gaussian quadrature-based model produces higher accuracy in long-time prediction than the Kramers–Moyal method and the numerics-informed likelihood method on the stochastic Kubo oscillator as well as other two stochastic systems with non-polynomial Hamiltonian functions. Moreover, the Hamiltonian learning error ${\epsilon }_{\mathcal{H}}$ arising from the Gaussian quadrature-based model is lower than that from Simpson’s quadrature-based model. These demonstrate the superiority of our approach in learning accuracy and long-time prediction ability compared to certain existing methods and exhibit its potential to improve learning accuracy via applying precise quadrature formulae. Full article

When the time-dependent boundary element method, also termed the pseudo-initial condition method, is employed for solving transient heat conduction problems, the numerical evaluation of domain integrals is necessitated. Consequently, the accurate calculation of the domain integrals is of crucial importance for analyzing transient heat conduction. However, as the time step decreases progressively and approaches zero, the integrand of the domain integrals is close to singular, resulting in large errors when employing standard Gaussian quadrature directly. To solve the problem and further improve the calculation accuracy of the domain integrals, an (α, β) distance transformation is presented. Distance transformation is a simple and efficient method for eliminating near-singularity, typically applied to nearly singular integrals. Firstly, the (α, β) coordinate transformation is introduced. Then, a new distance transformation for the domain integrals is constructed by replacing the shortest distance with the time step. With the new method, the integrand of the domain integrals is substantially smoothed, and the singularity arising from small time steps in the domain integrals is effectively eliminated. Thus, more accurate results can be obtained by the (α, β) distance transformation. Different sizes of time steps, positions of source point, and shapes of integration elements are considered in numerical examples. Comparative studies of the numerical results for the domain integrals using various methods demonstrate that higher accuracy and efficiency are achieved by the proposed method. Full article

The paper presents a technique of motion planning for autonomous vehicles (AV) based on simultaneous trajectory and speed optimization. The method includes representing the trajectory by a finite element (FE), determining trajectory parameters in Frenet coordinates, composing a model of vehicle kinematics, defining optimization criteria and a cost function, forming a set of constraints, and adapting the Gaussian N-point scheme for quadrature numerical integration. The study also defines a set of minimum optimization parameters sufficient for making motion predictions with smooth functions of the trajectory and speed. For this, piecewise functions with three degrees of freedom (DOF) in FE’s nodes are implemented. Therefore, the high differentiability of the trajectory and speed functions is ensured to obtain motion criteria such as linear and angular speeds, acceleration, and jerks used in the cost function and constraints. To form the AV roadway position, the Frenet coordinate system and two variable parameters are used: the reference path length and the lateral displacement perpendicular to reference line’s tangent. The trajectory shape, then, depends only on the final position of the AV’s mass center and the final reference’s curvature. The method uses geometric, kinematic, dynamic, and physical constraints, some of which are related to hard restrictions and some to soft restrictions. The planning technique involves parallel forecasting for several variants of the AV maneuver followed by selecting the one corresponding to a specified criterion. The sequential quadratic programming (SQP) technique is used to find the optimal solution. Graphs of trajectories, speeds, accelerations, jerks, and other parameters are presented based on the simulation results. Finally, the efficiency, rapidity, and prognosis quality are evaluated. Full article

In this work, our primary focus is on the numerical computation of highly oscillatory integrals involving the Airy function. Specifically, we address integrals of the form ${\int }_0^b{x}^{\alpha }f\left(x\right)\mathrm{Ai}(-\omega x)dx$ over a finite or semi-infinite interval, where the integrand exhibits rapid oscillations when $\omega \gg 1$. The inherent high oscillation and algebraic singularity of the integrand make traditional quadrature rules impractical. In view of this, we strategically partition the interval into two segments: $[0,1]$ and $[1,b]$. For integrals over the interval $[0,1]$, we introduce a Filon-type method based on a two-point Taylor expansion. In contrast, for integrals over $[1,b]$, we transform the Airy function into the first kind of Bessel function. By applying Cauchy’s integration theorem, the integral is then reformulated into several non-oscillatory and exponentially decaying integrals over $[0,+\infty )$, which can be accurately approximated by the generalized Gaussian quadrature rule. The proposed methods are accompanied by rigorous error analyses to establish their reliability. Finally, we present a series of numerical examples that not only validate the theoretical results but also showcase the accuracy and efficacy of the proposed method. Full article

The study aims to improve the technique of motion planning for all-wheel drive (AWD) autonomous vehicles (AVs) by including torque vectoring (TV) models and extended physical constraints. Four schemes for realizing the TV drive were considered: with braking internal wheels, using a rear-axle sport differential (SD), with braking front internal wheel and rear-axle SD, and with SDs on both axles. The mathematical model combines 2.5D vehicle dynamics model and a simplified drivetrain model with the self-locking central differential. The inverse approach implies optimizing the distribution of kinematic parameters by imposing a set of constraints. The optimization procedure uses the sequential quadratic programming (SQP) technique for the nonlinear constrained minimization. The Gaussian N-point quadrature scheme provides numerical integration. The distribution of control parameters (torque, braking moments, SDs’ friction moment) is performed by evaluating linear and nonlinear algebraic equations inside of optimization. The technique proposed demonstrates an essential difference between forecasts built with a pure kinematic model and those considering the vehicle’s drive/control features. Therefore, this approach contributes to the predictive accuracy and widening model properties by increasing the number of references, including for actuators and mechanisms. Full article

In this paper, we present a new highly efficient numerical algorithm for nonlinear variable-order space fractional reaction–diffusion equations. The algorithm is based on a new method developed by using the Gaussian quadrature pole rational approximation. A splitting technique is used to address the issues related to computational efficiency and the stability of the method. Two linear systems need to be solved using the same real-valued discretization matrix. The stability and convergence of the method are discussed analytically and demonstrated through numerical experiments by solving test problems from the literature. The variable-order diffusion effects on the solution profiles are illustrated through graphs. Finally, numerical experiments demonstrate the superiority of the presented method in terms of computational efficiency, accuracy, and reliability. Full article

The interaction between solitary waves and underwater barriers is investigated using our in-house code, entitled VPM (volume-average/point-value multi-moment)–THINC/QQ (THINC method with quadratic surface representation and Gaussian quadrature)-coupled model. The stability and accuracy of the proposed model are validated by comparing the numerical results with those of the well established two-phase flow solver interFoam. All the results indicate that the presented coupled model has the advantage of high fidelity in simulating solitary wave propagation. Subsequently, solitary waves passing over a single underwater barrier are simulated by the present model. Numerical results are compared with experimental results in terms of the free surface elevation, velocity profile, vorticity field, and wave forces. Great agreements are obtained. In the end, the interactions between solitary waves and double underwater barriers are investigated numerically. The results reveal that the reflection coefficient increases first, and then decreases, with the increasing space between the two barriers. For cases with different wave heights, the transmission coefficient decreases monotonically, and the dissipation coefficient is opposed to the transmission coefficient. Full article

In this work, we present a comparative analysis of different numerical methods to obtain the focusing properties of the zone plates based on Fibonacci and Cantor sequences. The Fresnel approximation was solved numerically in order to obtain the axial irradiance provided by these diffractive lenses. Two different methods were applied. The first one is based on numerical integration, specifically the Simpson integration method and the two-dimensional Gaussian quadrature. The second consisted in the implementation of the Fast Fourier Transform in both one and two dimensions. The axial irradiance of the lenses, the relative error with respect to the analytical solution, and the calculation time required by each method are analyzed and compared. From this analysis it was concluded that the Gauss method presents the best balance between accuracy and computation time. This analysis could be useful to decide the most convenient numerical method to be used for the study of more complex diffractive structures. Full article

Twisted light beams such as optical angular momentum (OAM) with numerous possible orthogonal states have drawn the prodigious contemplation of researchers. OAM multiplexing is a futuristic multi-access technique that has not been scrutinized for optical satellite communication (OSC) systems thus far, and it opens up a new window for ultra-high-capacity systems. This paper presents the 4.8 Tbps (5 wavelengths × 3 OAM beams × 320 Gbps) ultra-high capacity OSC system by incorporating polarization division multiplexed (PDM) 256-Quadrature amplitude modulation (256-QAM) and OAM beams. To realize OAM multiplexing, Laguerre Gaussian (LG) transverse mode profiles such as LG00, LG140, and LG400 were used in the proposed study. The effects of the receiver’s digital signal processing (DSP) module were also investigated, and performance improvement was observed using DSP for its potential to compensate for the effects of dispersion, phase errors, and nonlinear effects using the blind phase search (BPS), Viterbi phase estimation (VPE), and the constant modulus algorithm (CMA). The results revealed that the proposed OAM-OSC system successfully covered the 22,000 km OSC link distance and, out of three OAM beams, fundamental mode LG00 offered excellent performance. Further, a detailed comparison of the proposed system and reported state-of-the-art schemes was performed. Full article