Simultaneous Approximation Research Articles

Multivariate Approximation Using Symmetrized and Perturbed Hyperbolic Tangent-Activated Multidimensional Convolution-Type Operators

In this article, we introduce, for the first time, multivariate symmetrized and perturbed hyperbolic tangent-activated convolution-type operators in three forms. We present their approximation properties, that is, their quantitative convergence to the unit operator via the multivariate modulus of continuity. We continue with the multivariate global smoothness preservation of these operators. We present, in detail, the related multivariate iterative approximation, as well as, multivariate simultaneous approximation, and their combinations. Using differentiability in our research, we produce higher rates of approximation, and multivariate simultaneous global smoothness preservation is also achieved.

Another Look at the Security Analysis of the Modulus N = p2q by Utilizing an Approximation Approach for ϕ(N)

Newly developed techniques have been recently documented, which capitalize on the security provided by prime power modulus denoted as N = prqs where2 ≤ s < r. Previousresearchprimarilyconcentrated on the factorization of the modulus of type at minimum N = p3q2. In contrast, within the context of 2 ≤ s < r, we address scenarios in the modulus N = p2q (i.e. r = 2 and s = 1) still need to be covered, showing a significant result to the field of study. This work presents two factorization approaches for the multiple moduli Ni = p2 iqi, relying on a good approximation of the Euler’s totient function ϕ(Ni). The initial method for factorization deals with the multiple moduli Ni = p2 iqi derived from m public keys (Ni,ei) and is interconnected through the equation eid − kiϕ(Ni) = 1. In contrast, the second factorization method is associated with the eidi − kϕ(Ni) = 1. By reorganizing the equations as a simultaneous Diophantine approximation problem and implementing the LLL algorithm, it becomes possible to factorize the list of moduli Ni = p2 iqi concurrently, given that the unknowns d, di, k, and ki are suff iciently small. The key difference between our results and the referenced work is that we cover a real-world cryptosystem that uses the modulus N =p2q. In contrast, the previous work covers a hypothetical situation of modulus in the form of N = prqs.

Approximation by Symmetrized and Perturbed Hyperbolic Tangent Activated Convolution-Type Operators

In this article, for the first time, the univariate symmetrized and perturbed hyperbolic tangent activated convolution-type operators of three kinds are introduced. Their approximation properties are presented, i.e., the quantitative convergence to the unit operator via the modulus of continuity. It follows the global smoothness preservation of these operators. The related iterated approximation as well as the simultaneous approximation and their combinations, are also extensively presented. Including differentiability and fractional differentiability into our research produced higher rates of approximation. Simultaneous global smoothness preservation is also examined.

Complex Generalized Stancu-Schurer Operators

Abstract In the present study, we determine a Schurer type of complex generalized Stancu operators and examine its some approximation properties. Firstly, we obtain upper quantitative estimates for the convergence and then we get lower estimates from a qualitative Voronovskaja-type result for these specified operators attached to the analytic functions in an open disk centered at the origin. Hence, we present the exact order of simultaneous approximation by using obtained upper and lower estimates.

Collaborative train timetabling and passenger flow control in oversaturated metro lines considering state-dependence

Congestion downgrades the moving speed of passengers in trains and platforms. This in turn affects congestion. That is, the moving speed of passengers interacts with congestion. It is state-dependent and obeys the fundamental diagram. This interaction makes the system unstable and can result in substantial passenger delays and heightened security risks in oversaturated metro lines. This study aims to rectify this by developing a more realistic collaborative optimization approach. Firstly, we formulate a deterministic fluid queuing network model to accurately capture the dynamic propagation of congestion in oversaturated metro lines. This model incorporates the fundamental diagram, thereby effectively considering the interaction between passenger movement speeds and congestion. A recursive algorithm is devised to solve the fluid queuing network model. The complexity of the algorithm is independent of the capacities of the stations and trains. This provides the possibility of analyzing large-scale metro networks. Based on the queuing network model, a mixed-integer nonlinear programming model aimed at minimizing the total passenger waiting time is built. To solve the model effectively, we employ an alternating direction method of multipliers framework to decompose the model into nonlinear programming and integer nonlinear programming problems, which are respectively solved by improved simultaneous perturbation stochastic approximation and adaptive large neighborhood search algorithms. Finally, a small-scale case and a real-world case with data from Chengdu metro lines are implemented to demonstrate the performance and effectiveness of the proposed approach. The experimental results provide some interesting insights. (1) Without consideration of state-dependence, the queuing model underestimates the passenger waiting time and overestimates the service levels. (2) As demand increases, excluding extreme oversaturation scenarios, the optimized results of the state-independent model progressively deteriorate compared to our state-dependent model, with the largest percentage decline being 50.23%. This is especially noticeable when the train capacity is limited. (3) The contribution ratio of passenger flow control in mitigating congestion gradually surpasses that of train timetabling with increasing demand. In the small-scale case, the former contribution ratio to optimization rises from negligible to 98.19%.

Умови екстремальності допустимого елемента для задачі відшу-кання узагальненого чебишовського центра кількох точок деякого полінормованого простору відносно множини цього простору

The problems related to the need to approximate complex mathematical objects in the best possible way with simpler and more convenient ones arise in various sections of mathematical science. An important class of approximation theory problems is the best simultaneous approximation of several elements. The problem of finding the Chebyshev center of several points of a linear normalized space relative to the set of this space can be attributed to the problems of best simultaneous approximation of several elements. This task consists in finding in a given set of linear normed space such a point (the relative Chebyshev center) the maximum distance to which from several fixed points of space would be the smallest, in other words not exceeding the maximum distance from the given points to any other point of this set. The problems of the best simultaneous approximation of several elements of a linear normed space by convex sets of this space from single positions were considered, in particular, in works [1, 2]. In practice, one has to deal with such problems, in which, when finding the Chebyshev's center of several given points of a linear normed space relative to the set of this space, appear weighted distances. The task of finding the weighted distances of the Chebyshev’s center was considered, in particular, in the paper [3]. In this work, the criteria of generalized Chebyshov’s center in the sense of the weighted distances of the of several points of a linear normed space relative to the convex set of this space, based on the duality ratio for the corresponding extremal problem, are established. If in the problem of the Chebyshev's center of several points of a linear normed space, in which the distances between points are determined by weighted norms, the weighted norms are replaced, generally speaking, by different norms given on the corresponding linear space, then we obtain the problem of the Chebyshev's center of several points of some polynormed space, which is considered in this work. It is clear that the problems about the Chebyshev's center of several points of a linear normed space, which were discussed above, are partial cases of the problem about the Chebyshev center of several points of some polynormed space.

Maximization of Flight Performance of Eight-Rotor Multirotor with Differentiated Hub Angle

The aim of this article is to design a rotary wing aircraft autopilot system that improves flight performance by changing the body shape during flight. The method is to obtain values that stabilize the longitudinal and lateral flight of the aircraft, where the amount of metamorphosis and Proportional-Integral-Derivative (PID) coefficients are determined using the simultaneous perturbation stochastic approximation (SPSA) optimization algorithm. The rotary wing aircraft has a deformable structure with eight rotors. Shape-changing rotary-wing aircraft are aircraft that can fly with the lift generated by propellers. Aerial platform; It consists of arms and trunk. The angle between mechanism A and the arm to which the rotors are connected can be changed with the horizontal plane and different configurations are obtained. When the angle between the arms is 45°, the octo configuration turns into a stable structure, while when the angle between the arms is 0°, the X8 configuration provides high maneuverability and increased controllability. Metamorphosis, its effect on longitudinal and lateral flight stability and improvement studies were carried out in a simulation environment and the results are presented in this study. As a result of the shape change, longitudinal and lateral narrowing occurred by 26.8° percent. Simulation tests were modeled in a closed environment, free from atmospheric effects. The obtained flight performance values are presented in Tables.

Simple Inverse Kinematics Computation Considering Joint Motion Efficiency.

Inverse kinematics (IK) is an important and challenging problem in the operation of industrial manipulators. This study proposes a simple IK calculation scheme for an industrial serial manipulator. The proposed technique can calculate appropriate values of the joint variables to realize the desired end-effector position and orientation while considering the motion costs of each joint. Two scalar functions are defined for the joint variables: one is to evaluate the end-effector position and orientation, whereas the other is to evaluate the motion efficiency of the joints. By combining the two scalar functions, the IK calculation of the manipulator is formulated as a numerical optimization problem. Furthermore, a simple algorithm for solving the IK via the aforementioned optimization is constructed on the basis of the simultaneous perturbation stochastic approximation with a norm-limited update vector (NLSPSA). The proposed scheme considers not only the accuracy of the position and orientation of the end-effector but also the efficiency of the robot movement. Therefore, it yields a practical result of the inverse problem. Moreover, the proposed algorithm is simple and easy to implement owing to the high-calculation efficiency of NLSPSA. Finally, the effectiveness of the proposed method is verified through numerical examples using a redundant manipulator.

Identification of nonlinear system model and inverse model based on conditional invertible neural network

In applications such as adaptive inverse control, internal model control, and active noise control, the identification accuracy of the system model and the inverse model directly affects the performance. However, it is not easy to identify inverse models for nonlinear systems. Moreover, existing methods require two identification calculations to obtain the system model and the inverse model. Therefore, an identification method of nonlinear system model and inverse model based on conditional invertible neural network (cINN) is proposed. The invertible structure of cINN enables simultaneous approximation of complex nonlinear functions and simultaneous acquisition of the corresponding inverse functions. Consequently, both the nonlinear system model and the inverse model can be identified concurrently through cINN. Moreover, the identification performance of the cINN-based method is validated and applied to disturbance cancellation in a classical nonlinear simulation system. Finally, the inverse model of the actuator is identified by cINN, and the inverse model is applied to the nonlinear compensation of the actuator.

Real-Time Estimation of Origin–Destination Matrices Using a Deep Neural Network for Digital Twins

The digital twin, a real-time replica of physical systems, has garnered attention as a promising tool to strategize and evaluate solutions for complex real-world issues. However, developing digital twins in the field of transportation faces significant challenges related to the real-time estimation of dynamic origin–destination (OD) matrices constrained by computation time. To bridge this gap, microscopic traffic simulations with real-time synchronization are being researched. Nonetheless, the computational issue persists, emphasizing the need for more efficient OD estimation methods. In this regard, our objective is to reduce computation time in simulation-based methods by developing a data-driven metamodel using a deep neural network. The proposed model serves to map the correlation between the OD matrix and detector data. This model simplifies the computational process using hidden layers, rather than calculating complex interactions between vehicles in the traffic simulation. Compared to conventional methods, we evaluate the capability to estimate a reasonable OD matrix within a restricted time and validate our approach using detector data from Daejeon City, South Korea. The findings indicate that by combining our data-driven metamodel with the simultaneous perturbation stochastic approximation, it becomes feasible to estimate a reasonable OD matrix within a stipulated time frame, compared to the conventional method. Given a 1-min time frame, the proposed method outperforms the conventional simulation-based method by improving the calibration performance of traffic flow by 44.5 percentage points. This paper proposes a practical and versatile approach for real-time OD estimation, laying the foundation for extending microscopic traffic simulation into the digital twin.

On Simultaneous Approximation of Algebraic Power Series over a Finite Field

On Simultaneous Approximation of Algebraic Power Series over a Finite Field

A new approach for fracture parameters optimization of multi-fractured horizontal wells: Virtual boundary method

Horizontal well fracturing technology stands as the cornerstone for achieving large-scale commercial production in shale reservoirs. Yet, the complexity of fracture variables within horizontal wells, comprising both discrete and continuous factors, presents a formidable challenge for optimization. Existing methods struggle to holistically optimize these mixed fracture parameters, including discrete fracture number, continuous fracture location, and half-length. This study pioneers a fracture deployment optimization approach tailored for fractured horizontal wells in shale reservoirs, leveraging the innovative virtual boundary method (VBM). Initially, a virtual boundary model is developed by extending the length of the fracturing horizontal wellbore, defining the extended area as a virtual reservoir devoid of hydrocarbon. Subsequently, the net present value (NPV) is adopted as the objective function, with continuous fracture location and half-length serving as optimization variables. The modified simultaneous perturbation stochastic approximation (mSPSA) algorithm is then employed for optimization. Following optimization, the optimized fractures within the virtual area are identified and deleted via position identification, enabling the comprehensive optimization of fracture number, location, and half-length. Notably, the computational efficiency of this method surpasses existing approaches by a factor of 5.1 compared to the bisection nested SPSA optimization method. Validation through three calculation cases, encompassing homogeneous reservoir, distinct sweet spot reservoir, and strongly heterogeneous reservoir, further underscores the method's reliability. Post-optimization, NPV increases by 1.1 times, 1.4 times, and 0.34 times for the respective calculation cases compared to the initial plan. When applied to an actual shale oil reservoir, the method yields a 1.67 times increase in NPV compared to existing deployment plans. This research lays a solid theoretical foundation for the design of horizontal fracturing well construction plans in shale reservoirs.

Upwind Summation-by-parts Finite Differences: error Estimates and WENO methodology

High order upwind summation-by-parts finite difference operators have recently been developed. When combined with the simultaneous approximation term method to impose boundary conditions, the method converges faster than using traditional summation-by-parts operators. We prove the convergence rate by the normal mode analysis for such methods for a class of hyperbolic partial differential equations. Our analysis shows that the penalty parameter for imposing boundary conditions affects the convergence rate for stable methods. In addition, to solve problems with discontinuous data, we extend the method to also have the weighted essentially nonoscillatory property. The overall method is stable, achieves high order accuracy for smooth problems, and is capable of solving problems with discontinuities.

The influence of basis sets and ansatze building to quantum computing in chemistry.

Quantum computing is an exciting area, which has grown at an astonishing rate in the last decade. It is especially promising for the computational and theoretical chemistry area. One algorithm has received a lot of attention lately, the variational quantum eigensolver (VQE). It is used to solve electronic structure problems and it is suitable to the noisy intermediate-scale quantum (NISQ) hardware. VQE calculations require ansatze and one of the most known is the unitary coupled cluster (UCC). It uses the chosen basis set to generate a quantum computing circuit which will be iteratively minimized. The present work investigates the circuit depth and the number of gates as a function of basis sets and molecular size. It has been shown that for the current quantum devices, only the smallest molecules and basis sets are tractable. The H molecule with the cc-pVTZ and aug-cc-pVTZ basis sets have circuit depths in the order of 10 to 10 gates and the C H molecule with 3-21G basis set has a circuit depth of gates. At the same time the analysis demonstrates that the H molecule with STO-3G basis set, requires at least 500 shots to reduce the error and that, although error mitigation schemes can diminish the error, they were not able to completely negate it. The quantum computing and electronic structure calculations were performed using the Qiskit package from IBM and the PySCF package, respectively. The ansatze were generated using the UCCSD method as implemented in Qiskit, using the basis sets STO-3G, 3-21G, 6-311G(d,p), def2-TZVP, cc-pVDZ, aug-cc-pVDZ, cc-pVTZ, and aug-cc-pVTZ. The operators and the Hamiltonian were mapped using the Jordan-Wigner scheme. The classical optimizer chosen was the simultaneous perturbation stochastic approximation (SPSA). The quantum computers used were the Nairobi and Osaka, with 7 and 127 qubits respectively.

Flight control system design of UAV with wing incidence angle simultaneously and stochastically varied

PurposeThis study aims to simultaneously and stochastically maximize autonomous flight performance of a variable wing incidence angle having an unmanned aerial vehicle (UAV) and its flight control system (FCS) design.Design/methodology/approachA small UAV is produced in Iskenderun Technical University Drone Laboratory. Its wing incidence angle is able to change before UAV flight. FCS parameters and wing incidence angle are simultaneously and stochastically designed to maximize autonomous flight performance using an optimization method named simultaneous perturbation stochastic approximation. Obtained results are also benefitted during UAV flight simulations.FindingsApplying simultaneous and stochastic design approach for a UAV having passively morphing wing incidence angle and its flight control system, autonomous flight performance is maximized.Research limitations/implicationsPermission of the Directorate General of Civil Aviation in Turkish Republic is necessary for real-time flights.Practical implicationsSimultaneous stochastic variable wing incidence angle having UAV and its flight control system design approach is so useful for maximizing UAV autonomous flight performance.Social implicationsSimultaneous stochastic variable wing incidence angle having UAV and its flight control system design methodology succeeds confidence, excellent autonomous performance index and practical service interests of UAV users.Originality/valueCreating an innovative method to recover autonomous flight performance of a UAV and generating an innovative procedure carrying out simultaneous stochastic variable wing incidence angle having UAV and its flight control system design idea.

Simultaneous Approximation for an Exponential Operator Connected with x4/3

Very recently, the authors in [5] proposed the exponential-type operator connected with and studied its convergence estimates. In the present research, we extend the study and obtain the general form of its 𝑝-th order moment; 𝑝 ∈ ℕ ∪ {0}. Further, we establish the simultaneous approximation for the operator under consideration.

Stochastic optimal tuning for flight control system of morphing arm octorotor

PurposeThis study aims to discuss the simultaneous longitudinal and lateral flight control of the octorotor, a rotary wing unmanned aerial vehicle (UAV), for the first time under the effect of morphing and to improve autonomous flight performance.Design/methodology/approachThis study aims to design and control the octorotor flight control system with stochastic optimal tuning under morphnig effect. For this purpose, models of different arm lengths of the octorotor were drawn in the Solidworks program. The morphing was carried out by simultaneously lengthening or shortening the arm lengths of the octorotor. The morphing rate was estimated by using simultaneous perturbation stochastic approximation (SPSA). The stochastic gradient descent algorithm, which is frequently used in machine learning, was used to estimate the changing moments of inertia with the change of arm lengths. The proportional integral derivative (PID) controller has been preferred as an octorotor control algorithm because of its simplicity of structure. The PID gains required to control both longitudinal and lateral flight were also estimated with SPSA.FindingsWith SPSA, three longitudinal flight PID gains, three lateral flight PID gains and one morphing ratio were estimated. PID gains remained within the limits set for SPSA, giving satisfactory results. In addition, the cost index created was 93% successful. The gradient descent algorithm used for the moment of inertia estimation achieved the optimum result in 1,570 iterations. However, in the simulations made with the obtained data, longitudinal and lateral flight was successfully carried out.Originality/valueOctorotor longitudinal and lateral flight control was performed quickly and effectively with the proposed method. In addition, the desired parameters were obtained with the optimization methods used, and the longitudinal and lateral flight of the octorotor was successfully carried out in the desired trajectory.

Enhancing buildings' energy efficiency prediction through advanced data fusion and fuzzy classification

This study proposes a method to predict buildings' energy efficiency based on available descriptive information and without a physical visit, by merging diverse datasets and employing advanced classification techniques. By integrating geographical, structural, legal, and socio-economic data with Energy Performance Certificate (EPC) observations, our approach yields a rich learning set. Through variable selection methods like forward selection with KNN and simultaneous perturbation stochastic approximation for fuzzy KNN, we refine model variables. Comparing fuzzy and hard classification using KNN, Kriging or Random Forest approaches, we find fuzzy classification more adept at capturing nuanced energy inefficiency indicators. Our study highlights the importance of mass energy efficiency prediction for sustainable renovation efforts.

Anti-derivatives approximator for enhancing physics-informed neural networks

This study presents a novel strategy for constructing an approximator for arbitrary univariate functions. The proposed approximation utilizes the anti-derivatives of a Fourier series expansion for the presumed piecewise function, resulting in a remarkable feature that enables the simultaneous approximation of an arbitrary function and its anti-derivatives. These anti-derivatives can be employed to discover solution curves for systems of ordinary differential equations based on an optimization scheme, even in the presence of chaotic dynamics. Additionally, the anti-derivatives approximator is extended as an adaptive activation function for physics-informed neural networks, leveraging the high-order differentiability of the anti-derivatives. Systematic experiments have demonstrated the outstanding merits of the proposed anti-derivatives-based approximator, including its ability to construct regression models for volatile data and their anti-derivatives, solve differential equations, and enhance the capabilities of physics-informed neural networks.

HK-SPSA based performance optimization method for steam generator liquid level control

Steam generator level control systems play a crucial role in ensuring safe, economical, and stable operation of nuclear power plants. However, traditional control systems exhibit limited efficiency in commissioning and are susceptible to human error. Additionally, traditional parameter tuning methods are typically experienced-based, cumbersome, and time-consuming, making it challenging to obtain optimal parameters. To address these issues, we propose a hybrid knowledge-guided improved simultaneous perturbation stochastic approximation algorithm (HK-SPSA) to optimize the control parameters of steam generator level control systems. Firstly, the algorithm utilizes iteration point adjacency information to approximate the current optimization process status and guide the adaptive adjustment of the iteration step size. Secondly, it utilizes composite gradient information generated by the estimated historical and current gradients to guide the optimization direction and the tuning of iteration step size. Simulation experiments have shown that HK-SPSA improves optimization performance and reduces the costs of optimization and adjustment compared to traditional methods.