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Abstract The general synthetic iterative scheme (GSIS) has proven its efficacy in modeling rarefied gas dynamics, where the steady-state solutions are obtained after dozens of iterations of the Boltzmann equation, with minimal numerical dissipation even using large spatial cells. In this paper, the fast convergence and asymptotic-preserving properties of the GSIS are harnessed to remove the limitations of the direct simulation Monte Carlo (DSMC) method. The GSIS, which leverages high-order constitutive relations derived from DSMC, is applied intermittently, which not only rapidly steers the DSMC towards steady state, but also eliminates the requirement that the cell size must be smaller than the molecular mean free path. Several numerical tests have been conducted to validate the accuracy and efficiency of this hybrid GSIS-DSMC approach.

The general synthetic iterative scheme (GSIS) has proven its efficacy in modeling rarefied gas dynamics, where the steady-state solutions are obtained after dozens of iterations of the Boltzmann equation, with minimal numerical dissipation even using large spatial cells. In this paper, the fast convergence and asymptotic-preserving properties of the GSIS are harnessed to remove the limitations of the direct simulation Monte Carlo (DSMC) method. The GSIS, which leverages high-order constitutive relations derived from DSMC, is applied intermittently, which not only rapidly steers the DSMC towards steady state, but also eliminates the requirement that the cell size must be smaller than the molecular mean free path. Several numerical tests have been conducted to validate the accuracy and efficiency of this hybrid GSIS-DSMC approach. [ABSTRACT FROM AUTHOR]

This paper studies the exponential finite-time formation problems for a group of locomotive agents that maintain order on a circle. In an ideal environment, that is, without external disturbances, a combined protocol with linear continuous-time states and their nonlinear continuous functions is proposed. Each agent can adjust the movement speed itself by changing the exponent of the nonlinear function. The obtained closed-loop system will converge with an exponential speed when the initial states of the agents are far from the target location, and with a finite-time speed nearby. For the case with external disturbances, a combined protocol with linear continuous-time states and their sign functions is presented. Due to the discontinuity of the sign functions, the Filippov solutions are employed. The exponential finite-time circle formation will be achieved even in the existence of external disturbances. Moreover, the above result is extended to the application in circle containment, where the defenders will form a circle formation to surround the swarm of protectees. Finally, several computational simulations illustrate the validity of the proposed protocols. [ABSTRACT FROM AUTHOR]

We present a new gradient-like dynamical system related to unconstrained convex smooth multiobjective optimization which involves inertial effects and asymptotic vanishing damping. To the best of our knowledge, this system is the first inertial gradient-like system for multiobjective optimization problems including asymptotic vanishing damping, expanding the ideas previously laid out in [H. Attouch and G. Garrigos, Multiobjective Optimization: An Inertial Dynamical Approach to Pareto Optima, preprint, arXiv:1506.02823, 2015]. We prove existence of solutions to this system in finite dimensions and further prove that its bounded solutions converge weakly to weakly Pareto optimal points. In addition, we obtain a convergence rate of order O(t-2) for the function values measured with a merit function. This approach presents a good basis for the development of fast gradient methods for multiobjective optimization. [ABSTRACT FROM AUTHOR]

This paper presents a design for timing mismatch calibration in a TIADC (Time-Interleaved Analog-to-Digital Converter) with wideband inputs. By exploiting the approximately linear relationship between the autocorrelation properties of sub-ADCs and timing mismatch, we achieve rapid convergence of error estimation. A low-cost detection method is proposed based on the convergent monotonicity of the Least Mean Square (LMS) algorithm, which can automatically correct the calibration direction when the input signal goes beyond the Nyquist zone. Physical test results indicate that the spurs caused by timing mismatch can be suppressed by 26–30 dB using the proposed method. [ABSTRACT FROM AUTHOR]

This work introduces a novel adaptive fixed-time control strategy for nonlinear systems subject to external disturbances. The focus pertains to the introduction of the fixed-time terminal sliding mode control (FxSMC) technique. The proposed scheme exhibits rapid convergence, chatter-free and smooth control inputs, and stability within a fixed time. The utilization of an adaptive methodology in combination with the FxSMC yields the proposed strategy. This approach is employed to address the dynamic system in the presence of external disturbances. The results obtained from the Lyapunov analysis will provide insights into the stability of the closed-loop system in a fixed time. In the end, the simulation results are presented in order to assess and demonstrate the effectiveness of the methodology.

In this article, an adaptive critic scheme with a novel performance index function is developed to solve the tracking control problem, which eliminates the tracking error and possesses the adjustable convergence rate in the offline learning process. Under some conditions, the convergence and monotonicity of the accelerated value function sequence can be guaranteed. Combining the advantages of the adjustable and general value iteration schemes, an integrated algorithm is proposed with a fast guaranteed convergence, which involves two stages, namely the acceleration stage and the convergence stage. Moreover, an effective approach is given to adaptively determine the acceleration interval. With this operation, the fast convergence of the new value iteration scheme can be fully utilized. Finally, compared with the general value iteration, the numerical results are presented to verify the fast convergence and the tracking performance of the developed adaptive critic design. [ABSTRACT FROM AUTHOR]

This work introduces a novel adaptive fixed-time control strategy for nonlinear systems subject to external disturbances. The focus pertains to the introduction of the fixed-time terminal sliding mode control (FxSMC) technique. The proposed scheme exhibits rapid convergence, chatter-free and smooth control inputs, and stability within a fixed time. The utilization of an adaptive methodology in combination with the FxSMC yields the proposed strategy. This approach is employed to address the dynamic system in the presence of external disturbances. The results obtained from the Lyapunov analysis will provide insights into the stability of the closed-loop system in a fixed time. In the end, the simulation results are presented in order to assess and demonstrate the effectiveness of the methodology. [ABSTRACT FROM AUTHOR]

In a Hilbert space setting $ {\mathcal H} $ H , for convex optimization, we analyse the fast convergence properties as $ t \rightarrow +\infty $ t → + ∞ of the trajectories $ t \mapsto u(t) \in {\mathcal H} $ t ↦ u (t) ∈ H generated by a third-order in time evolution system. The function $ f: {\mathcal H} \to {\mathbb R} $ f : H → R to minimize is supposed to be convex, continuously differentiable, with $ {\rm argmin}_{{\mathcal H}} f \neq \emptyset $ argmin H f ≠ ∅. It enters into the dynamic through its gradient. Based on this new dynamical system, we improve the results obtained by Attouch et al. [Fast convex optimization via a third-order in time evolution equation. Optimization. 2020;71(5):1275–1304]. As a main result, when the damping parameter α satisfies $ \alpha \gt 3 $ α > 3 , we show that $ f(u(t)) - \inf _{{\mathcal H}} f = o(1/t^3) $ f (u (t)) − inf H f = o (1 / t 3) as $ t \rightarrow +\infty $ t → + ∞ , as well as the convergence of the trajectories. We complement these results by introducing into the dynamic an Hessian-driven damping term, which reduces the oscillations. In the case of a strongly convex function f, we show an autonomous evolution system of the third-order in time with an exponential rate of convergence. All these results have natural extensions to the case of a convex lower semicontinuous function $ f: {\mathcal H} \to \mathbb R\cup \{+\infty \} $ f : H → R ∪ { + ∞ }. Just replace f with its Moreau envelope. [ABSTRACT FROM AUTHOR]

We present a novel hybrid method that integrates the spectral integral method (SIM) with finite element for pressure acoustics in an inhomogeneous medium surrounded by spherical or incomplete spherical surfaces. Our approach, called the finite element spectral integral (FESI) method, enables efficient calculations of the pressure field in near-field and far-field zones. It is more efficient than the conventional finite element method hybridized with the boundary element method (i.e., the FEBI method). Furthermore, we construct a symmetric coefficient matrix using the proposed symmetric coupling to speed up the FESI method. We demonstrate through several examples that our FESI method offers significantly improved efficiency and lower memory consumption compared to the FEM combined with the perfect matching layer method while achieving higher accuracy. • Novel FESI for complete/incomplete spherical boundaries with symmetric coupling. • Efficient exterior and far-field solutions using spectral domain solution and NDSFT. • FESI outperforms FEM-PML scheme in COMSOL, with improved efficiency and accuracy. [ABSTRACT FROM AUTHOR]

Clustering is a technique used to partition a dataset into groups of similar elements. In addition to traditional clustering methods, clustering for probability density functions (CDF) has been studied to capture data uncertainty. In CDF, automatic clustering is a clever technique that can determine the number of clusters automatically. However, current automatic clustering algorithms update the new probability density function (pdf) fi(t)$$ {f}_i(t) $$ based on the weighted mean of all previous pdfs fj(t−1),j=1,2,...,N$$ {f}_j\left(t-1\right),j=1,2,\dots, N $$, resulting in slow convergence. This paper proposes an efficient automatic clustering algorithm for pdfs. In the proposed approach, the update of fi(t)$$ {f}_i(t) $$ is based on the weighted mean of f1(t),f2(t),...,fi−1(t),fi(t−1),fi+1(t−1),...,fN(t−1)$$ \left\{{f}_1(t),{f}_2(t),\dots, {f}_{i-1}(t),{f}_i\left(t-1\right),{f}_{i+1}\left(t-1\right),\dots, {f}_N\left(t-1\right)\right\} $$, where N$$ N $$ is the number of pdfs and i=1,2,...,N$$ i=1,2,\dots, N $$. This technique allows for the incorporation of recently updated pdfs, leading to faster convergence. This paper also pioneers the applications of certain CDF algorithms in the field of surface image recognition. The numerical examples demonstrate that the proposed method can result in a rapid convergence at some early iterations. It also outperforms other state‐of‐the‐art automatic clustering methods in terms of the Adjusted Rand Index and the Normalized Mutual Information. Additionally, the proposed algorithm proves to be competitive when clustering material images contaminated by noise. These results highlight the applicability of the proposed method in the problem of surface image recognition. [ABSTRACT FROM AUTHOR]

State-of-charge balance is vital for allowing multiple energy storage units (ESUs) to make the most of stored energy and ensure safe operation. Concerning scenarios wherein boost converters are used as the interfaces between ESUs and loads, this paper proposes a balancing strategy for realizing consistent state-of-charge (SoC) levels and equal currents among different ESUs. This strategy is valid for both parallel and series applications. Its advantages also include its high precision of SoC equalization without extra sensors and fast convergence. A common outer voltage loop is used to accomplish tight voltage regulation, while multiple inner current loops are utilized to achieve current control and SoC balance simultaneously. Firstly, by introducing SoC-based current distribution ratios (CDRs) to modify current references online, the currents are gradually adjusted to eliminate SoC deviations. Secondly, to expedite the balancing process, current saturations are further adopted. Thirdly, the influences of accelerating factor and current limits in CDR expressions are analyzed, and their selection guidelines are subsequently provided. Fourthly, the controller design, consisting of a dual loop, is illustrated to guarantee sufficient stability margins. Fifthly, an experimental platform consisting of three battery ESUs is developed to verify the proposed strategy. [ABSTRACT FROM AUTHOR]

This research proposes a load flow (LF) solver with DC/DC converter models for DC distribution. The solver is based on modified-augmented-nodal-analysis (MANA) formulation and is implemented with the Newton–Raphson (NR) algorithm. The conditions to guarantee the convergence range of the MANA are demonstrated with Kantorovich’s theorem. The proposed formulation is generic and flexible, allowing it to accommodate any arbitrary DC grid configuration. In addition, the formulation is extendable due to MANA’s modular structure, which enables the incorporation of any device without changing the prior formulation. Instead of bus power injections, the summation of line flows and their associated derivatives are used when considering connections with various kinds of buses equipped with various converters. The proposed method is implemented in MATLAB and tested on the modified IEEE 33 bus distribution test feeder. Electromagnetic Transient (EMT) simulations confirmed the accuracy of the proposed method. Moreover, the results are also compared with the recently proposed efficient Laplacian Matrix (eLM) based method. Results suggest that the proposed method is robust and offers superior convergence characteristics with a faster simulation speed.

The center of distances of central Cantor sets is investigated. Any central Cantor set is an achievement set of exactly one fast convergent sequence (a n) . We show that the center of distances of the central Cantor set is the union of all centers of distances of F n where F n is the set of all n-initial subsums of the series ∑ a n . Moreover, we give a necessary and sufficient condition for central Cantor sets to have not the minimal possible center of distances. [ABSTRACT FROM AUTHOR]

Time-consuming objective functions are inevitable in practical engineering optimization problems. This kind of function makes the implementation of metaheuristic methodologies challenging since the designer must compromise between the quality of the final solution and the overall runtime of the optimization procedure. This paper proposes a novel multi-objective optimization algorithm called FC-MOEO/AEP appropriate for highly time-consuming objective functions. It is based on an equilibrium optimizer equipped with an archive evolution path (AEP) mechanism. The AEP mechanism considers the evolutionary trajectory of the decision space and anticipates the potential regions for optimal solutions. Besides having a high convergence rate, the newly proposed approach also possesses an intelligent balance between exploration and exploitation capabilities, enabling the algorithm to effectively avoid getting stuck in the local Pareto. To assess the efficacy of the FC-MOEO/AEP, a range of mathematical optimization problems and three real-world structural design problems were employed. In order to gauge the method's performance against other approaches, a novel performance indicator known as convergence speed was introduced and utilized, in addition to standard metrics. The numerical findings demonstrate the robust and consistent performance of the FC-MOEO/AEP in tackling complex multi-objective problems. [ABSTRACT FROM AUTHOR]

Let H be a real Hilbert space, and f : H → R be a convex twice differentiable function whose solution set argmin f is nonempty. We investigate the long time behavior of the trajectories of the vanishing damped dynamical system with Tikhonov regularizing term and Hessian-driven damping x ¨ (t) + α x ˙ (t) + δ ∇ 2 f (x (t)) x ˙ (t) + β (t) ∇ f (x (t)) + c x (t) = 0 where α , c , δ are three positive constants, and the time scale parameter β is a positive nondecreasing function such that lim t → + ∞ β (t) = + ∞ . Under some assumptions on the parameter β , we will show rapid convergence of values, strong convergence toward the minimum norm element of argmin f , and rapid convergence of the gradients toward zero. Note that the Hessian-driven damping significantly reduces the oscillatory aspects, and the time scale parameter β improves the rate of convergences mentioned above. As particular cases of β , we set β (t) = t p ln q (t) , for (p , q) ∈ (R +) 2 ∖ { (0 , 0) } , and β (t) = e γ t p , for p ∈ ] 0 , 1 [ and γ > 0 . The manuscript concludes with two numerical examples and comments on their performance. [ABSTRACT FROM AUTHOR]

There are two common approaches to design a time synchronization protocol as reference-based and consensus-based. The first approach is mainly employed in protocols operating in large-scale wireless sensor networks (WSNs) because of its fast convergence. However, reference-based time synchronization protocols are vulnerable to node-failure in real-world applications. Although consensus-based time synchronization protocols do not experience the aforementioned problem, they are not suitable for operation in large-scale WSNs owing to their slow convergence. In this study, we propose a hybrid time synchronization protocol (HTSP) to utilize the advantages of both approaches. In particular, HTSP exploits a temporary reference node to significantly reduce the convergence time and employs average-based consensus during normal operation to cope with node failures. It is notable that each node in HTSP can seamlessly switch between the reference and consensus modes during its operation. Moreover, HTSP is a fully distributed protocol as it does not rely on any leader-election mechanism. Our simulation results demonstrated that HTSP outperforms flooding, gradient, and virtual link-based time synchronization protocols considering various types of network topologies and scales.

Distribution networks feature distinct topologies than transmission networks, such as radial or weakly meshed structures with tens of thousands of nodes. They have more points of power injection owing to the integration of distributed generators and high R/X ratios. Furthermore, there has recently been a surge of interest in DC distribution networks. In the planning and operation of modern distribution systems, load flow needs to be executed in series considering short intervals of time in the order of minutes or even less. Hence, these networks require a load flow solver that can converge fast with low computational burden. In this paper, we propose a unique iterative power flow solver based on graph theory for DC distribution networks. The proposed formulation is flexible and can handle both radial and mesh configurations with just one connectivity matrix. To validate the proposed method, we used the IEEE 33 bus test feeder and compared the results with an existing methodology. Results suggest that the proposed method is robust and possesses fast convergence.

The primary objective of anti-jamming techniques is to ensure that the transmitted data arrives at the intended receiver without being disturbed or jammed with by any jamming signal or other hostile activities to ensuring the security of the communication system. Deep reinforcement learning (DRL) has been extensively utilized in solving the dynamic spectrum anti-jamming problem. However, most of existing DRL-based algorithms require lots of training time, which fails to adapt the fast-channging jamming environment. Our objective is to find a practical and fast-convergence anti-jamming learning solution. To achieve this, we redesign the DRL algorithm in the following two ways. First, we split the cycle of reinforcement learning into two parts: applying process and training process. Second, we use soft labels instead of rewards which bring more information. We further theoretically show that the information gain can help our proposed algorithm converge faster. Moreover, we also show that our labeled DRL algorithm is better than the idealized DRL-based scheme which can obtain the same information as the soft labels. Simulation results demonstrate that compared with existing DRL-based algorithms, our proposed algorithm reduces the number of iterations by up to 90%. [ABSTRACT FROM AUTHOR]

State-of-charge balance is vital for allowing multiple energy storage units (ESUs) to make the most of stored energy and ensure safe operation. Concerning scenarios wherein boost converters are used as the interfaces between ESUs and loads, this paper proposes a balancing strategy for realizing consistent state-of-charge (SoC) levels and equal currents among different ESUs. This strategy is valid for both parallel and series applications. Its advantages also include its high precision of SoC equalization without extra sensors and fast convergence. A common outer voltage loop is used to accomplish tight voltage regulation, while multiple inner current loops are utilized to achieve current control and SoC balance simultaneously. Firstly, by introducing SoC-based current distribution ratios (CDRs) to modify current references online, the currents are gradually adjusted to eliminate SoC deviations. Secondly, to expedite the balancing process, current saturations are further adopted. Thirdly, the influences of accelerating factor and current limits in CDR expressions are analyzed, and their selection guidelines are subsequently provided. Fourthly, the controller design, consisting of a dual loop, is illustrated to guarantee sufficient stability margins. Fifthly, an experimental platform consisting of three battery ESUs is developed to verify the proposed strategy.

This brief proposes a signal-independent background calibration in pipeline-SAR analog-to-digital converters (ADCs) with a convergence-accelerated technique. To achieve signal independence, an auxiliary capacitor array CA is introduced to pre-inject a pseudo-random noise (PN) in the sampling phase to cancel out the opposite PN injection of the calibrated capacitor in the conversion phase, and CA is also used to realize the D/A function of the calibrated capacitor in the conversion phase. In this way, no matter what the signal is, the residue headroom remains unchanged even with PN injection. Moreover, the first sub-ADC is designed with extended conversion bits to quantize its own residue after delivering the conversion bits required by the first stage. Afterwards, this result is provided to the calibration algorithm to reduce the signal component and accelerate the convergence. Based on the simulation, the signal-to-noise and distortion ratio (SNDR) and spur-free dynamic range (SFDR) improve from 45.3 dB and 56.4 dB to 68.2 dB and 88.4 dB, respectively, after calibration. In addition, with the acceleration technique, convergence cycles decrease from 1.7 × 108 to 5.8 × 106. Moreover, no matter whether the input signal is DC, sine wave or band-limited white noise, the calibration all works normally. [ABSTRACT FROM AUTHOR]

Formation building for multi-small-AUV systems with on-board cameras is crucial under the limited communication underwater environment. A hybrid coordination strategy is proposed for the rapid convergence to a leader-follower pattern. The strategy consists of two parts: a time-optimal local-position-based controller (TOLC) and a distributed asynchronous discrete weighted consensus controller (ADWCC). The TOLC controller is designed to optimize the assignation of AUVs’ destinations in the given pattern and guide each AUV to its destination by the shortest feasible distance. The ADWCC controller is developed to direct the AUVs blocked by obstacles to reach their destinations with the information from the perceived neighbors by on-board cameras. The rapidity of the proposed strategy is theoretically discussed. The effectiveness of the proposed algorithm has been verified in the simulation environments in both MATLAB and Blender.

This paper proposed an optimal control strategy based on Q-learning for output regulation of discrete multi-agent systems. For the output regulation of the traditional multi-agent system, obtaining the exact dynamics model of the system and solving the HJB equation based on this is the main obstacle. The strategy achieved global optimal control of each agent′s output without relying on the dynamic model of the system through local communication between agents. In order to optimize the system response rate, this paper proposed a new finite time local error formula, which not only ensures the original global optimal performance of the algorithm, but also reduced the output synchronization time by nearly 50%. It analyzed the stability of the proposed algorithm. The simulation results show that this strategy achieves the optimal control of the system without building the complex system model and solving the discrete HJB equation, and the updated finite time local error formula can effectively shorten the convergence time. [ABSTRACT FROM AUTHOR]

There are two common approaches to design a time synchronization protocol as reference-based and consensus-based. The first approach is mainly employed in protocols operating in large-scale wireless sensor networks (WSNs) because of its fast convergence. However, reference-based time synchronization protocols are vulnerable to node-failure in real-world applications. Although consensus-based time synchronization protocols do not experience the aforementioned problem, they are not suitable for operation in large-scale WSNs owing to their slow convergence. In this study, we propose a hybrid time synchronization protocol (HTSP) to utilize the advantages of both approaches. In particular, HTSP exploits a temporary reference node to significantly reduce the convergence time and employs average-based consensus during normal operation to cope with node failures. It is notable that each node in HTSP can seamlessly switch between the reference and consensus modes during its operation. Moreover, HTSP is a fully distributed protocol as it does not rely on any leader-election mechanism. Our simulation results demonstrated that HTSP outperforms flooding, gradient, and virtual link-based time synchronization protocols considering various types of network topologies and scales. [ABSTRACT FROM AUTHOR]

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Abstract The Precise Point Positioning (PPP) technique uses a single Global Navigation Satellite System (GNSS) receiver to collect carrier-phase and code observations and perform centimeter-accuracy positioning together with the precise satellite orbit and clock corrections provided. According to the observations used, there are basically two approaches, namely, the ionosphere-free combination approach and the raw observation approach. The former eliminates the ionosphere effects in the observation domain, while the latter estimates the ionosphere effects using uncombined and undifferenced observations, i.e., so-called raw observations. These traditional techniques do not fix carrier-phase ambiguities to integers, if the additional corrections of satellite hardware biases are not provided to the users. To derive the corrections of hardware biases in network side, the ionosphere-free combination operation is often used to obtain the ionosphere-free ambiguities from the L1 and L2 ones produced even with the raw observation approach in earlier studies. This contribution introduces a variant of the raw observation approach that does not use any ionosphere-free (or narrow-lane) combination operator to derive satellite hardware bias and compute PPP ambiguity float and fixed solution. The reparameterization and the manipulation of design matrix coefficients are described. A computational procedure is developed to derive the satellite hardware biases on WL and L1 directly. The PPP ambiguity-fixed solutions are obtained also directly with WL/L1 integer ambiguity resolutions. The proposed method is applied to process the data of a GNSS network covering a large part of China. We produce the satellite biases of BeiDou, GPS and Galileo. The results demonstrate that both accuracy and convergence are significantly improved with integer ambiguity resolution. The BeiDou contributions on accuracy and convergence are also assessed. It is disclosed for the first time that BeiDou only ambiguity-fixed solutions achieve the similar accuracy with that of GPS/Galileo combined, at least in mainland China. The numerical analysis demonstrates that the best solutions are achieved by GPS/Galileo/BeiDou solutions. The accuracy in horizontal components is better than 6 mm, and in the height component better than 20 mm (one sigma). The mean convergence time for reliable ambiguity-fixing is about 1.37 min with 0.12 min standard deviation among stations without using ionosphere corrections and the third frequency measurements. The contribution of BDS is numerically highlighted.

This article studies the output regulation of discrete-time multi-agent systems with an unknown model by a finite-time optimal control algorithm based on Q-learning that uses the method of the linear quadratic regulator (LQR). The algorithm uses the Bellman optimality principle to deduce the Q-function under global optimality. It obtains the distributed optimal control law that minimizes the value of Q-function by policy iteration. Through local communication among agents, the optimal global control of each agent’s output can be realized without relying on the dynamic model of the system. Secondly, by designing a novel finite-time local error formula, the output regulation synchronization time is reduced by 50%. Finally, a MATLAB simulation example shows the capability of the nominated algorithm.

We study a recently discovered metric invariant - the center of distances. The center of distances of a nonempty subset A of a metric space (X , d) is defined by S (A) : = { α ∈ [ 0 , + ∞) : ∀ x ∈ A ∃ y ∈ A d (x , y) = α } . Given a nonincreasing sequence (a n) of positive numbers converging to 0, the set E (a n) : = x ∈ R : ∃ A ⊂ N x = ∑ n ∈ A a n is called the achievement set of the sequence (a n) . This new invariant is particularly useful in investigating achievability of sets on the real line. We concentrate on computing the centers of distances of central Cantor sets. Any central Cantor set C is an achievement set of exactly one fast convergent series ∑ a n , and consequently S (C) ⊃ 0 ∪ a n : n ∈ N . We try to check which central Cantor sets have the minimal possible center of distances and which have not. [ABSTRACT FROM AUTHOR]

Stochastic Gradient Descent (SGD) series optimization methods play the vital role in training neural networks, attracting growing attention in science and engineering fields of the intelligent system. The choice of learning rates affects the convergence rate of SGD series optimization methods. Currently, learning rate adjustment strategies mainly face the following problems: (1) The traditional learning rate decay method mainly adopts manual manner during training iterations, the small learning rate produced from which causes slow convergence in training neural networks. (2) Adaptive method (e.g., Adam) has poor generalization performance. To alleviate the above issues, we propose a novel automatic learning rate decay strategy for SGD optimization methods in neural networks. On the basis of the observation that the convergence rate's upper bound enjoys minimization in a specific iteration concerning the current learning rate, we first present the expression of the current learning rate determined by historical learning rates. And merely one extra parameter is initialized to generate automatic decreasing learning rates during the training process. Our proposed approach is applied to SGD and Momentum SGD optimization algorithms, and concrete theoretical proof explains its convergence. Numerical simulations are conducted on the MNIST and Cifar‐10 data sets with different neural networks. Experimental results show that our algorithm outperforms existing classical ones, achieving faster convergence rate, better stability, and generalization performance in neural network training. It also lays a foundation for large‐scale parallel search of initial parameters in intelligent systems. [ABSTRACT FROM AUTHOR]

Precise point positioning (PPP) with augmentation by a network of continuously operating reference stations (CORS), known as PPP-RTK, enables fast ambiguity resolution with PPP. However, it suffers from the dependency of a CORS network for augmentation corrections generation. This paper proposes a new real-time PPP-RTK method that generates the augmentation corrections using undifferenced and uncombined PPP at a single reference station. This method enables high accuracy and fast convergence real-time PPP-RTK for both multi-frequency and single-frequency receivers without the need of PPP estimate convergence at the reference station. It provides potential flexibility to include any number of more local reference stations for augmentation correction generation. The method can be regarded as a state-space-representation based RTK approach that offers equivalent performance but can overcome the limitations associated with the observation-space-representation based positioning techniques, such as the high transmission rate and rigorous baseline length requirements. It can also be considered as a special case of the network-based PPP-RTK but it offers increased flexibility in system implementation and opens doors to develop new correction services for PPP-RTK applications. Field tests results validate the positioning performance of the proposed method. The static positioning test indicates that the method could instantly output cm-level positioning solutions even before the corrections estimate convergence at the reference station. And the kinematic vehicle-based test reveals that, the method can provide the rover station receiver cm-level positioning solutions with instant initialization and ambiguity resolution either using a dual-frequency Global Navigation Satellite System (GNSS) receiver or a low-cost single-frequency GNSS receiver with 100% ambiguity fix rate. Also, the test results with different baseline lengths prove that the proposed method will be gradually degraded into standard AR-enabled PPP when the baseline length grows longer, and the ambiguity can be effectively fixed even at the 1258 km baseline length. [ABSTRACT FROM AUTHOR]

SIGNSGD is able to dramatically improve the performance of training large neural networks by transmitting the sign of each minibatch stochastic gradient, which achieves gradient communication compression and keeps standard stochastic gradient descent (SGD) level convergence rate. Meanwhile, the learning rate plays a vital role in training neural networks, but existing learning rate optimization strategies mainly face the following problems: (1) for learning rate decay method, small learning rates produced lead to converge slowly, and extra hyper-parameters are required except for the initial learning rate, causing more human participation. (2) Adaptive gradient algorithms have poor generalization performance and also utilize other hyper-parameters. (3) Generating learning rates via two-level optimization models is difficult and time-consuming in training. To this end, we propose a novel adaptive learning rate schedule for neural network training via SIGNSGD optimizer for the first time. In our method, based on the theoretical inspiration that the convergence rate's upper bound has minimization with the current learning rate in each iteration, the current learning rate can be expressed by a mathematical expression that is merely related to historical learning rates. Then, given an initial value, learning rates in different training stages can be adaptively obtained. Our proposed method has following advantages: (1) it is a novel automatic method without additional hyper-parameters except for one initial value, thus reducing the manual participation. (2) It has faster convergence rate and outperforms the standard SGD. (3) It makes neural networks achieve better performance with fewer gradient communication bits. Three numerical simulations are conducted on different neural networks with three public datasets: MNIST, Cifar-10 and Cifar-100 datasets, and several numerical results are presented to demonstrate the efficiency of our proposed approach. [ABSTRACT FROM AUTHOR]

Among the flooding-based time synchronization protocols in wireless sensor networks, rapid-flooding outperforms slow-flooding in terms of synchronization speed. However, sometimes it is not realistic to apply rapid-flooding in dense or low-duty cycle networks due to its operational constraint that timing messages must be forwarded immediately after being received. On the contrary, slow-flooding protocols are free from this operational constraint, but they have a fundamental limitation that their convergence time increases with the size of the network. In this letter, we prove that this limitation can be overcome by sharing the compensated skew and timing information to the next hop nodes so that they can all be synchronized in the same round. The evaluation results demonstrate that we can achieve a comparable convergence speed of rapid-flooding protocols with the slow-flooding-based mechanism and can provide better accuracy for network-wide and per-hop synchronization. [ABSTRACT FROM AUTHOR]

In this work, we find that the matrix representing the curl–curl operator in a partial differential equation solver of Maxwell’s equations can be analytically decomposed into a gradient divergence operator and a Laplacian, both of which can be constructed from the mesh information without any need for computation. The curl–curl operator can hence be replaced by the Laplacian to find the divergence-free component of the field solution. The Laplacian is positive definite and well-conditioned. As a result, the convergence of an iterative solution of Maxwell’s equations can be guaranteed, and also significantly accelerated. Based on the finding, we represent the divergence-free component of the unknown field solution by deducting its curl-free component. The curl-free component resides in the nullspace of the curl–curl operator, which is also analytically known from the mesh information no matter it is a regular grid or an unstructured mesh. After the divergence-free component is rapidly solved from a Laplacian counterpart of the original system matrix, the curl-free component can also be solved from a Laplacian matrix, and hence having fast and guaranteed convergence. The total computational cost of the proposed method is simply a small number of sparse matrix-vector multiplications. The proposed method has been successfully applied to solve ill-conditioned on-chip, packaging, and antenna radiation problems at both low and high frequencies, involving both inhomogeneous dielectrics and lossy conductors. Numerical experiments have demonstrated its fast and guaranteed convergence, as well as trivial computational cost independent of problem size. [ABSTRACT FROM AUTHOR]

A frequency diverse array (FDA) multiple‐input multiple‐output (MIMO) radar employs a small frequency increment across transmit elements to produce an angle‐range‐dependent beampattern for target angle and range detection. The joint angle and range estimation problem is a trilinear model. The traditional trilinear alternating least square (TALS) algorithm involves high computational load due to excessive iterations. We propose a fast‐convergence trilinear decomposition (FC‐TD) algorithm to jointly estimate FDA‐MIMO radar target angle and range. We first use a propagator method to obtain coarse angle and range estimates in the data domain. Next, the coarse estimates are used as initialized parameters instead of the traditional TALS algorithm random initialization to reduce iterations and accelerate convergence. Finally, fine angle and range estimates are derived and automatically paired. Compared to the traditional TALS algorithm, the proposed FC‐TD algorithm has lower computational complexity with no estimation performance degradation. Moreover, Cramér‐Rao bounds are presented and simulation results are provided to validate the proposed FC‐TD algorithm effectiveness.

In this brief, a heterogeneous diffusion-based communication protocol is proposed to achieve a significant improvement in the rate of convergence of the average consensus among the multi-agents. We prove that using the proposed communication protocol, star network, complete network, complete bipartite, $n-$ cube hypergraph, $d-$ regular bipartite graph, and $d-$ regular Ramanujan graph has a faster consensus as compared to the considered state of the art consensus protocol on these networks. Numerical simulations also show the validity of the results on real networks and the model networks. [ABSTRACT FROM AUTHOR]

This brief proposes a signal-independent background calibration in pipeline-SAR analog-to-digital converters (ADCs) with a convergence-accelerated technique. To achieve signal independence, an auxiliary capacitor array CA is introduced to pre-inject a pseudo-random noise (PN) in the sampling phase to cancel out the opposite PN injection of the calibrated capacitor in the conversion phase, and CA is also used to realize the D/A function of the calibrated capacitor in the conversion phase. In this way, no matter what the signal is, the residue headroom remains unchanged even with PN injection. Moreover, the first sub-ADC is designed with extended conversion bits to quantize its own residue after delivering the conversion bits required by the first stage. Afterwards, this result is provided to the calibration algorithm to reduce the signal component and accelerate the convergence. Based on the simulation, the signal-to-noise and distortion ratio (SNDR) and spur-free dynamic range (SFDR) improve from 45.3 dB and 56.4 dB to 68.2 dB and 88.4 dB, respectively, after calibration. In addition, with the acceleration technique, convergence cycles decrease from 1.7 × 108 to 5.8 × 106. Moreover, no matter whether the input signal is DC, sine wave or band-limited white noise, the calibration all works normally.

In this paper, a visual servoing robotic control scheme of fast convergence and calibration-free is proposed. The objective of this paper is to achieve a better convergence around the equilibrium for uncalibrated visual servoing systems. Moreover, there are threefold unknown parameters, the visual kinematic parameters, the dynamic parameters and the parameters of the feature points, considered in the design of the proposed control scheme. In order to achieve the above control objective, the finite-time tracking controller and three adaptive laws are proposed. The adaptability to the unknown parameters is guaranteed by the online adaptive laws. The finite-time convergence is achieved by the continuously non-smooth fractional order function in the controller. The rigorously mathematic proof of stability is given by homogeneous theory and the Lyapunov function formalism. Three real-time experiments are conducted to demonstrate the practical performance of the proposed scheme.

Firefly Algorithm (FA) is one of the most recently introduced stochastic, nature-inspired, meta-heuristic approaches used for solving optimization problems. The conventional FA use randomization factor during generation of solution search space and fireflies position changing, which results in imbalanced relationship between exploration and exploitation. This imbalanced relationship causes in incapability of FA to find the most optimum values at termination stage. In the proposed model, this issue has been resolved by incorporating PS at the termination stage of standard FA. The optimized values obtained from the FA are set as the initial starting points for the PS algorithm and the values are further optimized by PS to get the most optimal values or at least better values than the values obtained by conventional FA during its maximum number of iterations. The performance of the newly developed FA-PS model has been tested on eight minimization functions and six maximization functions by considering various performance evaluation parameters. The results obtained have been compared with other optimization algorithms namely genetic algorithm (GA), standard FA, artificial bee colony (ABC), ant colony optimization (ACO), differential equations (DE), bat algorithm (BA), grey wolf optimization (GWO), Self-Adaptive Step Firefly Algorithm (SASFA), and FA-Cross algorithm in terms of convergence rate and various numerical performance evaluation parameters. A significant improvement has been observed in the solution quality by embedding PS in the standard FA at the termination stage. The result behind this improvement is the better exploration and exploitation of the solution search space at this stage.

In this paper, we focus on the transition control of a ducted fan vertical take-off and landing (VTOL) unmanned aerial vehicle (UAV). To achieve a steady transition from hover to high-speed flight, a neural-networks-based controller is proposed to learn the system dynamics and compensate for the tracking error between the aircraft dynamics and the desired dynamic performance. In prior, we derive the nonlinear system model of the aircraft full-envelope dynamics. Then, we propose a novel neural-networks-based control scheme and apply it on the underactuated aircraft system. Key features of the proposed controller consist of projection operator, state predictor and dynamic-formed adaptive input. It is proved and guaranteed that the tracking errors of both state predictor and neural-networks weights are upper bounded during the whole neural-networks learning procedure. The very adaptive input is formed into a dynamic structure that helps achieve a reliable fast convergence performance of the proposed controller, especially in high-frequency disturbance conditions. Consequently, the closed-loop system of the aircraft is able to track a certain trajectory with desired dynamic performance. Satisfactory results are obtained from both simulations and practical flight test in accomplishing the designed flight course.

High‐precision ionospheric correction is an important prerequisite for undifferenced and uncombined precise point positioning (PPP) with fast convergence capability. Different from the traditional use of pseudoranges to extract ionospheric information, the ionospheric accuracy obtained by PPP with fixed ambiguity is higher. Methods: A wide ‐ area vertical total electron content(VTEC) ionospheric model is developed based on a reference network, where the VTEC of each station is extracted from the undifferenced and uncombined PPP approach. The proposed VTEC model is expressed in the between‐satellite single ‐differenced polynomial function of the sites' latitude and solar hour angle. A data processing procedure and software system are developed for both processing server and user PPP client. In the server part, the VTEC ionospheric delay of the whole network is estimated, the VTEC polynomial function is established and the model parameters are broadcasted to the PPP clients. In the user PPP client, real‐time positioning using the broadcasted wide‐area VTEC parameters are performed.Results: Experiments using the GNSS network in the European region are performed, and results show that the internal RMSE （root mean squared error） and external RMSE of the derived VTEC model is around 3 TECus and 1 TECu for the GLONASS （global navigation satellite system） and GPS, respectively. User kinematic PPP test results using the new VTEC ionospheric model show that, 78 of the whole 136 (57.35%) kinematic undifferenced and uncombined PPP hourly samples converge in 5 minutes, where for 97 samples (71.32%) converge in 10 minutes, 122 samples （89.7%） converge in 15 minutes and 132 samples （97.06%） converge without 25 minutes. In the unconstrained PPP experiment, the results in the above convergence time are 15 (11.03%), 64 (47.06%), 91 (66.91%), and 110 (80.88%), respectively. Conclusions: Experiments show that single‐difference VTEC ionospheric model has a huge improvement in the accuracy of extracting the ionosphere and shortening the real‐time PPP convergence time in a wide‐area. [ABSTRACT FROM AUTHOR]

The generalized linear models (GLMs) are widely used in statistical analysis and the related design issues are undoubtedly challenging. The state‐of‐the‐art works mostly apply to design criteria on the estimates of regression coefficients. The prediction accuracy is usually critical in modern decision‐making and artificial intelligence applications. It is of importance to study optimal designs from the prediction aspects for GLMs. In this work, we consider Elastic I‐optimality as a prediction‐oriented design criterion for GLMs, and develop an efficient algorithm for such EI‐optimal designs. By investigating theoretical properties for the optimal weights of any set of design points and extending the general equivalence theorem to the EI‐optimality for GLMs, the proposed efficient algorithm adequately combines the Fedorov–Wynn algorithm and the multiplicative algorithm. It achieves great computational efficiency with guaranteed convergence. Numerical examples are conducted to evaluate the feasibility and computational efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

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A frequency diverse array (FDA) multiple‐input multiple‐output (MIMO) radar employs a small frequency increment across transmit elements to produce an angle‐range‐dependent beampattern for target angle and range detection. The joint angle and range estimation problem is a trilinear model. The traditional trilinear alternating least square (TALS) algorithm involves high computational load due to excessive iterations. We propose a fast‐convergence trilinear decomposition (FC‐TD) algorithm to jointly estimate FDA‐MIMO radar target angle and range. We first use a propagator method to obtain coarse angle and range estimates in the data domain. Next, the coarse estimates are used as initialized parameters instead of the traditional TALS algorithm random initialization to reduce iterations and accelerate convergence. Finally, fine angle and range estimates are derived and automatically paired. Compared to the traditional TALS algorithm, the proposed FC‐TD algorithm has lower computational complexity with no estimation performance degradation. Moreover, Cramér‐Rao bounds are presented and simulation results are provided to validate the proposed FC‐TD algorithm effectiveness. [ABSTRACT FROM AUTHOR]

Since the initial publication of Kelly et al.'s seminal paper on resource allocation in wired networks, many studies based on the cross-layer design philosophy have been conducted in both wired and wireless networks. The Lagrangian duality technique has been widely adopted to solve the cross-layer optimization problem, but it is slowly convergent and sensitive to the iterative step size, especially for large networks. In this paper, a joint congestion control and power allocation second-order algorithm (JCCPA) is proposed, in which the joint optimization problem is modeled as a network utility maximization (NUM) framework, and the primal-dual interior-point method is used to solve the NUM problem. The JCCPA updates the primal variables and dual variables simultaneously according to their Newton directions; as a result,resulting in a faster convergence speed is reached. Moreover, the matrix-splitting technology is utilized to decompose the computation of the Hessian matrix and its inverse into different nodes and links so that the distributed update of source rate and node power can be implemented. The simulation results demonstrate that the JCCPA not only significantly improves energy efficiency but also has a faster convergence speed and is insensitive to step size.