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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]

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]

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]

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]

Recovering 6D object pose has gained much focus, because of its application in robotic intelligent manipulation to name but a few. This paper presents an approach for 6D object pose refinement from noisy depth images obtained from a consumer depth sensor. Compared to the state of the art aimed at the same goal, the proposed method has high precision, high robustness to partial occlusions and noise, low computation cost and fast convergence. This is achieved by using an iterative scheme that only employs Random Forest to minimize a cost function of object pose which can quantify the misalignment between the ground truth and the estimated one. The random forest in our algorithm is learnt only using synthetic depth images rendered from 3D model of the object. Several experimental results show the superior performance of the proposed approach compared to ICP-based algorithm and optimization-based algorithm, which are generally used for 6D pose refinement in depth images. Moreover, the iterative process of our algorithm can be much faster than the state of the art by only using one CPU core. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of two-dimensional (2D) direction of arrival (DOA) estimation of multiple signals for generalized coprime planar arrays consisting of two rectangular uniform planar subarrays. We propose a fast-convergence trilinear decomposition approach, which uses propagator method (PM) as the initialization of the angle estimation to speed the convergence of trilinear decomposition. The received signal of each subarray can be fitted into a trilinear model or parallel factor (PARAFAC) model so that the trilinear alternating least square algorithm can be used to estimate the angle information. Meanwhile, the necessary initialization of DOA estimates can be achieved via PM, which endows the proposed approach a fast convergence and subsequently results in a low complexity. Specifically, we eliminate the ambiguous estimates by utilizing the coprime property and the true DOA estimates can be achieved by selecting the nearest ones of all DOA estimates. The proposed approach can obtain the same estimation performance as the conventional PARAFAC algorithm, but with a low computational cost. Numerical simulation results are provided to validate the effectiveness and superiority of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this work, we study the convergence of an efficient iterative method, the fast sweeping method (FSM), for numerically solving static convex Hamilton-Jacobi equations. First, we show the convergence of the FSM on arbitrary meshes. Then we illustrate that the combination of a contraction property of monotone upwind schemes with proper orderings can provide fast convergence for iterative methods. We show that this mechanism produces different behavior from that for elliptic problems as the mesh is refined. An equivalence between the local solver of the FSM and the Hopf formula under linear approximation is also proved. Numerical examples are presented to verify our analysis. [ABSTRACT FROM AUTHOR]

In this paper, a new finite-time dynamic surface control (DSC) method is proposed to enhance the convergence rate and robustness of a DSC system on the basis of the traditional infinite-time stability viewpoint for application to nonlinear large-scale systems. By defining a novel recursive finite-time virtual error variable, a virtual controlled stabilizing functions are derived, and the finite-time control performance is ensured by means of the finite-timebased Lyapunov stability theorem. An adaptive fuzzy logic system is adopted to approximate the unknown functions. The applicability of the proposed control strategy is verified by simulation and an experiment for a mobile manipulator and an articulated manipulator. [ABSTRACT FROM AUTHOR]