Convergence Rate Research Articles

A Note on the Distribution of the Extreme Degrees of a Random Graph via the Stein-Chen Method

We offer an alternative proof, using the Stein-Chen method, of Bollobás’ theorem concerning the distribution of the extreme degrees of a random graph. Our proof also provides a rate of convergence of the extreme degree to its asymptotic distribution. The same method also applies in a more general setting where the probability of every pair of vertices being connected by edges depends on the number of vertices.

Legendre-Galerkin spectral algorithm for fractional-order BVPs: Application to the Bagley-Torvik equation

<p>Herein, we provide an efficient spectral Galerkin algorithm, according to a special type of shifted Legendre basis for finding a semi-analytic solution to the Liouville-Caputo fractional boundary value problem. The algorithm’s main goal is to transform the fractional differential problem into a linear system with efficiently invertible, well-structured matrices. The convergence rates of the algorithm are carefully obtained as well as the error bound.</p>

Innovative Mathematical Insights through Artificial Intelligence (AI): Analysing Ramanujan Series and the Relationship between e and π\pi

This paper presents a comprehensive case study on the application of Artificial Intelligence (AI) in mathematics, focusing on the Ramanujan series and the intricate relationship between the mathematical constants e and π. The study explores how AI, particularly machine learning and pattern recognition techniques, can be harnessed to discover new mathematical series and patterns, thereby extending the pioneering work of the legendary mathematician Srinivasa Ramanujan. The paper begins with an overview of the Ramanujan series, illustrating their significance and applications in mathematical computations. It then delves into the specifics of AI methodologies employed to unearth new series for e and π, highlighting the algorithms and models used. Through detailed analysis and experimentation, the study demonstrates how AI can generate new series expansions for e and π, offering enhanced convergence rates and computational efficiencies. Furthermore, the paper examines the relationship between these two constants, providing insights into their interconnected nature through AI-discovered series and patterns. Practical applications of these new series in fields such as numerical methods, cryptography, and theoretical physics are also discussed. By showcasing the successful integration of AI in the realm of mathematical research, this case study underscores the potential of AI to revolutionize traditional mathematical approaches, fostering the discovery of new knowledge and the refinement of existing theories. The findings contribute to a deeper understanding of the interplay between e and π, reinforcing the profound impact of Ramanujan's work in modern mathematics and the transformative power of AI in advancing this legacy.

EMG-YOLO: road crack detection algorithm for edge computing devices.

Road cracks significantly shorten the service life of roads. Manual detection methods are inefficient and costly. The YOLOv5 model has made some progress in road crack detection. However, issues arise when deployed on edge computing devices. The main problem is that edge computing devices are directly connected to sensors. This results in the collection of noisy, poor-quality data. This problem adds computational burden to the model, potentially impacting its accuracy. To address these issues, this paper proposes a novel road crack detection algorithm named EMG-YOLO. First, an Efficient Decoupled Header is introduced in YOLOv5 to optimize the head structure. This approach separates the classification task from the localization task. Each task can then focus on learning its most relevant features. This significantly reduces the model's computational resources and time. It also achieves faster convergence rates. Second, the IOU loss function in the model is upgraded to the MPDIOU loss function. This function works by minimizing the top-left and bottom-right point distances between the predicted bounding box and the actual labeled bounding box. The MPDIOU loss function addresses the complex computation and high computational burden of the current YOLOv5 model. Finally, the GCC3 module replaces the traditional convolution. It performs global context modeling with the input feature map to obtain global context information. This enhances the model's detection capabilities on edge computing devices. Experimental results show that the improved model has better performance in all parameter indicators compared to current mainstream algorithms. The EMG-YOLO model improves the accuracy of the YOLOv5 model by 2.7%. The mAP (0.5) and mAP (0.9) are improved by 2.9% and 0.9%, respectively. The new algorithm also outperforms the YOLOv5 model in complex environments on edge computing devices. The EMG-YOLO algorithm proposed in this paper effectively addresses the issues of poor data quality and high computational burden on edge computing devices. This is achieved through optimizing the model head structure, upgrading the loss function, and introducing global context modeling. Experimental results demonstrate significant improvements in both accuracy and efficiency, especially in complex environments. Future research can further optimize this algorithm and explore more lightweight and efficient object detection models for edge computing devices.

Efficient High-Order Space-Angle-Energy Polytopic Discontinuous Galerkin Finite Element Methods for Linear Boltzmann Transport

We introduce an hp-version discontinuous Galerkin finite element method (DGFEM) for the linear Boltzmann transport problem. A key feature of this new method is that, while offering arbitrary order convergence rates, it may be implemented in an almost identical form to standard multigroup discrete ordinates methods, meaning that solutions can be computed efficiently with high accuracy and in parallel within existing software. This method provides a unified discretisation of the space, angle, and energy domains of the underlying integro-differential equation and naturally incorporates both local mesh and local polynomial degree variation within each of these computational domains. Moreover, general polytopic elements can be handled by the method, enabling efficient discretisations of problems posed on complicated spatial geometries. We study the stability and hp-version a priori error analysis of the proposed method, by deriving suitable hp-approximation estimates together with a novel inf-sup bound. Numerical experiments highlighting the performance of the method for both polyenergetic and monoenergetic problems are presented.

Structural Variation Along the Southern Hikurangi Subduction Zone, Aotearoa New Zealand, From Seismic Reflection and Retro‐Deformation Analysis

AbstractThe southern Hikurangi subduction zone exhibits significant along‐strike variation in convergence rate and obliquity, sediment thickness and, uniquely, the increasing proximity of southern Hikurangi to, and impingement on, the incoming continental Chatham Rise, an ancient Gondwana accretionary complex. There are corresponding changes in the morphology and structure of the Hikurangi accretionary prism. We combine widely spaced multichannel seismic reflection profiles with high resolution bathymetry and previous interpretations to characterize the structure and the history of the accretionary prism since 2 Ma. The southern Hikurangi margin can be divided into three segments. A northeastern segment (A) characterized by a moderately wide (∼70 km), low taper (∼5°) prism recording uninhibited outward growth in the last ∼1 Myr. Deformation resolvable in seismic reflection data accounts for ∼20 % of plate convergence, comparable with the central Hikurangi margin further North. A central segment (B) characterized by a narrow (∼30 km), moderate taper (∼8°) prism, with earlier (∼2‐∼1 Ma) shortening than segment A. Outward prism growth ceased coincidentally with development of major strike‐slip faults in the prism interior, reduced margin‐normal convergence rate, and the onset of impingement on the incoming Chatham Rise to the south. A southwestern segment (C) marks the approximate southern termination of subduction but widens to ∼50 km due to rapid outward migration of the deformation front via fault reactivation within the now‐underthrusting corner of the Chatham Rise. Segment C exhibits minimal shortening as margin‐normal subduction velocity decreases and plate motion is increasingly taken up by interior thrusts and strike‐slip faults.

Improvements for matrix-type implicit temporal scheme regarding entropy fix and local time step

The matrix version of Symmetric Successive Over Relaxation (matrix-SSOR) scheme has been proved to be more efficient than the standard Lower-Upper Symmetric Gauss-Seidel (LU-SGS), but less robust for high-speed flows. In order to ulteriorly improve the convergence rate as well as numerical stability of matrix-SSOR, two improvements regarding entropy fix and local time step have been proposed and validated. Firstly, an augmented entropy fix method is imposed on the inviscid Jacobian matrix and proved to be effective in two high-speed flows, in which the key parameter in entropy fix is discussed and found to be insensitive within appropriate range of values. Since the time step also has great effects on the numerical stability and convergence rate, a modified cell residual adapted local time step method with consideration of the residual history is developed, which is found to be effective for increasing the convergence rate when the matrix-SSOR is applied, but invalid when the LU-SGS is used. The proposed modified local time step method is also insensitive to the key parameter within appropriate range of values. The two modifications can be conveniently implanted into analogous matrix-type implicit schemes to improve the numerical performance.

Using experience classification for training non-Markovian tasks

Unlike standard Reinforcement Learning (RL) model, many real-world tasks are non-Markovian, which requires long-term memory and dependency. Hence solving a non-Markovian task is more difficult and challenging than solving a Markovian one. In this paper, we propose a novel RL approach for training non-Markovian tasks expressed in temporal logic LTLf (Linear Temporal Logic over Finite Traces). To this end, an encoding of linear complexity from LTLf into MDPs (Markov Decision Processes) is introduced in order to take advantage of advanced RL algorithms. We further propose an experience classification method based on the automaton structure (theoretically equivalent to LTLf specifications). An automatic reward shaping technique and a prioritized experience replay mechanism are developed to cooperate with the classification method to improve the performance of RL algorithms. We provide empirical evaluations on two widely used benchmark problems, Waterworld and Cartpole, both augmented with complex non-Markovian tasks. The evaluations are conducted with respect to the metrics of training speed, policy quality, convergence rates, computational efficiency and scalability etc. The experimental results show that our approach achieves superior performance over other relevant studies, specially with an average improvement of 133% in convergence rate and a reduction of 11% in training time.

ConGCNet: Convex geometric constructive neural network for Industrial Internet of Things

The intersection of the Industrial Internet of Things (IIoT) and artificial intelligence (AI) has garnered ever-increasing attention and research interest. Nevertheless, the dilemma between the strict resource-constrained nature of IIoT devices and the extensive resource demands of AI has not yet been fully addressed with a comprehensive solution. Taking advantage of the lightweight constructive neural network (LightGCNet) in developing fast learner models for IIoT, a convex geometric constructive neural network with a low-complexity control strategy, namely, ConGCNet is proposed in this article via convex optimization and matrix theory, which enhances the convergence rate and reduces the computational consumption in comparison with LightGCNet. First, a low-complexity control strategy is proposed to reduce the computational consumption during the hidden parameters training process. Second a novel output weights evaluated method based on convex optimization is proposed to guarantee the convergence rate. Finally, The universal approximation property of ConGCNet is proved by the low-complexity control strategy and convex output weights evaluated method. Simulation results, encompassing four benchmark datasets and the real ore grinding process, demonstrate that ConGCNet effectively reduces computational consumption in the modelling process and improves the model’s convergence rate.

Shuffling-type gradient method with bandwidth-based step sizes for finite-sum optimization

Shuffling-type gradient method is a popular machine learning algorithm that solves finite-sum optimization problems by randomly shuffling samples during iterations. In this paper, we explore the convergence properties of shuffling-type gradient method under mild assumptions. Specifically, we employ the bandwidth-based step size strategy that covers both monotonic and non-monotonic step sizes, thereby providing a unified convergence guarantee in terms of step size. Additionally, we replace the lower bound assumption of the objective function with that of the loss function, thereby eliminating the restrictions on the variance and the second-order moment of stochastic gradient that are difficult to verify in practice. For non-convex objectives, we recover the last iteration convergence of shuffling-type gradient algorithm with a less cumbersome proof. Meanwhile, we also establish the convergence rate for the minimum iteration of gradient norms. Under the Polyak-Łojasiewicz (PL) condition, we prove that the function value of last iteration converges to the lower bound of the objective function. By selecting appropriate boundary functions, we further improve the previous sublinear convergence rate results. Overall, this paper contributes to the understanding of shuffling-type gradient method and its convergence properties, providing insights for optimizing finite-sum problems in machine learning. Finally, numerical experiments demonstrate the efficiency of shuffling-type gradient method with bandwidth-based step size and validate our theoretical results.

Least total logistic distance metric algorithm and its variable step-size version

Classical mean-square error (MSE)-based adaptive filtering algorithms are useful for noise-free inputs. However, if the input signal of the adaptive filter with such an adaptive filtering algorithm is corrupted by noise, it will suffer from serious degradation of steady-state performance. To address the problem, a new adaptive filtering algorithm is proposed in this work. This algorithm not only uses the total method to compensate the bias caused by noisy input but also minimizes the cost function based on logistic distance metric (LDM) to achieve robustness against impulsive noise. Compared with the existing algorithms, the proposed least total logistic distance metric (LTLDM) algorithm has less misalignment when the input signal is disturbed by noise and has good robustness to impulsive noise. This work also tackles the crucial trade-off between convergence rate and misalignment by developing a variable step-size for LTLDM. In addition, to give deeply insight into the stochastic behavior of the proposed LTLDM, its mean-square deviation is analyzed at steady state. The advantage of LTLDM and the accuracy of theoretical expressions are verified by extensive simulations.

Optimizing kernel width for new risk-sensitive loss: A generalized algorithmic approach

This paper outlines a novel approach to the development of adaptive filtering algorithms, which minimizes susceptibility to the effects of non-Gaussian measurement noise. Using the generalized Gaussian density (GGD) function as the kernel function, we developed a generalized kernel risk-sensitive loss (GKRSL) criterion by which to derive the generalized minimum kernel risk-sensitive loss (GMKRSL) algorithm. We also developed a variable kernel width algorithm for GMKRSL (VKW-GMKRSL) to address the trade-off between convergence rate and steady-state misalignment resulting from a constant kernel width. In simulations, GMKRSL and VKW-GMKRSL outperformed existing works in system identification performance.

GIJA:Enhanced geyser‐inspired Jaya algorithm for task scheduling optimization in cloud computing

AbstractTask scheduling optimization plays a pivotal role in enhancing the efficiency and performance of cloud computing systems. In this article, we introduce GIJA (Geyser‐inspired Jaya Algorithm), a novel optimization approach tailored for task scheduling in cloud computing environments. GIJA integrates the principles of the Geyser‐inspired algorithm with the Jaya algorithm, augmented by a Levy Flight mechanism, to address the complexities of task scheduling optimization. The motivation for this research stems from the increasing demand for efficient resource utilization and task management in cloud computing, driven by the proliferation of Internet of Things (IoT) devices and the growing reliance on cloud‐based services. Traditional task scheduling algorithms often face challenges in handling dynamic workloads, heterogeneous resources, and varying performance objectives, necessitating innovative optimization techniques. GIJA leverages the eruptive dynamics of geysers, inspired by nature's efficiency in channeling resources, to guide task scheduling decisions. By combining this Geyser‐inspired approach with the simplicity and effectiveness of the Jaya algorithm, GIJA offers a robust optimization framework capable of adapting to diverse cloud computing environments. Additionally, the integration of the Levy Flight mechanism introduces stochasticity into the optimization process, enabling the exploration of solution spaces and accelerating convergence. To evaluate the efficacy of GIJA, extensive experiments are conducted using synthetic and real‐world datasets representative of cloud computing workloads. Comparative analyses against existing task scheduling algorithms, including AOA, RSA, DMOA, PDOA, LPO, SCO, GIA, and GIAA, demonstrate the superior performance of GIJA in terms of solution quality, convergence rate, diversity, and robustness. The findings of GIJA provide a promising solution quality for addressing the complexities of task scheduling in cloud environments (95%), with implications for enhancing system performance, scalability, and resource utilization.

An error analysis for deep binary classification with sigmoid loss

Deep neural networks have demonstrated remarkable efficacy in diverse classification tasks. In this paper, we specifically focus on the predictive performance in deep binary classification problems with the sigmoid loss. Given that sigmoid loss is categorized as a non-convex and bounded loss function, it exhibits potential resilience against the disruptive impact of outlier noises. We first derive the convergence rate of the excess misclassification risk for deep ReLU neural networks with the sigmoid loss, a result that attains minimax optimality. To the best of our acknowledge, we are the first to derive the convergence rate for the sigmoid loss. Moreover, we extend our analysis to derive a faster convergence rate under margin assumptions. This achievement renders our findings comparable to those of commonly employed convex loss functions operating under analogous assumptions. Lastly, we undertake a comprehensive validation of the robustness inherent in the sigmoid loss across diverse datasets.

The marriage of operations research and reinforcement learning: Integration of NEH into Q-learning algorithm for the permutation flowshop scheduling problem

The permutation flowshop scheduling problem (PFSP) attracted much interest from the operations research (OR) community, resulting in various heuristic and metaheuristic methods over the past half-century. However, given the hard nature of the PFSP, efficient algorithms rely on the configuration of initial solutions and sophisticated heuristics. Combining OR and artificial intelligence (AI), this paper investigates the marriage of OR and reinforcement learning for the PFSP. A novel method integrating NEH into the Q-learning algorithm is proposed for solving the PFSP with makespan criterion. With the refinement of action selection by strengthening good actions and weakening bad actions, the proposed Q-NEH algorithm shows a better learning and convergence rate, and outperforms the Q-learning algorithm for all 120 instances in Taillard’s benchmark dataset, with a significantly decreased average relative learning error (RLE) from 13.05 % to 1.73 %. Furthermore, by comparing the performances of the Q-NEH algorithm with related state-of-art benchmark algorithms in a wide range set of instances, it further confirms the superiority and stability of the Q-NEH algorithm in statistics. With a total of 2570 independent tests in two phases of experimental evaluations, the superiority of the Q-NEH algorithm for the PFSP suggests that the integration of OR and AI is a promising research direction for solving complex scheduling problems.

Application of uniform experimental design theory to multi-strategy improved sparrow search algorithm for UAV path planning

The sparrow search algorithm (SSA) is a meta-heuristic optimization algorithm based on the predatory behavior of sparrows. However, SSA tends to fall into the local optimum when solving optimization problems with complex constraints. To improve its optimization efficiency and overall performance, this paper develops a multi-strategy improved SSA (ISSA) based on uniform experimental design theory. Specifically, the wrap-around L2-discrepancy (WD), as a uniformity metric for uniform design of experiments, is fully utilized with macro-regulation, adaptive dynamic management strategy, and boundary redistribution management mechanism to quantify the population uniformity in each iteration of ISSA. Inspired by the concept of uniform design, WD is initially adopted to gauge population uniformity, and the threshold acceptance (TA) algorithm is employed to produce an initial population with improved uniformity, hence augmenting the population’s diversity and quickening the rate of convergence. Secondly, a macroscopical individual iterative strategy of the producer is adjusted to avoid the population converging to the origin. Then, a dynamic population uniformity affiliation function based on WD is introduced to adjust the population uniformity affiliation function according to the relative amount of change in the global optimum, and the number of danger perceivers is adjusted according to the affiliation function. What’s more, a new boundary update strategy is also proposed in ISSA based on population uniformity. By comparing ISSA on 23 standard test functions and recently updated validation function set CEC2022 with the original SSA, and some classical as well as newly developed algorithms, the superiority of the present ISSA is thoroughly confirmed. As an application case, the ISSA algorithm is utilized to solve the path planning problem of the complex environment of unmanned aerial vehicle (UAV) based on threat models by applying it to 2D maps containing circular and polygonal obstacles, as well as 3D maps containing mountain peaks and cylindrical obstacles. The simulation results show that ISSA can find more effective routes through various environments with obstacles.

Application of an efficient analytical technique based on Aboodh transformation to solve linear and non-linear dynamical systems of integro-differential equations

In literature, it is usually very difficult to investigate the analytical and numerical solutions of fractional integro-differential equations (FIDEs). In the current work, the solutions to linear and non-linear FIDEs and their systems have been analyzed by using the Aboodh transform decomposition method (ATDM). The Aboodh transformation is first utilized to simplify the given problem, and then the decomposition method is implemented to obtain the required results. The Adomian polynomials and Daftardar-Jafari polynomials are used to control the non-linear term in the system. The obtained results are compared with both polynomials and with different fractional orders. We show that Daftardar-Jafari polynomials work more accurately for the proposed method than Adomian polynomials. The graphical and tabular representations are made to show the effectiveness of the proposed technique. The accuracy of ATDM is remarkable while using the proposed technique. It is observed that ATDM solutions very closed to the exact solutions of the problems. ATDM has a lower computation cost and higher rate of convergence and thus can be used to solve other non-linear fractional order integro-differential equations and their systems. Finally, we propose some future research directions by utilising ATDM.

Quantitative relaxation towards equilibrium for solutions to the Boltzmann-Fermi-Dirac equation with cutoff hard potentials

We provide the first quantitative result of convergence to equilibrium in the context of the spatially homogeneous Boltzmann-Fermi-Dirac equation associated with hard potentials interactions under angular cut-off assumption, providing an explicit – algebraic – rate of convergence to Fermi-Dirac steady solutions. This result complements the quantitative convergence result of [15] and is based upon new uniform-in-time-and-εL∞ bound on the solutions.

High order numerical method for a subdiffusion problem

We consider a subdiffusive fractional differential problem characterized by an equation that incorporates a time Riemann-Liouville fractional derivative of order 1−α, α∈(0,1), on its right-hand side, while the diffusive coefficient is allowed to vary with both space and time. An high order numerical method for the subdiffusion problem is derived based on the fractional splines of degree β∈(1,2]. The main purpose of this work is to apply fractional splines for approximating the fractional integral in the definition of the Riemann-Liouville fractional derivative, and hence explain how to solve the subdiffusion problem using this approach. It is discussed how the rate of convergence of the numerical method depends on the solution, the degree of the spline and the order of the fractional derivative.

Zero dissipation limit of the anisotropic Boussinesq equations with Navier-slip and Neumann boundary conditions

In this paper, we study the vanishing dissipation limit of the 2D anisotropic Boussinesq equations with the Navier-slip boundary condition for velocity field and the fixed flux boundary condition for temperature in the upper half plane. By constructing boundary layer correctors to compensate for the discrepancies between dissipative equations and non-dissipative equations at the boundary, we prove that the solutions of the anisotropic Boussinesq equations converge to the solutions of the non-dissipative Boussinesq equations in L2-norm. Particularly, we find that the anisotropic dissipation coefficients only affect the rate of convergence, which is different from the phenomenon of the Dirichlet problem of the anisotropic Boussinesq equations in Wang & Xu (2021).