Convergence Rate Research Articles

R-linear and strong convergence of Tseng methods for solving certain split and non-split problems

In the recent past, authors have fondly assumed the co-coercive condition for solving certain class of split inverse problems. Recently, some attempts have been made to relax this very strict condition, however, only few results exists in literature in such case. In this paper, we present some new Mann-type Tseng methods for solving split monotone variational inclusion and common fixed point problems for a finite collection of quasi-pseudocontractive operators. We prove strong convergence and R − linear convergence rate results of our methods, while the co-coerciveness property is dispensed with. Our methods incorporate the relaxation, inertial extrapolation and self-adaptive step size techniques to achieve better performance and faster convergence. With the aid of relevant numerical experiments, we showcase the performance profile of our methods in comparison with recent methods in literature.

Convergence rate for a regularized scalar conservation law

Convergence rate for a regularized scalar conservation law

On diagonally structured scheme for nonlinear least squares and data-fitting problems

Recently, structured nonlinear least-squares (NLS) based algorithms gained considerable emphasis from researchers; this attention may result from increasingly applicable areas of these algorithms in different science and engineering domains. In this article, we coined a new efficient structured-based NLS algorithm. We developed a diagonal Hessian-based formulation for solving NLS problems. We derived the quasi-Newton update based on a diagonal matrix scheme subject to a modified structured secant condition. Also, we show that the algorithm search direction satisfies a sufficient descent condition under some standard assumptions. Subsequently, we also prove the global convergence of the algorithm and then eventually show the linear convergence rate for strongly convex functions. Furthermore, we numerically experimented with the proposed algorithm on benchmark test functions available in the literature. Finally, in the scheme, we apply the method to solve some data-fitting problems.

Trend to Equilibrium for Run and Tumble Equations with Non-uniform Tumbling Kernels

We study the long-time behaviour of a run and tumble model which is a kinetic-transport equation describing bacterial movement under the effect of a chemical stimulus. The experiments suggest that the non-uniform tumbling kernels are physically relevant ones as opposed to the uniform tumbling kernel which is widely considered in the literature to reduce the complexity of the mathematical analysis. We consider two cases: (i) the tumbling kernel depends on the angle between pre- and post-tumbling velocities, (ii) the velocity space is unbounded and the post-tumbling velocities follow the Maxwellian velocity distribution. We prove that the probability density distribution of bacteria converges to an equilibrium distribution with explicit (exponential for (i) and algebraic for (ii)) convergence rates, for any probability measure initial data. To the best of our knowledge, our results are the first results concerning the long-time behaviour of run and tumble equations with non-uniform tumbling kernels.

Optimal convergence analysis of two RPC-SAV schemes for the unsteady incompressible magnetohydrodynamics equations

Abstract In this paper, we present and analyze two linear and fully decoupled schemes for solving the unsteady incompressible magnetohydrodynamics equations. The rotational pressure-correction (RPC) approach is adopted to decouple the system, and the recently developed scalar auxiliary variable (SAV) method is used to treat the nonlinear terms explicitly and keep energy stability. One is the first-order RPC-SAV-Euler and the other one is generalized Crank–Nicolson-type scheme: GRPC-SAV-CN. For the RPC-SAV-Euler scheme, both unconditionally energy stability and optimal convergence are derived. The new GRPC-SAV-CN is constructed and can be regarded as a parameterized scheme, which includes PC-SAV-CN when the parameter $\beta =0$ and RPC-SAV-CN when $\beta \in (0,\frac {1}{2}]$; see Algorithm 3.2. However, Jiang and Yang (Jiang, N. & Yang, H. (2021) SIAM J. Sci. Comput., 43, A2869–A2896) point out that the SAV method has low accuracy by several commonly tested benchmark flow problem when solving Navier–Stokes equations. To improve the accuracy, we added two stabilization $-\alpha _{1}\varDelta t\nu \varDelta (\widetilde {\textbf {u}}^{n+1}-{\textbf {u}}^{n})$ and $\alpha _{2}\varDelta t\sigma ^{-1}\mbox {curl}\mbox {curl} (\textbf {H}^{n+1}-\textbf {H}^{n})$ in the GRPC-SAV-CN scheme, which play decisive roles in giving optimal error estimates. The unconditionally energy stability of the proposed scheme is given. We prove that the PC-SAV-CN scheme has second-order convergence speed, and the RPC-SAV-CN one has 1.5-order convergence rate. Finally, some numerical examples are presented to verify the validity and convergence of the numerical schemes.

Gas engine CCHP system optimization: An energy, exergy, economic, and environment analysis and optimization based on developed northern goshawk optimization algorithm

This paper aims to enhance the design and operation of a Combined Cooling, Heating, and Power (CCHP) system utilizing a gas engine as the primary energy source for a residential building in China. An Energy, Exergy, Economic, and Environment (4E) analysis is employed to assess the system's performance and impact based on energy, exergy, economic, and environmental criteria. The effectiveness of the DNGO algorithm is evaluated on a case study site and compared with Northern Goshawk Optimization (NGO) and Genetic Algorithm (GA). The findings demonstrate that the DNGO algorithm identifies the optimal gas engine size of 130 kW. The algorithm's search capabilities are greatly enhanced by this unique blend, surpassing what traditional methods can offer. The DNGO algorithm brings several advantages, including unparalleled energy efficiency, reduced exergy destruction, and a substantial decrease in CO2 emissions. This not only supports environmental sustainability but also aligns with global standards. Economically, the algorithm enhances the performance of the CCHP system, evident through a reduced payback period and increased annual profit. Additionally, the algorithm's rapid convergence rate allows it to reach the optimal solution faster than its counterparts, making it advantageous for time-sensitive applications. Incorporating innovative methods like chaos theory, the DNGO algorithm effectively avoids local optima, enabling a broader search for the best solution. The utilization of Lévy flight further enhances the algorithm's ability to escape local optima and navigate the search space more efficiently. Additionally, swarm intelligence is employed to simulate the collective behavior of decentralized systems, aiding in problem-solving. This research represents a significant advancement in optimization techniques for CCHP systems and offers a fresh perspective to the field of swarm-based optimization algorithms.

New two-microphone simplified sub-band forward algorithm based on separated variable step-sizes for acoustic noise reduction

This paper presents an extended two-microphone adaptive filtering algorithm implemented on the sub-band simplified forward technique using new separated variable step-sizes parameters (denoted SSF-SVS: Simplified Sub-band Forward Separated Variable Step-sizes). This algorithm is proposed to reduce the acoustic noise and enhance the quality of estimated speech signal. Firstly, a sub-signal decomposition processing strategy is used to facilitating effective noise reduction with very fast convergence rate using two analysis filter banks. Secondly, a two-microphone simplified forward blind source separation (BSS) structure updated by the normalized least-mean-square algorithm (NLMS) for extracting the estimated speech signal. To guarantee a very fast convergence rate with high level of enhanced speech quality, thirdly we proposed new separated variable step-sizes parameters for controlling the adaptation of sub-band filters. The concept involves suggesting individual variable step-size parameter for each adaptive sub-filter, which are updated independently using the input sub-signal and the estimated output sub-signal. Experimental results confirm significant improvements in terms of convergence rate based on the mean square error criterion (MSE) in diverse acoustical environments. In other hand, the good performance of the proposed SSF-SVS algorithm in terms of speech quality is verified by on output signal-to-noise ratio (Output-SNR). The proposed algorithm compared with six two-microphone adaptive filtering algorithms in term of convergence rate, speech quality and computational complexity.

Optimal parameters identification for PEMFC using autonomous groups particle swarm optimization algorithm

The precise model for Polymer Electrolyte Membrane Fuel Cells (PEMFCs) is vital for simulation, control, and performance analysis of PEMFCs. It is crucial to accurately estimate the model parameters. Over the past decade, the extraction of unknown parameters of PEMFCs model is formulated as optimization problem and several meta-heuristic techniques have emerged to solve this problem. Despite the development of these techniques to tackle the problem, the slow convergence rate and susceptibility to being trapped in local minima are regarded as weaknesses of these methods. Additionally, because the PEMFC model is a nonlinear and complex model, not all optimization algorithms are suitable for solving it. This article introduces a novel approach that utilizes the Autonomous Groups Particle Swarm Optimization (AGPSO) algorithm for extracting precise values for uncertain parameters inherent in PEMFC model of 250Wstack and BCS-500W stack utilized significantly in the literature. The optimization problem's fitness function is formulated as the total squared errors (TSEs) between the voltage datasets measured and estimated. Three different versions of AGPSO algorithm are presented. Statistical analysis is performed on these three versions to assess their robustness, and the most robust version is identified. Furthermore, the performance of the most robust version is extensively tested and analyzed through a complete comparison with recent optimization algorithms findings from the updated state-of-the-art literature such as BO, QOBO, TGA, HHO and ASO. Further, a statistical analysis was carried out that ensured the reliability and robustness of the introduced AGPSO. The results highlight the efficacy and feasibility of AGPSO-based approach across all compared algorithms, demonstrating an enhancement in the accuracy of the PEMFC model. A two-phase sensitivity analysis, conducted through Monte Carlo Simulation (MCS), was employed to assess how variations in optimized parameters affect the objective function (TSEs). Furthermore, the uncertainty in the maximum power reference current is identified.

Hunger Games Search for the elucidation of gravity anomalies with application to geothermal energy investigations and volcanic activity studies

Abstract Recent metaheuristic approaches are extensively and intensively being implemented to the interpretation of gravity anomalies due to their superior advantages. We emphasize the application of Hunger Games Search (HGS), a newly established metaheuristic inspired by hunger-driven instincts and behavioral choices of animals, to elucidate gravity data for geothermal energy exploration and volcanic activity study. After recognizing the modal features of the objective function tailored and tuning the algorithm control parameters involved, HGS has been trial-tested on simulated data sets of different scenarios and finally experienced in two field cases from India and Japan. Notably, a second moving average strategy has been successfully integrated into the objective function to eradicate the regional component from observed responses. Post-inversion uncertainty appraisal tests have been further implemented to comprehend the reliability of solutions obtained. The solutions retrieved by HGS have been unbiasedly compared in terms of convergence rate, accuracy, stability, and robustness with the solutions of the commonly used particle swarm optimization algorithm. Based on the results accessed, the theoretical and field cases presented could be recuperated more precisely, stably, robustly, and coherently with the available geophysical, geological, and borehole verification, as HGS is able to better explore the model space without compromising its capability to efficiently approach the global minimum. This novel global optimization method can thus be considered as a promising tool in geothermal energy investigations and the study of volcanic activities.

Quasi-Interpolation on Chebyshev Grids with Boundary Corrections

Quasi-interpolation is a powerful tool for approximating functions using radial basis functions (RBFs) such as Gaussian kernels. This avoids solving large systems of equations as in RBF interpolation. However, quasi-interpolation with Gaussian kernels on compact intervals can have significant errors near the boundaries. This paper proposes a quasi-interpolation method with Gaussian kernels using Chebyshev points and boundary corrections to improve the approximation near the boundaries. The boundary corrections use a linear approximation of the function beyond the interval to estimate the truncation error and add correction terms. Numerical studies on test functions show that the proposed method reduces errors significantly near boundaries compared to quasi-interpolation without corrections, for both equally spaced and Chebyshev points. The convergence and accuracy with the boundary corrections are generally better with Chebyshev points compared to equally spaced points. The proposed method provides an efficient way to perform quasi-interpolation on compact intervals while controlling the boundary errors. This study introduces a novel approach to quasi-interpolation modification, which significantly enhances convergence rates and minimizes errors at boundary points, thereby advancing the methods for boundary approximation.

Improve robustness of machine learning via efficient optimization and conformal prediction

AbstractThe advance of machine learning (ML) systems in real‐world scenarios usually expects safe deployment in high‐stake applications (e.g., medical diagnosis) for critical decision‐making process. To this end, provable robustness of ML is usually required to measure and understand how reliable the deployed ML system is and how trustworthy their predictions can be. Many studies have been done to enhance the robustness in recent years from different angles, such as variance‐regularized robust objective functions and conformal prediction (CP) for uncertainty quantification on testing data. Although these tools provably improve the robustness of ML model, there is still an inevitable gap to integrate them into an end‐to‐end deployment. For example, robust objectives usually require carefully designed optimization algorithms, while CP treats ML models as black boxes. This paper is a brief introduction to our recent research focusing on filling this gap. Specifically, for learning robust objectives, we designed sample‐efficient stochastic optimization algorithms that achieves the optimal (or faster compared to existing algorithms) convergence rates. Moreover, for CP‐based uncertainty quantification, we established a framework to analyze the expected prediction set size (smaller size means more efficiency) of CP methods in both standard and adversarial settings. This paper elaborates the key challenges and our exploration towards efficient algorithms with details of background methods, notions for robustness measure, concepts of algorithmic efficiency, our proposed algorithms and results. All of them further motivate our future research on risk‐aware ML that can be critical for AI–human collaborative systems. The future work mainly targets designing conformal robust objectives and their efficient optimization algorithms.

An Improved Golden Jackal Optimization Algorithm Based on Mixed Strategies

In an effort to overcome the problems with typical optimization algorithms’ slow convergence and tendency to settle on a local optimal solution, an improved golden jackal optimization technique is proposed. Initially, the development mechanism is enhanced to update the prey’s location, addressing the limitation of just relying on local search in the later stages of the algorithm. This ensures a more balanced approach to both algorithmic development and exploration. Furthermore, incorporating the instinct of evading natural predators enhances both the effectiveness and precision of the optimization process. Then, cross-mutation enhances population variety and facilitates escaping from local optima. Finally, the crossbar strategy is implemented to change both the individual and global optimal solutions of the population. This technique aims to decrease blind spots, enhance population variety, improve solution accuracy, and accelerate convergence speed. A total of 20 benchmark functions are employed for the purpose of comparing different techniques. The enhanced algorithm’s performance is evaluated using the CEC2017 test function, and the results are assessed using the rank-sum test. Ultimately, three conventional practical engineering simulation experiments are conducted to evaluate the suitability of IWKGJO for engineering issues. The results obtained demonstrate the beneficial effects of the altered methodology and illustrate that the expanded golden jackal optimization algorithm has superior convergence accuracy and a faster convergence rate.

An evaluation of computational methods for aggregate data meta-analyses of diagnostic test accuracy studies

BackgroundA Generalized Linear Mixed Model (GLMM) is recommended to meta-analyze diagnostic test accuracy studies (DTAs) based on aggregate or individual participant data. Since a GLMM does not have a closed-form likelihood function or parameter solutions, computational methods are conventionally used to approximate the likelihoods and obtain parameter estimates. The most commonly used computational methods are the Iteratively Reweighted Least Squares (IRLS), the Laplace approximation (LA), and the Adaptive Gauss-Hermite quadrature (AGHQ). Despite being widely used, it has not been clear how these computational methods compare and perform in the context of an aggregate data meta-analysis (ADMA) of DTAs.MethodsWe compared and evaluated the performance of three commonly used computational methods for GLMM - the IRLS, the LA, and the AGHQ, via a comprehensive simulation study and real-life data examples, in the context of an ADMA of DTAs. By varying several parameters in our simulations, we assessed the performance of the three methods in terms of bias, root mean squared error, confidence interval (CI) width, coverage of the 95% CI, convergence rate, and computational speed.ResultsFor most of the scenarios, especially when the meta-analytic data were not sparse (i.e., there were no or negligible studies with perfect diagnosis), the three computational methods were comparable for the estimation of sensitivity and specificity. However, the LA had the largest bias and root mean squared error for pooled sensitivity and specificity when the meta-analytic data were sparse. Moreover, the AGHQ took a longer computational time to converge relative to the other two methods, although it had the best convergence rate.ConclusionsWe recommend practitioners and researchers carefully choose an appropriate computational algorithm when fitting a GLMM to an ADMA of DTAs. We do not recommend the LA for sparse meta-analytic data sets. However, either the AGHQ or the IRLS can be used regardless of the characteristics of the meta-analytic data.

Online identification and feed-forward compensation of nonlinear friction in servo system based on RBF neural network model

In this paper, an online identification and compensation method of nonlinear friction based on radial basis function (RBF) neural network model is proposed for the influence of nonlinear friction on machining accuracy in the low speed process of servo feed system of CNC machine tools. First, a three-layer single-input-output RBF neural network model is established for describing the nonlinear friction of servo feeding system. Second, the neural network online learning algorithm is improved based on adaptive gain, which improves the stability and accuracy of the algorithm. Finally, experiments were carried out on a three-axis milling machine to compensate the friction in the servo feed system in real time based on the online identification results. The results show that the method can effectively improve the online identification accuracy and convergence rate, and effectively improved the low-speed performance of the servo feed system.

Maximum span determination and optimal sizing of cable for improved performance of droop-controlled DC microgrid

DC microgrids are seen as smart solutions to interface DC loads and distributed energy resources (DERs). However, in DC microgrids, the placement of sources and loads as well as the size of cable impact the system’s span, regulation of voltage, and losses. This paper proposes novel algorithms to determine the maximum span and optimal size of cable for the improved performance of a droop-controlled DC microgrid. The proposed algorithms utilize an improved power flow analysis (IPFA) method based on Newton-Raphson technique to determine the maximum span and the optimal size of cable, enhancing the voltage regulation and reducing the cost. Additionally, the impacts of power rating, droop constants, and size of cable on the DC microgrid’s maximum span are investigated and reported. Owing to the intricate nature of the problem concerning the ideal size of cable for the droop-controlled DC microgrid, a heuristic optimization approach is employed. Further to enhance the rate of convergence and computational performance, an improved particle swarm optimization (IPSO) is also proposed. The constraint of keeping the bus voltage variations below the allowable voltage regulation limits is applied to the objective function of the optimization problem. In the case of a droop-controlled, DC microgrid having a specified configuration, the suggested algorithm can determine the ideal size of cable, guaranteeing both the least cost and enhanced voltage regulation. Comprehensive numerical and modelling results have been presented for a droop-controlled DC microgrid with different loads and DERs to verify the effectiveness of the proposed methods. Furthermore, the analysis reveals that configuration III of the DC microgrid with an optimal cable size of 19 mm2 lowers the absolute voltage regulation to as low as 1.06 V. The results of the detailed analysis validate the enhanced performance of the proposed algorithms and are highly useful to both system designers and consumers of the DC microgrids, eventually paving the way for the widespread use of DC microgrids in the future.

A note on ideal C$$^*$$-completions and amenability

AbstractFor a discrete group G, we consider certain ideals $$\mathcal {I}\subset c_0(G)$$ I ⊂ c 0 ( G ) of sequences with prescribed rate of convergence to zero. We show that the equality between the full group C$$^*$$ ∗ -algebra of G and the C$$^*$$ ∗ -completion $$\textrm{C}^*_{\mathcal {I}}(G)$$ C I ∗ ( G ) in the sense of Brown and Guentner (Bull. London Math. Soc. 45:1181–1193, 2013) implies that G is amenable.

On V-Geometric Ergodicity Markov Chains of the Two-Inertia Systems

This study employs the diffusion process to construct Markov chains for analyzing the common two-inertia systems used in industry. Two-inertia systems are prevalent in commonly used equipment, where the load is influenced by the coupling of external force and the drive shaft, leading to variations in the associated output states. Traditionally, the control of such systems is often guided by empirical rules. This paper examines the equilibrium distribution and convergence rate of the two-inertia system and develops a predictive model for its long-term operation. We explore the qualitative behavior of the load end at discrete time intervals. Our findings are applicable not only in control engineering, but also provide insights for small-scale models incorporating dual-system variables.

Slime Mould Algorithm Based on a Gaussian Mutation for Solving Constrained Optimization Problems

The slime mould algorithm may not be enough and tends to trap into local optima, low population diversity, and suffers insufficient exploitation when real-world optimization problems become more complex. To overcome the limitations of SMA, the Gaussian mutation (GM) with a novel strategy is proposed to enhance SMA and it is named as SMA-GM. The GM is used to increase population diversity, which helps SMA come out of local optima and retain a robust local search capability. Additionally, the oscillatory parameter is updated and incorporated with GM to set the balance between exploration and exploitation. By using a greedy selection technique, this study retains an optimal slime mould position while ensuring the algorithm’s rapid convergence. The SMA-GM performance was evaluated by using unconstrained, constrained, and CEC2022 benchmark functions. The results show that the proposed SMA-GM has a more robust capacity for global search, improved stability, a faster rate of convergence, and the ability to solve constrained optimization problems. Additionally, the Wilcoxon rank sum test illustrates that there is a significant difference between the optimization outcomes of SMA-GM and each compared algorithm. Furthermore, the engineering problem such as industrial refrigeration system (IRS), optimal operation of the alkylation unit problem, welded beam and tension/compression spring design problem are solved, and results prove that the proposed algorithm has a better optimization efficiency to reach the optimum value.

Parametric Bernstein operators based on contagion distribution

This paper proposes the extended form of the ‐Bernstein operators with the Pólya–Eggenberger distribution, also known as contagion distribution, using the integral operators for . We will refer to these sequence of operators as the “Parametric‐Bernstein operators based on contagion distribution” or simply the modified form of ‐Bernstein operators. These operators are defined with the help of two parameters, namely, and . We observe the significance of both these parameters. Subsequently, we calculate their moments and on the basis of Korovkin's approximation theorem we assert that our defined operators converge for a real valued continuous function . We give some basic properties, study the Voronovskaya type asymptotic result and discuss the ‐statistical approximation of our operators. We also use the modulus of continuity to study the rate of convergence. Alongside our primary work, we have also defined the King‐type modification that preserves the operators at and . Further, using modulus of continuity, we compare the properties and behavior of both the operators.

Stokes problem with slip boundary conditions using stabilized finite elements combined with Nitsche

We discuss how slip conditions for the Stokes equation can be handled using Nitsche method, for a stabilized finite element discretization. Emphasis is made on the interplay between stabilization and Nitsche terms. Well-posedness of the discrete problem and optimal convergence rates, in natural norm for the velocity and the pressure, are established, and illustrated with various numerical experiments. The proposed method fits naturally in the context of a finite element implementation while being accurate, and allows an increased flexibility in the choice of the finite element pairs.