Convergence Rate Research Articles

An algebraic global linelet preconditioner for incompressible flow solvers

Given the wide range of scales present in fluid dynamics, highly anisotropic meshes are often employed to resolve boundary layer flow at high Reynolds numbers. However, such meshes result in distorted elements, which can deteriorate the convergence of the preconditioned conjugate gradient solver (PCG) used to solve the discrete pressure equations. To address this issue, the Linelet Preconditioner is commonly used to accelerate the convergence rate of the PCG by building line segments (linelets) along the direction of the strongest couplings and applying a specific operation to each linelet. Under the current state of the art, linelets are constrained to operate independently within each domain partition. In these scenarios, PCG convergence was observed to deteriorate when the number of domain partitions is increased. This work proposes a communication step to be integrated into the preconditioning step, enabling different linelets to exchange information. The proposed method is observed to improve the convergence rate of linear systems with high stiffness due to highly stretched elements. Also, an algorithm is developed to generate the preconditioning matrix by purely algebraic considerations, in contrast to the usual approach of building linelets based on geometrical considerations. Furthermore, since the current approach is agnostic to the domain partition, it does not impose any constraint on the domain decomposition. Hence, the proposed method is a robust preconditioner with a numerical performance almost independent of the quality of the domain decomposition, providing a way for a more balanced load distribution.

Convergence rate and exponential stability of backward Euler method for neutral stochastic delay differential equations under generalized monotonicity conditions

Convergence rate and exponential stability of backward Euler method for neutral stochastic delay differential equations under generalized monotonicity conditions

Convergence rate toward shock wave under periodic perturbation for generalized Korteweg–de Vries–Burgers equation

In this paper, a viscous shock wave under space-periodic perturbation of generalized Korteweg–de Vries–Burgers equation is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover the exponential time decay rate toward the viscous shock wave is also obtained for some certain perturbations.

Approximation theorems via Pp -statistical convergence on weighted spaces

Abstract In this paper, we obtain some Korovkin type approximation theorems for double sequences of positive linear operators on two-dimensional weighted spaces via statistical type convergence method with respect to power series method. Additionally, we calculate the rate of convergence. As an application, we provide an approximation using the generalization of Gadjiev-Ibragimov operators for Pp -statistical convergence. Our results are meaningful and stronger than those previously given for two-dimensional weighted spaces.

Prescribed Performance Spacecraft Attitude Control with Multiple Convergence Rates

This paper proposes a prescribed performance control policy for spacecraft attitude tracking control. The designed controller employs a kind of performance envelope function that can configure the convergence rate arbitrarily. Moreover, the controller design is based on the barrier Lyapunov function and indirect adaptive methods, which can suppress the fluctuation of external disturbances to obtain lower steady-state errors. Compared to prior studies on spacecraft attitude tracking, the proposed policy can arbitrarily configure the system convergence performance to achieve performance trade-offs in different scenarios. Moreover, the barrier Lyapunov function design and adaptive methods can achieve low steady-state errors with low computational resource consumption. Simulations that verify its effectiveness and superiority are provided herein.

A High-Speed Acoustic Echo Canceller Based on Grey Wolf Optimization and Particle Swarm Optimization Algorithms

Currently, the use of acoustic echo cancellers (AECs) plays a crucial role in IoT applications, such as voice control appliances, hands-free telephony and intelligent voice control devices, among others. Therefore, these IoT devices are mostly controlled by voice commands. However, the performance of these devices is significantly affected by echo noise in real acoustic environments. Despite good results being achieved in terms of echo noise reductions using conventional adaptive filtering based on gradient optimization algorithms, recently, the use of bio-inspired algorithms has attracted significant attention in the science community, since these algorithms exhibit a faster convergence rate when compared with gradient optimization algorithms. To date, several authors have tried to develop high-performance AEC systems to offer high-quality and realistic sound. In this work, we present a new AEC system based on the grey wolf optimization (GWO) and particle swarm optimization (PSO) algorithms to guarantee a higher convergence speed compared with previously reported solutions. This improvement potentially allows for high tracking capabilities. This aspect has special relevance in real acoustic environments since it indicates the rate at which noise is reduced.

Mobile Robot Navigation Based on Noisy N-Step Dueling Double Deep Q-Network and Prioritized Experience Replay

Effective real-time autonomous navigation for mobile robots in static and dynamic environments has become a challenging and active research topic. Although the simultaneous localization and mapping (SLAM) algorithm offers a solution, it often heavily relies on complex global and local maps, resulting in significant computational demands, slower convergence rates, and prolonged training times. In response to these challenges, this paper presents a novel algorithm called PER-n2D3QN, which integrates prioritized experience replay, a noisy network with factorized Gaussian noise, n-step learning, and a dueling structure into a double deep Q-network. This combination enhances the efficiency of experience replay, facilitates exploration, and provides more accurate Q-value estimates, thereby significantly improving the performance of autonomous navigation for mobile robots. To further bolster the stability and robustness, meaningful improvements, such as target “soft” updates and the gradient clipping mechanism, are employed. Additionally, a novel and powerful target-oriented reshaping reward function is designed to expedite learning. The proposed model is validated through extensive experiments using the robot operating system (ROS) and Gazebo simulation environment. Furthermore, to more specifically reflect the complexity of the simulation environment, this paper presents a quantitative analysis of the simulation environment. The experimental results demonstrate that PER-n2D3QN exhibits heightened accuracy, accelerated convergence rates, and enhanced robustness in both static and dynamic scenarios.

Exploiting locality in sparse polynomial approximation of parametric elliptic PDEs and application to parameterized domains

This work studies how the choice of the representation for parametric, spatially distributed inputs to elliptic partial differential equations (PDEs) affects the efficiency of a polynomial surrogate, based on Taylor expansion, for the parameter-to-solution map. In particular, we show potential advantages of representations using functions with localized supports. As model problem, we consider the steady-state diffusion equation, where the diffusion coefficient and right-hand side depend smoothly but potentially in a highly nonlinear way on a parameter y ∈ [−1,1]N. Following previous work for affine parameter dependence and for the lognormal case, we use pointwise instead of norm-wise bounds to prove lp-summability of the Taylor coefficients of the solution. As application, we consider surrogates for solutions to elliptic PDEs on parametric domains. Using a mapping to a nominal configuration, this case fits in the general framework, and higher convergence rates can be attained when modeling the parametric boundary via spatially localized functions. The theoretical results are supported by numerical experiments for the parametric domain problem, illustrating the efficiency of the proposed approach and providing further insight on numerical aspects. Although the methods and ideas are carried out for the steady-state diffusion equation, they extend easily to other elliptic and parabolic PDEs.

On the Connection Between Irrationality Measures and Polynomial Continued Fractions

AbstractLinear recursions with integer coefficients, such as the recursion that generates the Fibonacci sequence $${F}_{n}={F}_{n-1}+{F}_{n-2}$$ F n = F n - 1 + F n - 2 , have been intensely studied over millennia and yet still hide interesting undiscovered mathematics. Such a recursion was used by Apéry in his proof of the irrationality of $$\zeta \left(3\right)$$ ζ 3 , which was later named the Apéry constant. Apéry’s proof used a specific linear recursion that contained integer polynomials (polynomially recursive) and formed a continued fraction; such formulas are called polynomial continued fractions (PCFs). Similar polynomial recursions can be used to prove the irrationality of other fundamental constants such as $$\pi$$ π and $$e$$ e . More generally, the sequences generated by polynomial recursions form Diophantine approximations, which are ubiquitous in different areas of mathematics such as number theory and combinatorics. However, in general it is not known which polynomial recursions create useful Diophantine approximations and under what conditions they can be used to prove irrationality. Here, we present general conclusions and conjectures about Diophantine approximations created from polynomial recursions. Specifically, we generalize Apéry’s work from his particular choice of PCF to any general PCF, finding the conditions under which a PCF can be used to prove irrationality or to provide an efficient Diophantine approximation. To provide concrete examples, we apply our findings to PCFs found by the Ramanujan Machine algorithms to represent fundamental constants such as $$\pi$$ π , $$e$$ e , $$\zeta \left(3\right)$$ ζ 3 , and the Catalan constant. For each such PCF, we demonstrate the extraction of its convergence rate and efficiency, as well as the bound it provides for the irrationality measure of the fundamental constant. We further propose new conjectures about Diophantine approximations based on PCFs. Looking forward, our findings could motivate a search for a wider theory on sequences created by any linear recursions with integer coefficients. Such results can help the development of systematic algorithms for finding Diophantine approximations of fundamental constants. Consequently, our study may contribute to ongoing efforts to answer open questions such as the proof of the irrationality of the Catalan constant or of values of the Riemann zeta function (e.g., $$\zeta \left(5\right)$$ ζ 5 ).

An improved genetic algorithm for robot path planning

The conventional genetic algorithm (GA) for path planning exists several drawbacks, such as uncertainty in the direction of robot movement, circuitous routes, low convergence rates, and prolonged search time. To solve these problems, this study introduces an improved GA-based path-planning algorithm that adopts adaptive regulation of crossover and mutation probabilities. This algorithm uses a hybrid selection strategy that merges elite, tournament, and roulette wheel selection methods. An adaptive approach is implemented to control the speed of population evolution through crossover and mutation. Combining with a local search operation enhances the optimization capability of the algorithm. The proposed algorithm was compared with the traditional GA through simulations, demonstrating shorter path lengths and reduced search times.

Application of Financial Mathematical Models Combined with Root Algorithms in Finance

Investors in the financial markets must deal with various hazards, for which they must create prudent investment portfolios and risk management plans. A multi-objective optimisation approach is proposed using the root algorithm to create a multi-objective root system growth model based on many clusters. An investment risk management optimisation model based on root system growth is built into the study using distributed decision-making. To create a multi-objective root algorithm-based portfolio optimisation model, the Markowitz mean-variance model and a multi-objective root algorithm are employed. Accordingto the findings, the multi-group multi-objective root system method has a real Pareto frontier solution that is more accessible, has a faster convergence rate, and has lower fitness values. The root algorithm’s solutions are workable, and the final risk values of 0.0105, 0.0082, and 0.4623 for the 2, 4, and 6 investment objectives are all in the low-risk class range. The optimal set of solutions discovered by the algorithm had better distributivity and convergence. The Hypervolume values for the multi-group multi-objective root algorithm were 5.5298 and 3.9628 for the dual-objective portfolio and the tri-objective portfolio of investment return costs, respectively. The findings of this study can guide the development of portfolio and risk management strategies.

Heat and mass transfer analysis for bi-dimensional bioconvective MHD nanofluid with varying thermal traits

ABSTRACT Bioconvection phenomena offer practical applications in biomicrosystems, fuel cells, biosensors, biocomputing, electronic thermal management, and renewable energy systems, showcasing their relevance and potential impact across various engineering sectors. Consequently, this study delves into the examination of a bi-dimensional magnetohydrodynamic nanofluid flow model encompassing the intricate dynamics of gyrotactic microorganisms, subject to the effects of suction and thermal radiation. The primary objective of the investigation is to optimize the thermal energy transfer efficiency of the fluid while concurrently minimizing associated costs, with the secondary aim of diminishing the frictional resistance at the fluid-solid interface. The basis for energy and momentum equations incorporating radiation using Rosseland’s approximation is established by the Buongiorno nanofluid model. The foundational PDEs along with their BCs, are reformulated into ODEs utilizing similarity variables, then transformed into a set of first-order ODEs ensuring compatibility with MATLAB’s bvp4c solver, which offers high convergence rates and accuracy due to its efficient algorithms and precision handling. Heat transfer rate boosts up by increasing suction and radiation parameters, while it dwindles due to an increase in the magnetic field. Enhancing suction and the magnetic field lowers skin friction. Microbial concentration surges by increasing the magnetic field, while suction minimizes it. Intensifying the thermophoresis parameter increases the concentration of nanoparticles.

Precise error bounds for numerical approximations of fractional HJB equations

Abstract We prove precise rates of convergence for monotone approximation schemes of fractional and nonlocal Hamilton–Jacobi–Bellman equations. We consider diffusion-corrected difference-quadrature schemes from the literature and new approximations based on powers of discrete Laplacians, approximations that are (formally) fractional order and second-order methods. It is well known in numerical analysis that convergence rates depend on the regularity of solutions, and here we consider cases with varying solution regularity: (i) strongly degenerate problems with Lipschitz solutions and (ii) weakly nondegenerate problems where we show that solutions have bounded fractional derivatives of order $\sigma \in (1,2)$. Our main results are optimal error estimates with convergence rates that capture precisely both the fractional order of the schemes and the fractional regularity of the solutions. For strongly degenerate equations, these rates improve earlier results. For weakly nondegenerate problems of order greater than one, the results are new. Here we show improved rates compared to the strongly degenerate case, rates that are always better than $\mathcal{O}\big (h^{\frac{1}{2}}\big )$.

Approximations of the Euler–Maruyama Method of Stochastic Differential Equations with Regime Switching

This work focuses on a class of regime-switching diffusion processes with both continuous components and discrete components. Under suitable conditions, we adopt the Euler–Maruyama method to deal with the convergence of numerical solutions of the corresponding stochastic differential equations. More precisely, we first show the convergence rates in the Lp-norm of the stochastic differential equations with regime switching under Lipschitz conditions. Then, we also discuss L1 and L2 convergence under non-Lipschitz conditions.

An investigation into the convergence of economic growth among Indian States and the path ahead

This study offers a comprehensive examination of economic growth and convergence in the Indian States and Union Territories (U.T.) over the period from 1991 to 2020. It investigates absolute, sigma, and conditional convergence within this diverse set of states and utilizes the augmented Solow and extended Solow models to explore conditional convergence dynamics. The empirical findings reveal several significant insights. First, there is no substantial negative correlation between the initial per capita GDP ratio and the average annual growth rate, indicating the absence of absolute convergence across the Indian States and U.T. economies during the study period. These results align with those obtained from sigma convergence analysis, reinforcing the absence of widespread convergence. However, conditional convergence is observed, as evidenced by the rate of conditional convergence (coefficient of initial GDP per capita) estimated at 0.038 among the Indian States and U.T. The presence of conditional convergence implies that while initial conditions matter, other factors, including physical and human capital, population growth, and additional variables, significantly contribute to the growth and convergence of Indian regions. The study concludes that policies aimed at promoting economic growth in the Indian States should prioritize the expansion of the labor force, investments in physical and human capital, and prudent government consumption. Furthermore, fostering an environment that encourages access to new technologies and ideas, maintaining sound macroeconomic management, and increasing investments in human capital formation are essential for sustained growth. Effective resource allocation through prudent budgetary policies and heightened investments in the health sector are recommended. Incentives to reduce fertility rates and adept monetary policy management are also identified as crucial elements for ensuring stability and sustainable growth. In summary, this research underscores the importance of adopting a holistic approach to foster economic growth and convergence in the Indian States. The suggested policy measures create a conducive environment for sustained development and prosperity in this diverse and dynamic region.

FedQV: Leveraging Quadratic Voting in Federated Learning

Federated Learning (FL) permits different parties to collaboratively train a global model without disclosing their respective local labels. A crucial step of FL, that of aggregating local models to produce the global one, shares many similarities with public decision-making, and elections in particular. In that context, a major weakness of FL, namely its vulnerability to poisoning attacks, can be interpreted as a consequence of the one person one vote (henceforth 1p1v) principle that underpins most contemporary aggregation rules. In this paper, we introduce FedQV, a novel aggregation algorithm built upon the quadratic voting, recently proposed as a better alternative to 1p1v-based elections. Our theoretical analysis establishes that FedQV is a truthful mechanism in which bidding according to one's true valuation is a dominant strategy that achieves a convergence rate matching that of state-of-the-art methods. Furthermore, our empirical analysis using multiple real-world datasets validates the superior performance of FedQV against poisoning attacks. It also shows that combining FedQV with unequal voting "budgets'' according to a reputation score increases its performance benefits even further. Finally, we show that FedQV can be easily combined with Byzantine-robust privacy-preserving mechanisms to enhance its robustness against both poisoning and privacy attacks. This extended abstract is an abridged version of [3].

Uniform convergence of solutions to stochastic hybrid models of gene regulatory networks

In a recent paper by Kurasov et al. (Math Biosci 305:170–177, 2018), a hybrid gene regulatory network was proposed to model gene expression dynamics by using a stochastic system of coupled partial differential equations. In more recent work, the existence and strong convergence of the solutions to equilibrium were proven. In this article, we improve upon their result by showing that the convergence rate is independent of the initial state, therefore proving that the solutions converge not only strongly but even uniformly to equilibrium. To this end, we make use of a recent convergence theorem for stochastic, irreducible semigroups that contain partial integral operators.

A novel finite-time stability criteria and controller design for nonlinear impulsive systems

The issue of a novel finite-time stability and stabilization design for a class of nonlinear impulsive systems is discussed in this paper. In previous finite-time control designs for impulsive systems, the convergence rate of these control strategies is even less rapid than the exponential convergence rate when the initial state is distant from the origin. To overcome this problem, two sufficient conditions for the novel finite-time stability of impulsive systems are presented, in which the stabilizing impulses and the destabilizing impulses are considered, respectively. Meanwhile, the upper bound of the settling-time is evaluated. In addition, combining Lyapunov functions theory with impulsive control, we design a controller for impulsive dynamical systems to satisfy the novel finite-time stability. Finally, two simulation examples are given, in which the first example demonstrates the superiority of the proposed finite-time stability criterion for nonlinear impulsive systems by comparing it with traditional finite-time stability, and the second example verifies the effectiveness of the presented controller.

Optimal design of offshore jacket platform using enhanced colliding bodies optimization algorithm

Given the considerable volume of materials used in jacket platforms, structural optimization of these structures is always of interest. In the design optimization of offshore jacket platforms, the objective function is iteratively evaluated through a number of complex and time-consuming analyses making the optimization process computationally expensive. To reduce the computational costs, therefore, it is imperative to investigate efficient optimization algorithms with a high convergence rate to achieve optimal solutions for offshore jacket structures as a large-scale and complex problem. Accordingly, this research studies the application of a novel metaheuristic algorithm called Enhanced Colliding Bodies Optimization (ECBO) for the design optimization of a real jacket platform, SPD19A. The optimization constraints comprise stress and buckling in the members, horizontal displacements at the working point, and structural adequacy control of connections. The optimization results are subsequently compared to a design optimized by the Genetic Algorithm (GA), as an example, to evaluate the efficiency of the ECBO algorithm for the offshore jacket structure. The outcomes indicate that ECBO optimizes the jacket more effectively by 15%, while the optimization ratio of GA is 11%. Hence, the results confirm that ECBO has great and favorable efficiency and can potently escape from local optima to reach a better design for the jacket structure.

Multiple feedback recurrent neural network based super-twisting predefined-time nonsingular terminal sliding mode control for quad-rotor UAV

This research deals with the overall predefined-time stability (PTS) of Unmanned Aerial Vehicles (UAV) ensuring rapid convergence based on a novel sliding mode control (SMC). The strength of predefined-time sliding manifolds lies in the convergence rate can be adjusted by an explicit parameter. For the limitation of chattering encountered by predefined-time SMC (PTSMC), a variable gain super-twisting algorithm (STA) with additional linear items is designed as the switch controller. To conserve the restrained computational resources of quadrotors, the equivalent control input is approximated by a multiple feedback recurrent neural network (MFRNN) directly, which is challenging for general recurrent neural networks. The proposed MFRNN is characterized by the incorporation of double-loop feedback within the layers, augmenting its capacity for accurate approximation. To address the vanishing gradients commonly encountered with traditional activation functions, LeakyRelu is chosen. The Lyapunov theory is utilized to ensure the PTS of overall system and obtain the MFRNN weight update laws. An experiment is conducted to validate the proposed scheme.