Convergence Rate Research Articles

Generalized Integral Method for Solving the Neutron Transport Equation with the Hybridized Discontinuous Galerkin Method

Accurate modeling of the neutron transport equation (NTE) with anisotropic scattering is crucial to the understanding of neutron interactions within various mediums. The primary challenges in this domain are (1) the considerable computational resources demanded by anisotropic calculations and (2) the numerical instabilities that arise due to the transport correction approximation. This study introduces a novel, generalized integral method based on the hybridized discontinuous Galerkin framework for solving the second-order NTE with anisotropic scattering. This method employs a spherical harmonics expansion to define the partial current at the mesh interface and applies an angular integral approach to the flux treatment within the mesh. This dual approach facilitates an efficient computational process while preserving accuracy. The integral method has been validated through comparisons with the standard discrete ordinates method (SN) using two eigenvalue problems. The integral method showcases several significant improvements over the traditional SN method. First, it repositions the P0 scattering sources during the formulation process, effectively circumventing the convergence issues associated with transport correction. Second, this strategic repositioning substantially enhances the convergence rates of iterative calculations. Last, a standout feature of the integral method is its capability to perform angular integrals during assembling matrices, successfully reducing the floating-point operations for local flux retrieval and eliminating the ray effect.

Impact of Cohesion Funds on Convergence Club's Economic Growth

ABSTRACTThis paper empirically estimates the impact of the European Structural and Investment Funds (ESIF) on economic growth across European regions. The narrative of this paper is based on the convergence club hypothesis. In this context, we implement the data‐driven Phillips and Sul test to classify European regions into endogenously identified convergence clubs that tend to converge to different steady‐state equilibria. We find three substantially different convergence clubs in terms of both per capita output and spatial location: capital cities and metropolitan areas (along the so‐called “Blue Banana”), core countries, and the periphery. We observe a persistent core‐periphery pattern in terms of output per capita among European regions with different rates of convergence. The convergence club comprising capital cities and metropolitan areas converges almost four times faster than the rest of the EU. Subsequently, we estimate club‐specific growth regressions to investigate the impact of ESIF expenditures on short‐run economic growth. Our main identification strategy relies on two instrumental variables, namely the spatial lag of EFSI expenditures‐to‐GDP and the air distance to Brussels, to address a strong endogeneity problem in strongly biased relationship between ESIF expenditures‐to‐GDP and short‐run economic growth. Our results indicate a positive impact of ESIF expenditures‐to‐GDP on short‐run economic growth in the second (core) and third (periphery) convergence clubs, with the impact being twice as large in the latter compared to the former. These results remain robust when adjusting the growth regressions to use ESIF expenditures‐to‐population instead of ESIF expenditures‐to‐GDP, although the pronounced difference in effect magnitude among convergence clubs diminishes.

Quadrotor attitude control with a modified fixed‐time synergetic controller: A geometric analysis and experimental validation

AbstractThis article proposes a novel modified fixed‐time synergetic control (MFTSC) approach to address the quadrotor control problem in the presence of external disturbances. In contrast to previous fixed‐time synergetic control (FTSC) methods, the proposed approach employs an aggregate macrovariable expression with independent fractional powers. This expression expands the set of real numbers that the parameters of the fractional power can take, leading to a significantly higher rate of convergence toward optimal control parameters (more than 28‐fold), as demonstrated through geometric analysis. The stability of the proposed control system is established, and numerical simulations and experimental validation are conducted using a quadrotor test bench to corroborate the theoretical results.

A mesh-independent method for second-order potential mean field games

Abstract This article investigates the convergence of the Generalized Frank–Wolfe (GFW) algorithm for the resolution of potential and convex second-order mean field games. More specifically, the impact of the discretization of the mean-field-game system on the effectiveness of the GFW algorithm is analyzed. The article focuses on the theta-scheme introduced by the authors in a previous study. A sublinear and a linear rate of convergence are obtained, for two different choices of stepsizes. These rates have the mesh-independence property: the underlying convergence constants are independent of the discretization parameters.

An exponential variation based PSO for analog circuit sizing in constrained environment

This work presents an Exponential Variation based Particle Swarm Optimization (EV-PSO) algorithm to improve the convergence rate and find an optimal solution to analog circuit optimization problems in a constrained-driven environment. Existing evolutionary algorithms have a lower convergence rate leading to higher design time. This work introduces two novel parameters, ζ1 and ζ2, into the velocity update equation. These parameters dynamically vary with the number of iterations. The algorithm was implemented on the Python platform. The results have shown that, in comparison to the considered existing methods, the exponential variation of the parameters ζ1 and ζ2 in the proposed algorithms have a larger rate of convergence. The proposed EV-PSO has a convergence rate of 27 iterations, which is 57.8%, 65.38%, and 59.1% better than the conventional PSO, differential evolution (DE) and genetic algorithm (GA) respectively. The typical design obtained from the optimal solution is verified through the simulation using 45-nm CMOS technology. The optimal solution presented in this work meets the desired input specifications within the specified constrained environment.

Wasserstein-Kaplan-Meier Survival Regression

Survival analysis plays a pivotal role in medical research, offering valuable insights into the timing of events such as survival time. One common challenge in survival analysis is the necessity to adjust the survival function to account for additional factors, such as age, gender, and ethnicity. We propose an innovative regression model for right-censored survival data across heterogeneous populations, leveraging the Wasserstein space of probability measures. Our approach models the probability measure of survival time and the corresponding non-parametric Kaplan-Meier estimator for each subgroup as elements of the Wasserstein space. The Wasserstein space provides a flexible framework for modeling heterogeneous populations, allowing us to capture complex relationships between covariates and survival times. We address an underexplored aspect by deriving the non-asymptotic convergence rate of the Kaplan-Meier estimator to the underlying probability measure in terms of the Wasserstein metric. The proposed model is supported with a solid theoretical foundation including pointwise and uniform convergence rates, along with an efficient algorithm for model fitting. The proposed model effectively accommodates random variation that may exist in the probability measures across different subgroups, demonstrating superior performance in both simulations and two case studies compared to the Cox proportional hazards model and other alternative models.

A “Consistent” Quasidiffusion Method for Solving Particle Transport Problems

The quasidiffusion (QD) method is an established and efficient iterative technique for solving particle transport problems. Each QD iteration consists of a high-order SN sweep, followed by a low-order QD calculation. QD has two defining characteristics: (1) its iterations converge rapidly for any spatial grid and (2) the converged scalar fluxes from the high-order SN sweep and the low-order QD calculation differ, by spatial truncation errors, from each other and from the scalar flux solution of the SN equations. In this paper, we show that by including a transport consistency factor in the low-order equation, the converged high-order and low-order scalar fluxes become equal to each other and to the converged SN scalar flux. However, the inclusion of the transport consistency factor has a negative impact on the convergence rate. We present numerical results that demonstrate the effect of the transport consistency factor on stability.

An Exponential Map-based Whale Optimization Algorithm (Exp-WOA) for Optimization

This research work offers three variations of the Whale Optimization Algorithm (WOA) based on exponential chaotic maps, namely Logistic-Exponential-Logistic WOA (LEL-WOA), Logistic-Exponential-Sinusoidal WOA (LES-WOA), and Logistic-Exponential-Tent WOA (LET-WOA). The WOA with an exponential chaos-based mechanism is developed in this study to overcome the poor rate of convergence of the WOA and to prevent getting caught in local optimal solutions while dealing with the challenges. An exponential chaotic mechanism was employed in this research to initialize the agents and control the parameters of the exploration and exploitation phases of WOA. The proposed methodologies (Exp-WOA) are evaluated using twenty-three widely recognized test functions. The results demonstrate that the given solutions can enhance the performance of WOA by achieving optimal (minimum) values. The findings also indicate that LEL-WOA and LES-WOA exhibit faster convergence than WOA.

Convergence analysis of iteratively regularized Landweber iteration with uniformly convex constraints in Banach spaces

In Banach spaces, the convergence analysis of iteratively regularized Landweber iteration (IRLI) is recently studied via conditional stability estimates. But the formulation of IRLI does not include general non-smooth convex penalty functionals, which is essential to capture special characteristics of the sought solution. In this paper, we formulate a generalized form of IRLI so that its formulation includes general non-smooth uniformly convex penalty functionals. We study the convergence analysis and derive the convergence rates of the generalized method solely via conditional stability estimates in Banach spaces for both the perturbed and unperturbed data. We also discuss few examples of inverse problems on which our method is applicable.

A nonsmooth Levenberg–Marquardt method based on KKT conditions for real-time pricing in smart grid

The real-time pricing of smart grid, which is an important mean of demand side management, is an ideal method to adjust the power balance between the supply and demand in smart grid system. By transforming the social welfare maximization model into a nonsmooth system of equations based on KKT conditions and investigating the related nonsmooth-function’s generalized Jacobi, a nonsmooth Levenberg–Marquardt method with global convergence is proposed in this paper. Also, the local convergence rate is obtained under the local error bound, which is weaker than the nonsingular condition. Finally, numerical simulations show that the smaller the approximate parameter in smoothing Newton’s method, the closer the electricity price is to the value obtained by the proposed algorithm. And the price of the multi price based on the method is lower than that of the single price algorithm that does not consider user types. Meanwhile, compared with the distributed algorithm with different scales, it is found that the price is lower and the social welfare is higher, which shows that the algorithm is robust and effective.

Continuously indexed graphical models

Abstract Let X={Xu}u∈U be a real-valued Gaussian process indexed by a set U. We show that X can be viewed as a graphical model with an uncountably infinite graph, where each Xu is a vertex. This graph is characterized by the reproducing property of X’s covariance kernel, without restricting U to be finite or countable, allowing the modelling of stochastic processes in continuous time/space. Unlike traditional methods, this characterization is not based on zero entries of an inverse covariance, posing challenges for structure estimation. We propose a plug-in methodology that targets graph recovery up to a finite resolution and shows consistency for graphs which are sufficiently regular and that can be applied to virtually any measurement regime. Furthermore, we derive convergence rates and finite-sample guarantees for the method, and demonstrate its performance through a simulation study and two data analyses.

Data-driven optimal shrinkage of singular values under high-dimensional noise with separable covariance structure with application

We develop a data-driven optimal shrinkage algorithm, named extended OptShrink (eOptShrink), for matrix denoising with high-dimensional noise and a separable covariance structure. This noise is colored and dependent across samples. The algorithm leverages the asymptotic behavior of singular values and vectors of the noisy data's random matrix. Our theory includes the sticking property of non-outlier singular values, delocalization of weak signal singular vectors, and the spectral behavior of outlier singular values and vectors. We introduce three estimators: a novel rank estimator, an estimator for the spectral distribution of the pure noise matrix, and the optimal shrinker eOptShrink. Notably, eOptShrink does not require estimating the noise's separable covariance structure. We provide a theoretical guarantee for these estimators with a convergence rate. Through numerical simulations and comparisons with state-of-the-art optimal shrinkage algorithms, we demonstrate eOptShrink's application in extracting maternal and fetal electrocardiograms from single-channel trans-abdominal maternal electrocardiograms.

Latent growth mixture models as latent variable multigroup factor models: Comment on McNeish et al. (2023).

McNeish et al. argue for the general use of covariance pattern growth mixture models because these models do not involve the assumption of random effects, demonstrate high rates of convergence, and are most likely to identify the correct number of latent subgroups. We argue that the covariance pattern growth mixture model is a single random intercept model. It and other models considered in their article are special cases of a general model involving slope and intercept factors. We argue growth mixture models are multigroup invariance hypotheses based on unknown subgroups. Psychometric models in which trajectories are modeled using slope factor loadings which vary by latent subgroup are often conceptually preferable. Convergence rates for mixture models can be substantially improved by using a variance component start value taken from analyses with one fewer class and by specifying multifactor models in orthogonal form. No single latent growth model is appropriate across all research contexts and, instead, the most appropriate latent mixture model must be "right-sized" to the data under consideration. Reanalysis of a real-world longitudinal data set of posttraumatic stress disorder symptomatology reveals a three-group model involving exponential decline, further suggesting that the four-group "cat's cradle" pattern frequently reported is artefactual. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Improving the performance of Stein variational inference through extreme sparsification of physically-constrained neural network models

Most scientific machine learning (SciML) applications of neural networks involve hundreds to thousands of parameters, and hence, uncertainty quantification for such models is plagued by the curse of dimensionality. Using physical applications, we show that L0 sparsification prior to Stein variational gradient descent (L0+SVGD) is a more robust and efficient means of uncertainty quantification, in terms of computational cost and performance than the direct application of SGVD or projected SGVD methods. Specifically, L0+SVGD demonstrates superior resilience to noise, the ability to perform well in extrapolated regions, and a faster convergence rate to an optimal solution.

New Kantorovich-type Szász–Mirakjan Operators

In this paper, we present a Kantorovich-type Szász–Mirakjan operators. Initially, we establish the recurrence relationship for the moments of these operators and provide the central moments up to the fourth degree. Subsequently, we analyze the local approximation properties of these operators using Peetre’s K-function. We investigate the rate of convergence, by utilizing the ordinary modulus of continuity and Lipschitz-type maximal functions. Additionally, we prove weighted approximation theorems and Voronoskaja-type theorems specific to these new operators. Following this, we introduce bivariate extension of these operators and investigate some approximation properties. Lastly, we include several numerical illustrative examples.

The use of convolutional neural networks for abnormal behavior recognition in crowd scenes

This study introduces the Abnormality Converging Scene Analysis Method (ACSAM) to detect abnormal group behavior using monitored videos or CCTV images in crowded scenarios. Abnormal behavior recognition involves classifying activities and gestures in continuous scenes, which traditionally presents significant computational challenges, particularly in complex crowd scenes, leading to reduced recognition accuracy. To address these issues, ACSAM employs a convolutional neural network (CNN) enhanced with Abnormality and Crowd Behavior Training layers to accurately detect and classify abnormal activities, regardless of crowd density. The method involves extracting frames from the input scene and using CNN to perform conditional validation of abnormality factors, comparing current values with previous high values to maximize detection accuracy. As the abnormality factor increases, the identification rate improves with higher training iterations. The system was tested on 26 video samples and trained on 34 samples, demonstrating superior performance to other approaches like DeepROD, MSI-CNN, and PT-2DCNN. Specifically, ACSAM achieved a 12.55% improvement in accuracy, a 12.97% increase in recall, and a 10.23% enhancement in convergence rate. These results suggest that ACSAM effectively overcomes the computational challenges inherent in crowd scene detection, offering a robust solution for real-time abnormal behavior recognition in crowded environments.

A decentralized control approach in hypergraph distributed optimization decomposition cases

We study the main decomposition approaches (primal, dual and primal–dual) for a distributed optimization problem from a dynamical system perspective where the couplings among the variables of the optimization problem are described by an undirected, unweighted hypergraph. We conduct stability analysis for the respective dynamical systems of the decomposition cases by using non linear decentralized control theoretic techniques and spectral properties of the respective communication matrices, i.e., the incidence and the Laplacian matrices of the hypergraph. Finally, we provide numerical simulations under a specific coalitional setting that demonstrate the superiority of the hypergraph compared to its respective graph analogue, the clique expansion graph, for the given decomposition algorithms in terms of convergence rate and information transmission efficiency.

A Method for Transforming Non-Convex Optimization Problem to Distributed Form

We propose a novel distributed method for non-convex optimization problems with coupling equality and inequality constraints. This method transforms the optimization problem into a specific form to allow distributed implementation of modified gradient descent and Newton’s methods so that they operate as if they were distributed. We demonstrate that for the proposed distributed method: (i) communications are significantly less time-consuming than oracle calls, (ii) its convergence rate is equivalent to the convergence of Newton’s method concerning oracle calls, and (iii) for the cases when oracle calls are more expensive than communication between agents, the transition from a centralized to a distributed paradigm does not significantly affect computational time. The proposed method is applicable when the objective function is twice differentiable and constraints are differentiable, which holds for a wide range of machine learning methods and optimization setups.

Accelerated convergence via adiabatic sampling for adsorption and desorption processes.

Under isothermal conditions, phase transitions occur through a nucleation event when conditions are sufficiently close to coexistence. The formation of a nucleus of the new phase requires the system to overcome a free energy barrier of formation, whose height rapidly rises as supersaturation decreases. This phenomenon occurs both in the bulk and under confinement and leads to a very slow kinetics for the transition, ultimately resulting in hysteresis, where the system can remain in a metastable state for a long time. This has broad implications, for instance, when using simulations to predict phase diagrams or screen porous materials for gas storage applications. Here, we leverage simulations in an adiabatic statistical ensemble, known as adiabatic grand-isochoric ensemble (μ, V, L) ensemble, to reach equilibrium states with a greater efficiency than its isothermal counterpart, i.e., simulations in the grand-canonical ensemble. For the bulk, we show that at low supersaturation, isothermal simulations converge slowly, while adiabatic simulations exhibit a fast convergence over a wide range of supersaturation. We then focus on adsorption and desorption processes in nanoporous materials, assess the reliability of (μ, V, L) simulations on the adsorption of argon in IRMOF-1, and demonstrate the efficiency of adiabatic simulations to predict efficiently the equilibrium loading during the adsorption and desorption of argon in MCM-41, a system that exhibits significant hysteresis. We provide quantitative measures of the increased rate of convergence when using adiabatic simulations. Adiabatic simulations explore a wide temperature range, leading to a more efficient exploration of the configuration space.

Application of water cycle algorithm with demand follows green level and nonlinear power pattern of the product for an inventory system

It is commonly known that a number of variables, including price, supply levels, time, and green level, affect how quickly certain things are in demand. Furthermore, the inventory carrying cost is considered to be a nonlinear representation of time and is subject to variation throughout time. More precisely, it rises with time since longer storage times necessitate more costly warehouse space. This study presents a fully backlogged situation inventory system for a single commodity where the product’s selling price, green level, and time are used to simultaneously compute the demand rate in accordance with a power pattern. Purchase price is determined by the product’s nonlinear green level. Complete backorders are available for shortages. The impact of the product’s selling price, green level and time power function are combined to determine the product’s demand. Moreover, the holding cost also rises as the product is stored for a longer period of time. The primary goal is to determine the best inventory policy to maximise total profit per unit of time. Though the problem is highly nonlinear in nature. Hence, we cannot solve it analytically. To overcome these difficulties, we have applied several well-known popular metaheuristic algorithms (Water Cycle Algorithm (WCA), Artificial Electric Field Algorithm (AEFA), Teaching Learning Based Optimization Algorithm (TLBOA), Grey Wolf Optimizer Algorithm (GWOA), Sparrow Search Algorithm (SSA), Whale Optimizer Algorithm (WOA), Prairie Dog Optimization Algorithm (PDOA), Gazelle Optimization Algorithm (GOA), A Sinh Cosh Optimizer Algorithm (SCHOA) and White Sherk Optimizer Algorithm (WSOA), Archimedes Optimization Paradigm Algorithm (AOPA), Marine Predator Optimization Algorithm (MPOA), Geyser Inspired Algorithm (GIA), Runge Kutta Optimization Algorithm (RKOA), Lungs Performance-based Optimization Algorithm (LPOA) and Dwarf Mongoose Optimization Algorithm (DMOA)). It is observed that WCA perform better than other algorithms with respect to the convergence rate. A numerical example is taken in order to validate the proposed model. Finally, a post optimality analysis is performed in order to make a fruitful conclusion.