Reliable Convergence Research Articles

Equation-Oriented Meanline Method for Axial Turbine Performance Prediction Under Choking Conditions

Abstract Meanline models play a crucial role in turbine design and system-level analyses, facilitating rapid evaluation of design concepts and prediction of off-design performance. Most of the existing meanline methods are inadequate in predicting turbine performance under choking conditions. These models either neglect the impact of losses on choking, or increases the computational complexity significantly. This limitation is addressed in this work, presenting a novel meanline model. The choking state at each cascade is determined by maximizing the mass flow rate, while taking into account the effect of losses. Leveraging the method of Lagrange multipliers, the optimization problems are transformed into a set of equations that seamlessly integrate with the rest of the meanline model. The resulting system of equations is then solved simultaneously using efficient root-finding algorithms, resulting in fast and reliable convergence. Validation against experimental data from three different turbines demonstrates the model's ability to accurately predict mass flow rate, torque, and exit flow angles across single- and multi-stage turbines, with errors typically within ±2.5 % and ±5.0 % for mass flow rate and torque respectively, and within ±5 degrees for flow angles. The proposed approach represents a significant advancement in meanline modeling, offering improved accuracy and computational efficiency.

TRAA: a two-risk archive algorithm for expensive many-objective optimization

Many engineering problems are essentially expensive multi-/many-objective optimization problems, and surrogate-assisted evolutionary algorithms have gained widespread attention in dealing with them. As the objective dimension increases, the error of predicting solutions based on surrogate models accumulates. Existing algorithms do not have strong selection pressure in the candidate solution obtaining and adaptive sampling stages. These make the effectiveness and area of application of the algorithms unsatisfactory. Therefore, this paper proposes a two-risk archive algorithm, which contains a strategy for mining high-risk and low-risk archives and a four-state adaptive sampling criterion. In the candidate solution mining stage, two types of Kriging models are trained, then conservative optimization models and non-conservative optimization models are constructed for model searching, followed by archive selection to obtain more reliable two-risk archives. In the adaptive sampling stage, in order to improve the performance of the algorithms, the proposed criterion considers environmental assessment, demand assessment, and sampling, where the sampling approach involves the improvement of the comprehensive performance in reliable environments, convergence and diversity in controversial environments, and surrogate model uncertainty. Experimental results on numerous benchmark problems show that the proposed algorithm is far superior to seven state-of-the-art algorithms in terms of comprehensive performance.

Broad learning system based on maximum multi-kernel correntropy criterion

The broad learning system (BLS) is an effective machine learning model that exhibits excellent feature extraction ability and fast training speed. However, the traditional BLS is derived from the minimum mean square error (MMSE) criterion, which is highly sensitive to non-Gaussian noise. In order to enhance the robustness of BLS, this paper reconstructs the objective function of BLS based on the maximum multi-kernel correntropy criterion (MMKCC), and obtains a new robust variant of BLS (MKC-BLS). For the multitude of parameters involved in MMKCC, an effective parameter optimization method is presented. The fixed-point iteration method is employed to further optimize the model, and a reliable convergence proof is provided. In comparison to the existing robust variants of BLS, MKC-BLS exhibits superior performance in the non-Gaussian noise environment, particularly in the multi-modal noise environment. Experiments on multiple public datasets and real application validate the efficacy of the proposed method.

On the efficient non-linear solver for hydraulic fracturing and well cementing simulations based on Anderson acceleration

The aim of this study is to create a fast and stable iterative technique for numerical solution of a quasi-linear elliptic pressure equation. We developed a modified version of the Anderson acceleration (AA) algorithm to fixed-point (FP) iteration method. It computes the approximation to the solutions at each iteration based on the history of vectors in extended space, which includes the vector of unknowns, the discrete form of the operator, and the equation's right-hand side. Several constraints are applied to AA algorithm, including a limitation of the time step variation during the iteration process, which allows switching to the base FP iterations to maintain convergence. Compared to the base FP algorithm, the improved version of the AA algorithm enables a reliable and rapid convergence of the iterative solution for the quasi-linear elliptic pressure equation describing the flow of particle-laden yield-stress fluids in a narrow channel during hydraulic fracturing, a key technology for stimulating hydrocarbon-bearing reservoirs. In particular, the proposed AA algorithm allows for faster computations and resolution of unyielding zones in hydraulic fractures that cannot be calculated using the FP algorithm. The quasi-linear elliptic pressure equation under consideration describes various physical processes, such as the displacement of fluids with viscoplastic rheology in a narrow cylindrical annulus during well cementing, the displacement of cross-linked gel in a proppant pack filling hydraulic fractures during the early stage of well production (fracture flowback), and multiphase filtration in a rock formation. We estimate computational complexity of the developed algorithm as compared to Jacobian-based algorithms and show that the performance of the former one is higher in modelling of flows of viscoplastic fluids. We believe that the developed algorithm is a useful numerical tool that can be implemented in commercial simulators to obtain fast and converged solutions to the non-linear problems described above.

Hybrid Tiki Taka and Mean Differential Evolution based Weibull distribution: A comprehensive approach for solar PV modules parameter extraction with Newton-Raphson optimization

Effective parameter extraction is crucial in modeling and analyzing complex systems, particularly photovoltaic (PV) cells/modules. Therefore, it is essential to develop a robust optimization model that can effectively determine the optimal parameters of PV models. This study introduces a novel metaheuristic algorithm, Tiki Taka Algorithm (TTA), inspired by sports strategies, particularly the renowned tiki-taka style. The innovation lies in the hybridization of TTA with the Mean Differential Evolution based on Weibull distribution (MDEW). The resultant optimal parameters obtained from this hybridization serve as the initial guess for Newton-Raphson (NR), a critical step that ensures the estimated current closely aligns with the measured current, leading to a minimized Root Mean Square Error (RMSE). The hybrid approach leverages the excellent exploration capabilities of Weibull distribution and the exploitation strength of the Mean Differential Evolution (MeanDE) mutation, offering a comprehensive solution for navigating complex optimization landscapes. The proposed TTA-MDEW effectively identifies various parameters in PV models, including single diodes, double diodes, and PV modules. The findings show that TTA-MDEW holds its ground against the most sophisticated optimization methods in terms of reliability, accuracy, and speed of convergence. Additionally, data from real-world manufacturer datasheets reveal that our method delivers exceptionally precise results under varying irradiance and temperature conditions. Therefore, these outcomes establish the TTA-MDEW as an effective tool for extracting precise parameters from various PV models, enhancing the modeling of PV systems.

Integrated metaheuristic algorithms with extreme learning machine models for river streamflow prediction

Accurate river streamflow prediction is pivotal for effective resource planning and flood risk management. Traditional river streamflow forecasting models encounter challenges such as nonlinearity, stochastic behavior, and convergence reliability. To overcome these, we introduce novel hybrid models that combine extreme learning machines (ELM) with cutting-edge mathematical inspired metaheuristic optimization algorithms, including Pareto-like sequential sampling (PSS), weighted mean of vectors (INFO), and the Runge–Kutta optimizer (RUN). Our comparative assessment includes 20 hybrid models across eight metaheuristic categories, using streamflow data from the Aswan High Dam on the Nile River. Our findings highlight the superior performance of mathematically based models, which demonstrate enhanced predictive accuracy, robust convergence, and sustained stability. Specifically, the PSS-ELM model achieves superior performance with a root mean square error of 2.0667, a Pearson’s correlation index (R) of 0.9374, and a Nash–Sutcliffe efficiency (NSE) of 0.8642. Additionally, INFO-ELM and RUN-ELM models exhibit robust convergence with mean absolute percentage errors of 15.21% and 15.28% respectively, a mean absolute errors of 1.2145 and 1.2105, and high Kling-Gupta efficiencies values of 0.9113 and 0.9124, respectively. These findings suggest that the adoption of our proposed models significantly enhances water management strategies and reduces any risks.

Physical Prior Mean Function-Driven Gaussian Processes Search for Minimum-Energy Reaction Paths with a Climbing-Image Nudged Elastic Band: A General Method for Gas-Phase, Interfacial, and Bulk-Phase Reactions.

The climbing-image nudged elastic band (CI-NEB) method serves as an indispensable tool for computational chemists, offering insight into minimum-energy reaction paths (MEPs) by delineating both transition states (TSs) and intermediate nonstationary structures along reaction coordinates. However, executing CI-NEB calculations for reactions with extensive reaction coordinate spans necessitates a large number of images to ensure a reliable convergence of the MEPs and TS structures, presenting a computationally demanding optimization challenge, even with mildly costly electronic-structure methods. In this study, we advocate for the utilization of physically inspired prior mean function-based Gaussian processes (GPs) to expedite MEP exploration and TS optimization via the CI-NEB method. By incorporating reliable prior physical approximations into potential energy surface (PES) modeling, we demonstrate enhanced efficiency in multidimensional CI-NEB optimization with surrogate-based optimizers. Our physically informed GP approach not only outperforms traditional nonsurrogate-based optimizers in optimization efficiency but also on-the-fly learns the reaction path valley during optimization, culminating in significant advancements. The surrogate PES derived from our optimization exhibits high accuracy compared to true PES references, aligning with our emphasis on leveraging reliable physical priors for robust and efficient posterior mean learning in GPs. Through a systematic benchmark study encompassing various reaction pathways, including gas-phase, bulk-phase, and interfacial/surface reactions, our physical GPs consistently demonstrate superior efficiency and reliability. For instance, they outperform the popular fast inertial relaxation engine optimizer by approximately a factor of 10, showcasing their versatility and efficacy in exploring reaction mechanisms and surface reaction PESs.

Comparison of nonlinear solution methods for magnetic equivalent circuits of saturated induction motors

This work compares three nonlinear solution methods for the performance of an induction motor’s magnetic equivalent circuit model with magnetic saturation. The interrelation between magnetic flux density and permeability introduces nonlinearities in the differential system of equations. Three popular nonlinear solution methods are selected for comparison, namely (i) the Gauss–Seidel method, (ii) the Newton–Raphson method and (iii) the inverse Broyden’s method. While all three methods have been applied in this context before, no comparison study has been published to the authors’ best knowledge. The study finds that the inverse Broyden’s method is most performant in terms of the number of required iterations, the computation time per iteration and the resulting total computation time. However, for substantial saturation levels, the authors recommend a hybrid implementation of multiple solution methods to obtain robust and reliable convergence.

Optimal Parameter Estimation Techniques for Complex Nonlinear Systems

AbstractAccurate parameter estimation and state identification within nonlinear systems are fundamental challenges addressed by optimization techniques. This paper fills a critical gap in previous research by investigating tailored optimization methods for parameter estimation in nonlinear system modeling, with a particular emphasis on chaotic dynamical systems. We introduce and compare three optimization methods: a gradient-based iterative algorithm, the Levenberg-Marquardt algorithm, and the Nelder-Mead simplex method. These methods are strategically employed to simplify complex nonlinear optimization problems, rendering them more manageable. Through a comprehensive exploration of the performance of these methods in determining parameters across diverse systems, including the van der Pol oscillator, the Rössler system, and pharmacokinetic modeling, our study revealed that the accuracy and reliability of the Nelder-Mead simplex method were consistent. The Nelder-Mead simplex algorithm emerged as a powerful tool, that consistently outperforms alternative methods in terms of root mean squared error (RMSE) and convergence reliability. Visualizations of trajectory comparisons and parameter convergence under various noise levels further emphasize the algorithm’s robustness. These studies suggest that the Nelder-Mead simplex method has potential as a valuable tool for parameter estimation in chaotic dynamical systems. Our study’s implications extend beyond theoretical considerations, offering promising insights for parameter estimation techniques in diverse scientific fields reliant on nonlinear system modeling.

Strategies to improve the isochrone algorithm for ship voyage optimisation

ABSTRACT An optimisation algorithm is an essential part of voyage planning systems to achieve autonomous and intelligent operations. Various algorithms have been proposed for voyage planning to minimise fuel consumption and increase punctuality. Among them, the isochrone method has been recognised for its robustness and efficiency in voyage optimizations. This paper improves it to overcome its incompetence in multi-objective optimisation and reliable route convergence towards the destination. Five improved methods are proposed and compared, to investigate the most effective enhancing strategy to achieve robustness and practicality in real-time application. By changing search space after the middle stage of the voyage, and formulating an augmented cost function to refine search criteria according to different optimisation objectives, one improved isochrone method (named Isochrone-A*) shows more competitive capability, with potential in real-time implementations. The effectiveness and efficiency of these five improved strategies are compared using four ocean-crossing voyages collected by a chemical tanker.

Nonlinear Robust Adaptive Sliding Mode Control Strategies Involve a Fractional Ordered Approach to Reducing Dengue Vectors

This study presents an innovative approach by integrating adaptive sliding mode control strategies with fractional order modeling to address the challenge of reducing Aedes aegypti mosquito populations, the primary vector of Dengue - a widespread and debilitating disease. By employing the Atangana-Baleanu-Caputo fractional operator to model the dynamics of the mosquito population, we achieve a more precise representation of complex and non-linear behaviors. The motivation behind adopting the Adaptive Sliding Mode Control (ASMC) approach lies in the critical need to efficiently control Aedes aegypti mosquito populations, a key step in combating the prevalence of dengue. The ASMC method dynamically adjusts control parameters based on evolving conditions, enhancing its adaptability to the changing dynamics of mosquito populations.The Lyapunov stability theorem ensures the reliability of tracking convergence and control structure. Additionally, we implement the Toufik Atangana method to solve both state and adjoint fractional differential equations using the ABC derivative operator. This incorporation adds a novel dimension to the study, providing a comprehensive framework for addressing the intricate dynamics inherent in the Aedes aegypti mosquito population. To assess the effectiveness of the proposed strategy, a numerical performance index is introduced at the end of the abstract. This index justifies the controller’s efficacy by comparing it to other conventional controllers. The inclusion of this quantitative measure reinforces the significance of the proposed strategy in the context of dengue prevention and control efforts.

Trajectory optimization with hybrid probabilistic roadmap approach to achieve time efficient navigation of unmanned vehicles in unstructured environment

PurposeThis paper aims to focus on solving the path optimization problem by modifying the probabilistic roadmap (PRM) technique as it suffers from the selection of the optimal number of nodes and deploy in free space for reliable trajectory planning.Design/methodology/approachTraditional PRM is modified by developing a decision-making strategy for the selection of optimal nodes w.r.t. the complexity of the environment and deploying the optimal number of nodes outside the closed segment. Subsequently, the generated trajectory is made smoother by implementing the modified Bezier curve technique, which selects an optimal number of control points near the sharp turns for the reliable convergence of the trajectory that reduces the sum of the robot’s turning angles.FindingsThe proposed technique is compared with state-of-the-art techniques that show the reduction of computational load by 12.46%, the number of sharp turns by 100%, the number of collisions by 100% and increase the velocity parameter by 19.91%.Originality/valueThe proposed adaptive technique provides a better solution for autonomous navigation of unmanned ground vehicles, transportation, warehouse applications, etc.

Error‐based grid adaptation methods for plasma edge simulations with SOLPS‐ITER

AbstractOnly recently, plasma edge simulations up to the wall have been enabled with SOLPS‐ITER. This requires dedicated gridding techniques to reconcile grid alignment with the magnetic field and refinement toward the wall as grid quality is primordial to ensure fast and reliable convergence. Therefore, the gridding approach for the grids up to the wall is analyzed and improved. A truncation error analysis is performed on the discrete operators of the discretization scheme in SOLPS‐ITER, resulting in indicators of grid properties that are undesired. Based on these indicators, grid adaptation and grid smoothing algorithms are developed to reduce truncation errors and improve the overall grid quality. The resulting methods are applied on an AUG single‐null case. Here, the impact of the new gridding strategy is examined on the divertor heat load, a typical quantity of interest for plasma edge simulations. The new gridding methods allow to mitigate spurious numerical spikes in the target heat load profiles, reduce the convergence time with a factor 30, and improve the accuracy of the heat load with a factor 3 compared to original grids with similar total number of cells.

Notice of Violation of IEEE Publication Principles: A Hybrid ANFIS/ABC-based Online Selective Harmonic Elimination Switching Pattern for Cascaded Multi-level Inverters of Microgrids

Notice of Violation of IEEE Publication Principles A Hybrid ANFIS/ABC-based Online Selective Harmonic Elimination Switching Pattern for Cascaded Multi-level Inverters of Microgrids, by H. R. Baghaee; M. Mirsalim; G. B. Gharehpetian; H. A. Talebi; A. Niknam-Kumle, in the IEEE Transactions on Industrial Electronics After careful and considered review of the content and authorship of this paper by a duly constituted expert committee, this paper has been found to be in violation of IEEE’s Publication Principles. This authorship of this paper was revised without providing attribution to, or permission from, the coauthors of an earlier version of the manuscript. Alireza Niknam-Kumle was responsible for removing the original coauthor names and replacing them with new coauthors, and then resubmitting the content in a new manuscript to IEEE Transactions on Industrial Electronics Due to the nature of this violation, reasonable effort should be made to remove all past and future references to this paper. Control of electronically-coupled distributed energy resources and topology/switching strategy of power converters have the essential roles to provide high quality power for microgrids including unbalanced and nonlinear loads. Multi-level inverters (MLIs) are used to act as high quality converters with more conversion efficiency, and less switching-frequency/switching-losses. Nevertheless their nonlinearity, complexity, and solvability features of selective harmonic elimination strategy that has affected its industrial applications, it can provide desirable output waveform by keeping fundamental component in its desired value as well as eliminating/minimizing of low-order harmonics. This paper presents adaptive neuro-fuzzy inference system (ANFIS) as a switching angle estimator for harmonic optimization problem in cascade MLIs, which is trained based on the database including optimal switching angles obtained for inverter with non-equal DC sources by artificial bee colony (ABC) algorithm. The reliable search ability, accuracy and convergence time of ABC have been compared with particle swarm optimization algorithm. Also, performance of ANFIS is compared with artificial neural network based on correlation coefficient and root mean square error. Simulation results, experimental tests and real-time simulations reveals the effectiveness of proposed estimator for online elimination of low-order harmonics of MLIs in microgrids and its superiority over other methods.

Imitation learning by state-only distribution matching

Imitation Learning from observation describes policy learning in a similar way to human learning. An agent’s policy is trained by observing an expert performing a task. Although many state-only imitation learning approaches are based on adversarial imitation learning, one main drawback is that adversarial training is often unstable and lacks a reliable convergence estimator. If the true environment reward is unknown and cannot be used to select the best-performing model, this can result in bad real-world policy performance. We propose a non-adversarial learning-from-observations approach, together with an interpretable convergence and performance metric. Our training objective minimizes the Kulback-Leibler divergence (KLD) between the policy and expert state transition trajectories which can be optimized in a non-adversarial fashion. Such methods demonstrate improved robustness when learned density models guide the optimization. We further improve the sample efficiency by rewriting the KLD minimization as the Soft Actor Critic objective based on a modified reward using additional density models that estimate the environment’s forward and backward dynamics. Finally, we evaluate the effectiveness of our approach on well-known continuous control environments and show state-of-the-art performance while having a reliable performance estimator compared to several recent learning-from-observation methods.

Reliability and Validity of the Multidimensional Scale of Perceived Social Support Among Women and Adolescent Girls With Disabilities in Selected Sub-districts of Bangladesh.

Background Adequate community-based or societal collaboration and cooperation are considerably important for the overall welfare of women and adolescent girls with disabilities. "The Multidimensional Scale of Perceived Social Support (MSPSS)" has not been evaluated for reliability and validity amid women and adolescent girls with disabilities in the Bangladeshi context. Methods A Bangla-translated form of the MSPSS was constructed, and the survey was conducted among 152 women and adolescent girls with disabilities who were purposefully recruited from Bogura Sadar and Chapainawabganj Sadar sub-districts of Bangladesh. Results The Cronbach's alpha of the entire scale was 0.868, indicating high internal consistency. Cronbach's alpha for the family sub-scale was 0.763, the friends sub-scale was 0.820, and the significant others scale was 0.776. The composite reliability for the family sub-scale was 0.849677, the friends sub-scale was 0.881248, and the significant others sub-scale was 0.859668. Convergence reliability was established following sub-scale-wise scores. It affirms the consistency of measurements. The content validity score was >0.62, following the Lawshe approach. The three-factor model was adopted during confirmatory factor analysis when the three-factor model run in SPSS Amos (version 21) CFI (comparative fit index) was 0.919. Conclusions In Bangladesh, to the best of our knowledge, our study is initially to calculate the perceived societal assistance of women and adolescent girls with disabilities. We validated the Bangla-translated form of the MSPSS from the Bangladeshi perspective. Researchers and clinicians may rely on our accurate and validated MSPSS translation into Bangla when working with this group. Based on our findings, this study endorses implementing the MSPSS for assessing professed community-based collaboration using the three-factor model, especially among women and adolescent girls with disabilities.

Learning adaptive coarse basis functions of FETI-DP

Domain decomposition methods are successful and highly parallel scalable iterative solution methods for discretized partial differential equations. Nevertheless, for many classes of problems, for example, elliptic partial differential equations with arbitrary coefficient distributions, adaptive coarse spaces are necessary to obtain robustness or, in other words, to guarantee a reliable and fast convergence. Adaptive coarse spaces are usually computed by solving many localized eigenvalue problems related to edges or faces of the domain decomposition. This results in a computationally expensive setup of the domain decomposition preconditioner or system operator. In earlier work, to which the authors have contributed, a deep learning based classification model was used to classify the critical edges or faces where eigenvalue problems have to be solved. In the present paper, we suggest to directly learn the adaptive constraints using a deep feedforward neural network regression model and thus completely skip the computationally most expensive part of the setup, i.e., the solution of local eigenvalue problems. We consider a specific adaptive FETI-DP (Finite Tearing and Interconnecting – Dual Primal) approach and concentrate on stationary diffusion problems in two dimensions with arbitrary coefficient functions with large jumps. As an input for the neural network, we use an image representation of the coefficient function which resolves the structure of the coefficient distribution but is not necessarily identical to the discretization of the partial differential equation. Therefore, our approach is independent of the finite element mesh and can, in principle, be easily extended to other adaptive coarse spaces, problems, and domain decomposition methods. We show the robustness of our method for different problems and the generalization property of our trained neural networks by considering different coefficient distributions not contained in the training set. We also combine the learned constraints with computationally cheap frugal constraints to further improve our approach.

Ab initio calculation of electron-phonon linewidths and molecular dynamics with electronic friction at metal surfaces with numeric atom-centred orbitals

Molecular motion at metallic surfaces is affected by nonadiabatic effects and electron-phonon coupling. The ensuing energy dissipation and dynamical steering effects are not captured by classical molecular dynamics simulations, but can be described with the molecular dynamics with electronic friction method and linear response calculations based on density functional theory. Herein, we present an implementation of electron-phonon response based on an all-electron numeric atomic orbital description in the electronic structure code FHI-aims. After providing details of the underlying approximations and numerical considerations, we present significant scalability and performance improvements of the new code compared to a previous implementation (Maurer et al 2016 Phys. Rev. B 94 115432). We compare convergence behaviour and results of our simulations for exemplary systems such as H2 adsorption on Cu(111), and CO on Ru(0001) against existing plane wave implementations. We examine different expressions to calculate electronic friction and vibrational lifetimes for their reliability and ease of convergence. Finally, we show the capabilities of the new code by studying the contribution of interband and intraband excitations to the vibrational lifetime of aperiodic adsorbate motion in large, previously unfeasible, periodic surface models.

Optimising Time-Frequency Distributions: A Surface Metrology Approach

Time-frequency signal processing offers a significant advantage over temporal or frequency-only methods, but representations require optimisation for a given signal. Standard practice includes choosing the appropriate time-frequency distribution and fine-tuning its parameters, usually via visual inspection and various measures—the most commonly used ones are based on the Rényi entropies or energy concentration by Stanković. However, a discrepancy between the observed representation quality and reported numerical value may arise when the filter kernel has greater adaptability. Herein, a performance measure derived from the Abbot–Firestone curve similar to the volume parameters in surface metrology is proposed as the objective function to be minimised by the proposed minimalistic differential evolution variant that is parameter-free and uses a population of five members. Tests were conducted on two synthetic signals of different frequency modulations and one real-life signal. The multiform tiltable exponential kernel was optimised according to the Rényi entropy, Stanković’s energy concentration and the proposed measure. The resulting distributions were mutually evaluated using the same measures and visual inspection. The optimiser demonstrated a reliable convergence for all considered measures and signals, while the proposed measure showed consistent alignment of reported numerical values and visual assessments.

Computational Performance of Deep Reinforcement Learning to Find Nash Equilibria

We test the performance of deep deterministic policy gradient—a deep reinforcement learning algorithm, able to handle continuous state and action spaces—to find Nash equilibria in a setting where firms compete in offer prices through a uniform price auction. These algorithms are typically considered “model-free” although a large set of parameters is utilized by the algorithm. These parameters may include learning rates, memory buffers, state space dimensioning, normalizations, or noise decay rates, and the purpose of this work is to systematically test the effect of these parameter configurations on convergence to the analytically derived Bertrand equilibrium. We find parameter choices that can reach convergence rates of up to 99%. We show that the algorithm also converges in more complex settings with multiple players and different cost structures. Its reliable convergence may make the method a useful tool to studying strategic behavior of firms even in more complex settings.