Minimum Number Of Iterations Research Articles

Quadratic Programming Problems and Its Solution Techniques With Neural Network Modeling

ABSTRACTThis paper presents a study on quadratic programming problems (QPP) and introduces a new neural network model with feasibility analysis. The stability analysis of QPP using neural network modeling is also discussed. The proposed neural network model has a simple form, and its optimal feasibility for both primal and dual QP problems is established. The model demonstrates a good convergence rate with a minimal number of iterations, achieving a very fast convergence to the exact solutions of both the primal and dual QP problems. The optimal solutions for the original QP problem and its dual QPP are obtained. Finally, two simple numerical examples are simulated to illustrate the findings.

Comparison the Performances for Distributed Machine Learning: Evidence from XGboost and DNN

Abstract. As a matter of fact, distributed machine learning approaches are widely adopted to enhance the training speed. The purpose of this study is to compare the performance of distributed machine learning models, specifically XGBoost and Deep Neural Networks (DNNs), using a telecommunications customer dataset. The dataset consists of 320,000 samples and 25 features and the analysis focuses on model performance under different data heterogeneity conditions. According to the analysis, XGBoost achieves excellent performance, rapidly reaching a high AUC of 97.7% with a minimal number of iterations, proving its effectiveness on structured, small and medium-sized datasets. In contrast, DNN struggled with this dataset and failed to outperform XGBoost due to its low dimensionality and small size. The paper also discusses the main limitations, including the lack of model diversity, the low dimensionality of the dataset, and the problem of model interpretability, especially for DNN. The results suggest that XGBoost is more suited to small and structured datasets, whereas DNN excels at high dimensionality and complex datasets. Future research should focus on improving model diversity, tuning, and addressing interpretability challenges.

Assessment of Multigrid Schemes with Fixed Patterns for a Two-Dimensional Heat Transfer Problem

Heat transfer problems and their solutions are of critical importance in almost all areas of engineering and technology, while many real-world problems are inherently three-dimensional. Simplifying them to 2D models offers practical advantages with reasonable models. With being the fundamental class of problem of heat transfer, the 2D thermal diffusion problem was selected for the study. Multigrid methods were referred to as standing out in terms of cost reduction while keeping solution accuracy. The effectiveness of multigrid methods employing fixed pattern schemes was subject to investigation. To set up numerical experimentation, an authentic code generation effort was given that implements a basic finite volume method, intergrid operations and iterative solvers. A reference case with an analytical Laplace solution was selected and properly validated by the results. A variety of multigrid schemes with fixed patterns were explored around parameters, iterations per sweep and maximum coarsening level. Results were compiled on the focal points of cost and performance. A comparison of the direct iterative methods with multigrid schemes proved the effectiveness of multigrid schemes. With the assessment of the cost and performance outcomes, it is concluded that any multigrid scheme should visit the maximum coarsest level possible while keeping a minimum number of iterations on each grid resolution.

Optimizing Wireless Sensor Network Localization: Hybrid Natural Inspired with Bat Swarm Algorithm

In wireless sensor networks, the data received from sensor nodes is processed and communicated to the next node or cluster head. Sensed data are meaningful only when location information is embedded with the data. Therefore, the sensor should be able to estimate its location information and embed the location information with the sensed data. This research work proposes three hybrid nature-inspired localization optimization algorithms for location estimation of sensor nodes, namely the Hybrid genetic –Bat Localization Algorithm (GA-BAT LA), Hybrid Bat- Particle Swarm Optimization Localization Algorithm (BAT-PSO LA), and Hybrid genetic -Bat- Particle Swarm Optimization Localization Algorithm (GA-BAT-PSO LA). The localization algorithms are developed, implemented, and compared to ensure the accuracy of the estimation of location information. A fixed number of location-known anchor nodes with a variable location-unknown sensor node are considered. The algorithms are compared for a minimum number of iterations, average number of iterations, standard deviation, maximum number of iterations, time complexity, and accuracy to estimate the location of the sensor nodes.

Comparison of Bioinspired Techniques for Tracking Maximum Power under Variable Environmental Conditions

This paper presents a comparative analysis of bioinspired algorithms employed on a PV system subject to standard conditions, under step-change of irradiance conditions, and a partial shading condition for tracking the global maximum power point (GMPP). Four performance analysis and comparison techniques are artificial bee colony, particle swarm optimization, genetic algorithm, and a new metaheuristic technique called jellyfish optimization, respectively. These existing algorithms are well-known for tracking the GMPP with high efficiency. This paper compares these algorithms based on extracting GMPP in terms of maximum power from a PV module running at a uniform (STC), nonuniform solar irradiation (under step-change of irradiance), and partial shading conditions (PSCs). For analysis and comparison, two modules are taken: 1Soltech-1STH-215P and SolarWorld Industries GmbH Sunmodule plus SW 245 poly module, which are considered to form a panel by connecting four series modules. Comparison is based on maximum power tracking, total execution time, and minimum number of iterations to achieve the GMPP with high tracking efficiency and minimum error. Minitab software finds the regression equation (objective function) for STC, step-changing irradiation, and PSC. The reliability of the data (P-V curves) was measured in terms of p value, R, R2, and VIF. The R2 value comes out to be near 1, which shows the accuracy of the data. The simulation results prove that the new evolutionary jellyfish optimization technique gives better results in terms of higher tracking efficiency with very less time to obtain GMPP in all environmental conditions, with a higher efficiency of 98 to 99.9% with less time of 0.0386 to 0.1219 sec in comparison to ABC, GA, and PSO. The RMSE value for the proposed method JFO (0.59) is much lower than that of ABC, GA, and PSO.

Theoretical and Empirical Analysis of Cost-Function Merging for Implicit Hitting Set WCSP Solving

The Implicit Hitting Set (HS) approach has shown very effective for MaxSAT solving. However, only preliminary promising results have been obtained for the very similar Weighted CSP framework. In this paper we contribute towards both a better theoretical understanding of the HS approach and a more effective HS-based solvers for WCSP. First, we bound the minimum number of iterations of HS thanks to what we call distinguished cores. Then, we show a source of inefficiency by introducing two simple problems where HS is unfeasible. Next, we propose two reformulation methods that merge cost-functions to overcome the problem. We provide a theoretical analysis that quantifies the magnitude of the improvement of each method with respect to the number of iterations of the algorithm. In particular, we show that the reformulations can bring an exponential number of iterations down to a constant number in our working examples. Finally, we complement our theoretical analysis with two sets of experiments. First, we show that our results are aligned with real executions. Second, and most importantly, we conduct experiments on typical benchmark problems and show that cost-function merging may be heuristically applied and it may accelerate HS algorithms by several orders of magnitude. In some cases, it even outperforms state-of-the-art solvers.

A mathematically rigorous algorithm to define, compute and assess relevance of the probable dissociation constants in characterizing a biochemical network

Metabolism results from enzymatic- and non-enzymatic interactions of several molecules, is easily parameterized with the dissociation constant and occurs via biochemical networks. The dissociation constant is an empirically determined parameter and cannot be used directly to investigate in silico models of biochemical networks. Here, we develop and present an algorithm to define, compute and assess the relevance of the probable dissociation constant for every reaction of a biochemical network. The reactants and reactions of this network are modelled by a stoichiometry number matrix. The algorithm computes the null space and then serially generates subspaces by combinatorially summing the spanning vectors that are non-trivial and unique. This is done until the terms of each row either monotonically diverge or form an alternating sequence whose terms can be partitioned into subsets with almost the same number of oppositely signed terms. For a selected null space-generated subspace the algorithm utilizes several statistical and mathematical descriptors to select and bin terms from each row into distinct outcome-specific subsets. The terms of each subset are summed, mapped to the real-valued open interval 0,∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left(0,\\infty \\right)$$\\end{document} and used to populate a reaction-specific outcome vector. The p1-norm for this vector is then the probable dissociation constant for this reaction. These steps are continued until every reaction of a modelled network is unambiguously annotated. The assertions presented are complemented by computational studies of a biochemical network for aerobic glycolysis. The fundamental premise of this work is that every row of a null space-generated subspace is a valid reaction and can therefore, be modelled as a reaction-specific sequence vector with a dimension that corresponds to the cardinality of the subspace after excluding all trivial- and redundant-vectors. A major finding of this study is that the row-wise sum or the sum of the terms contained in each reaction-specific sequence vector is mapped unambiguously to a positive real number. This means that the probable dissociation constants, for all reactions, can be directly computed from the stoichiometry number matrix and are suitable indicators of outcome for every reaction of the modelled biochemical network. Additionally, we find that the unambiguous annotation for a biochemical network will require a minimum number of iterations and will determine computational complexity.

Intelligent E-commerce Logistics Supply Chain Management and Scheduling Optimisation

Abstract In the digital era, logistics supply chain scheduling has become the key to enhancing the competitiveness of e-commerce, and the fine management and scheduling optimization of the e-commerce logistics supply chain is particularly important. Based on the three-layer supply chain scheduling model, the study combines the production cost, transportation cost, and transportation time of each member of the supply chain with other influencing indexes. It establishes a multi-stage supply chain scheduling model (SCISM) with dual-objective. The rotational algorithm in linear programming is used to solve the model after it has been constructed. The effectiveness of the SCISM model for intelligent e-commerce logistics supply chains is explored through algorithmic and arithmetic validation of the SCISM model. The results show that the SCISM model algorithm determines the optimal solution σ∗ = (1,2,4,6,3,5) and the optimal objective generalized function value J(σ∗) = −49328 in only 0.95 seconds, which significantly improves the solution efficiency. Compared to Genetic Algorithm (GA) and Particle Swarm Algorithm (PSO), the SCISM algorithm quickly achieves the global optimal solution with the minimum number of iterations (100).

Reservoir computing with higher-order interactive coupled pendulums.

The reservoir computing approach utilizes a time series of measurements as input to a high-dimensional dynamical system known as a reservoir. However, the approach relies on sampling a random matrix to define its underlying reservoir layer, which leads to numerous hyperparameters that need to be optimized. Here, we propose a nonlocally coupled pendulum model with higher-order interactions as a novel reservoir, which requires no random underlying matrices and fewer hyperparameters. We use Bayesian optimization to explore the hyperparameter space within a minimal number of iterations and train the coupled pendulums model to reproduce the chaotic attractors, which simplifies complicated hyperparameter optimization. We illustrate the effectiveness of our technique with the Lorenz system and the Hindmarsh-Rose neuronal model, and we calculate the Pearson correlation coefficients between time series and the Hausdorff metrics in the phase space. We demonstrate the contribution of higher-order interactions by analyzing the interaction between different reservoir configurations and prediction performance, as well as computations of the largest Lyapunov exponents. The chimera state is found as the most effective dynamical regime for prediction. The findings, where we present a new reservoir structure, offer potential applications in the design of high-performance modeling of dynamics in physical systems.

Bayesian optimization-based topology optimization using moving morphable bars for flexible structure design problems

This paper presents a Bayesian optimization (BO) based topology optimization to solve flexible structure design problems in highly nonlinear design spaces. Our primary interest is to avoid topology optimization’s strong dependency on initial designs, which is inevitable when employing gradient-based optimizers. In order to do this, BO is used as a gradient-free optimizer that uses a probabilistic model to trade off exploration and exploitation in order to balance global and local search with a minimal number of iterations. Because of the curse of dimensionality in BO, a moving morphable bar (MMB) is used as a primitive member for a low-dimensional surrogate. We verified that BO could handle dozens of design parameters using the MMB through mean compliance minimization problems with known global optimum layouts. The effect of the BO trade-off parameter in the acquisition function, balancing the optimizer’s exploration, is also investigated. Numerical examples show that the proposed BO-based topology optimization method has suggested better optimum solutions for flexible structures than the MMA-based topology optimization. The performance of the proposed method is also compared with other global optimization methods, showing better performance in terms of minimized objective functions as well as the number of analyses for convergence.

Growth models in aquaculture for hybrid and natural groupers based on early development stage

Grouper has been highlighted as current hot species for marine aquaculture in China, so it is vital to master its growth performance in early development stage and understand growth trend of it. However, there are few detailed reports on the growth model and biological parameters in the early growth of grouper. The aim of this study is to discuss and compare the applicability of growth indexes and 10 biological growth models in larval and juvenile stage of hybrid and natural groupers, then to evaluate the optimum model. Pearl gentian grouper (PG) and kelp grouper (KG) were selected. The growth data of grouper larvae and juvenile were chosen. Absolute growth (AG), relative growth rate (RGR) and specific growth rate (SGR), and the thermal-unit growth coefficient (TGC) were calculated. Ten nonlinear models, such as Logistic, Gompertz,Von Bertalanffy, Generalized Weibull, Janoschek, Schnute and et al., were used to fit growth. Models were evaluated and selected by Akaike information criterion (AIC) method, the values of RMSE, MAE and R2, parameter significance level and iterations. Fitting results showed that compared with AG, RGR and SGR, TGC has higher fitting degree, especially for fitting weight (R2 > 0.99). In larval stage, Schnute model had the highest best fit ratio (75%), and minimum number of iterations (< 71), with more than two significant parameters (P < 0.001). In juvenile stage, the beat fitting ratio of Janoschek model evaluated by AIC was 25%, and that by RMSE, MAE and R2 was 50%, which was higher than other models. It also had low iteration times and high significant level (P < 0.05). For model evaluation, best fitting result evaluated by AIC method was different from that evaluated by RMSE, MAE and R2. It is more reliable to combine the two methods for model evaluation. The growth curves generated by optimum models showed that the growth characteristics of larvae is exponential, and in juvenile stage, PG maintained accelerated growth but the growth rate of KG gradually decreased. In summary, the optimum model of larval stage was Schnute model and that of juvenile stage was Janoschek model; 4-parameter models have high robustness and accuracy for fitting growth; Compared with KG, PG grows faster and is more suitable for aquaculture.

Policy Optimization Adaptive Dynamic Programming for Optimal Control of Input-Affine Discrete-Time Nonlinear Systems

In this article, a policy optimization adaptive dynamic programming (POADP) method is developed for optimal control of discrete-time unknown nonlinear systems, where the iterative control policy is parameterized to optimize the iterative <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$</tex-math> </inline-formula> -function directly. The relaxed condition for the learning rate is given to guarantee the convergence of the present algorithm. Furthermore, the Polyak–Łojasiewicz inequality is introduced to analyze the optimality, i.e., the iterative <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$</tex-math> </inline-formula> -function converges to the optimum within a given computational threshold under a finite iteration, and the rate of convergence (i.e., the required minimum number of iterations) for the developed POADP method is also illustrated. To ease real implementations, the iterative <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$</tex-math> </inline-formula> -function and the iterative control policy are approximated by employing an actor–critic structure. Then, an experiment-based method is developed to obtain the initial weights of actor–critic structure. Finally, numerical simulation results of two examples are provided to validate the effectiveness of the POADP algorithm.

An accuracy evolution method applied to five-axis machining of curved surfaces

Currently, some high-value-added applications involve the manufacturing of curved surfaces, where it is challenging to achieve surface accuracy, repeatability, and productivity simultaneously. Among free-form surfaces, curved surfaces are commonly used in blades and airfoils (with a teardrop-shaped cross-section) and optical systems (with axial symmetry). In both cases, multi-axis milling accuracy directly affects the subsequent process step. Therefore, reducing even insignificant errors during machining can improve the accuracy in the final production stages. This study proposes an “evolution” method to improve the machining accuracy of curved surfaces. The key is to include compensation for the machining error after the first part through profile error measurement. Thus, correction can be applied directly after the manufacturing programming is fully developed, achieving the product with the minimum number of iterations. Accordingly, this method measures the machining error and changes only one key parameter after the process. This study considered two cases. First, an airfoil in which the clamping force was corrected; the results were quite good with only one modification in the blade machining case. Second is an aspherical surface where tool path correction in the Z-axis was applied; the error was effectively compensated along the normal vector of the workpiece surface. The experimental results showed that the surface accuracy increased from 44.4 to 4.5 μm, and the error was reduced by 89.9%, confirming that the accuracy of the machine tool and process had achieved “evolution.” This technical study is expected to help improve the quality and productivity of manufacturing highly accurate curved surfaces.

A numerically stable high-order Chebyshev-Halley type multipoint iterative method for calculating matrix sign function

&lt;abstract&gt;&lt;p&gt;A new eighth-order Chebyshev-Halley type iteration is proposed for solving nonlinear equations and matrix sign function. Basins of attraction show that several special cases of the new method are globally convergent. It is analytically proven that the new method is asymptotically stable and the new method has the order of convergence eight as well. The effectiveness of the theoretical results are illustrated by numerical experiments. In numerical experiments, the new method is applied to a random matrix, Wilson matrix and continuous-time algebraic Riccati equation. Numerical results show that, compared with some well-known methods, the new method achieves the accuracy requirement in the minimum computing time and the minimum number of iterations.&lt;/p&gt;&lt;/abstract&gt;

Obstacle Avoidance and Path Planning of a Wheeled Mobile Robot Using Hybrid Algorithm

In the mobile robot workplace, the path planning problem is crucial. Robotic systems employ intelligence algorithms to plan the robot's path from one point to another. This paper proposes the fastest and optimal path planning of the wheeled mobile robot with collision avoidance to find the optimal route during wheeled mobile robot navigation from the start point to the target point. It is done using a modern meta-heuristic hybrid algorithm called IPSOGWO by combining Improved Particle Swarm Optimization (IPSO) with Grey Wolf Optimizer (GWO). The principal idea is based on boosting the ability to exploit in PSO with the exploration ability in GWO to the better-automated alignment between local and global search capabilities towards a targeted, optimized solution. The proposed hybrid algorithm tackles two objectives: the protection of the path and the length of the path. During, Simulation tests of the route planning by the hybrid algorithm are compared with individual results PSO, IPSO, and GWO concepts about the minimum length of the path, execution time, and the minimum number of iterations required to achieve the best route. This work's effective proposed navigation algorithm was evaluated in a MATLAB environment. The simulation results indicated that the developed algorithm reduced the average path length and the average computation time, less than PSO by (1%, 1.7%), less than GWO by (1%, 1.9%), and less than IPSO by (0.05%, 0.4%), respectively. Furthermore, the superiority of the proposed algorithm was proved through comparisons with other famous path planning algorithms with different static environments.

Order-Statistic Based Target Detection with Compressive Measurements in Single-Frequency Multistatic Passive Radar

This paper considers the problem of compressive target detection with direct-path interference (DPI) and clutter in a single-frequency network (SFN) based multistatic passive radar system (MS-PRS). Specifically, a measurement matrix is designed to jointly obtain compressive observations and remove the DPI and clutter. We first analyze a compressive subspace detector which assumes the target support is known. When the target supports cannot be accurately obtained, an order-statistic (OS) based detector, referred to as the OSOMP, is proposed by using the orthogonal matching pursuit (OMP) algorithm to estimate the target support, and then projecting the compressive observations into the estimated subspace. Since OMP applies an iterative ranking process to select the components/atoms of the dictionary, the OSOMP test variable is an order statistic, which has a non-convergent distribution. To cope with this problem, a modified test statistic for the OSOMP detector is presented and an analytical expression for the probability of false alarm is obtained. We further discuss the minimum number of iterations required by the OSOMP algorithm to achieve the desired probability of detection and false alarm. Numerical simulations are conducted to verify the theoretical analysis and illustrate the performance of the proposed detector relative to several benchmark detectors.

Error-Controlled Iterative Algorithms for Digital Linearization of IMDD-Based Optical Fibre Transmission Systems

In intensity-modulation and direct-detection (IMDD) single-mode fibre (SMF) transmission systems, the nonlinear operation of square-law detection causes signal-signal beating interference (SSBI) that considerably limits the signal transmission capacity versus reach performance of the systems. To address this challenge, an error-controlled iterative algorithm (ECIA) with extra decision thresholds is proposed to digitally linearize the IMDD transmission systems. In the ECIA, instead of amplitude errors used in previously reported algorithms, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${{\boldsymbol{Q}}}^{ - 2}$</tex-math></inline-formula> factor is utilized as an objective function, along with the proposed stochastic gradient descent (SGD) optimization of the thresholds for symbol decision. The thresholds can be adaptively adjusted to reduce decision errors during iterations. To improve the algorithm's adaptivity to various system operation conditions, new constraints are also introduced including maximum step size and/or sign of threshold variation. By making use of the identified optimum key parameters of the ECIA, numerical investigations are conducted of the effectiveness of the proposed scheme in supporting 100 Gb/s 4-level pulse amplitude modulation (PAM4) transmissions over 80 km standard SMFs. Results show that compared with the previously reported decision-directed data-aided iterative algorithm (DD-DIA), the ECIA significantly reduces the power/optical signal-to-noise ratio (OSNR) penalty, required minimum number of iterations (convergence rate) and improves the wavelength drift tolerance. For the ECIA, an OSNR penalty of ≤3.8 dB (10% overhead) and a minimum iteration count of 50 are observed for up to 400 km SMF transmissions, showing its robustness to accumulated dispersions of long fibres. More importantly, a pilot-free operation (0% overhead) is also feasible at a cost of a slightly increased OSNR penalty of ≤0.5 dB.

Deep Learning in Financial Analytics : Exchange Rate Modelling

In finance, a major enthralling research question has been the accurate determination of future market and economic movements. A lot of researchers have tried to develop different models with different accuracies of prediction over the years. It appears that the full potential of deep learning has not been explored to study FX rates. The current study, therefore, explored the proficiency of deep neural networks in predictive modeling. I tested different models of artificial neural networks (using hyperparameters’ tuning like training algorithms, number of hidden layers, and hidden nodes) using neural network input-output fitting and tried to find the best fit model. The model was also validated by layered digital dynamic time series modeling using autoregression with two delays. The appraisal metrics used were regression R - value, MSE, time-series response plot, and error autocorrelation plot. It was concluded that the artificial neural network with a single hidden layer having 17 nodes and trained using the Levenberg– Marquardt algorithm gave the best performance in a minimum number of iterations. This study marks an extensive examination of ANN modeling. This model can be used by traders, investors, financiers, economists, bankers, speculators, hedgers, and governments to get insights into future forex rates and thus make profitable decisions. Various trading policies, import-export policies, and pricing of commodities in indigenous markets can be managed precisely. Future studies can use these models in simulated trading and help establish an alliance between statistical significance and economic significance.

A honey badger algorithm for optimal sizing of an AC coupled hybrid stand-alone photovoltaic system

An optimal design of a hybrid stand-alone photovoltaic system is essential to ensure reliable electricity supply to the loads and low cost of electricity generation. This paper presents an optimal design of an AC coupled hybrid stand-alone photovoltaic system which consists of photovoltaic modules, batteries, grid inverters, bi-directional inverters and diesel generator. A computational intelligence known as Honey Badger Algorithm (HBA) was employed to minimize either the loss of power supply probability or the levelized cost of electricity by selecting the optimal model of system components. For validation purpose, an iterative sizing algorithm (ISA) was developed to yield optimal design solution without the usage of computational intelligence. The HBA was found to be a faster algorithm than ISA in both design cases, i.e. minimizing loss of power supply probability and minimizing levelized cost of electricity. In addition, HBA was also discovered to be superior than Particle Swarm Optimization, Teaching Learning Based Optimization and Firefly Algorithm during the design optimization by producing minimum loss of power supply probability and minimum levelized cost of electricity with lower computational time, population size and minimum number of iterations for convergence.

Parallel adaptive Bayesian quadrature for rare event estimation

Various numerical methods have been extensively studied and used for reliability analysis over the past several decades. However, how to understand the effect of numerical uncertainty (i.e., numerical error due to the discretization of the performance function) on the failure probability is still a challenging issue. The active learning probabilistic integration (ALPI) method offers a principled approach to quantify, propagate and reduce the numerical uncertainty via computation within a Bayesian framework, which has not been fully investigated in context of probabilistic reliability analysis. In this study, a novel method termed ‘Parallel Adaptive Bayesian Quadrature’ (PABQ) is proposed on the theoretical basis of ALPI, and is aimed at broadening its scope of application. First, the Monte Carlo method used in ALPI is replaced with an importance ball sampling technique so as to reduce the sample size that is needed for rare failure event estimation. Second, a multi-point selection criterion is proposed to enable parallel distributed processing. Four numerical examples are studied to demonstrate the effectiveness and efficiency of the proposed method. It is shown that PABQ can effectively assess small failure probabilities (e.g., as low as 10−7) with a minimum number of iterations by taking advantage of parallel computing.