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Bayesian cubature (BC) has emerged to be one of most competitive approach for estimating the multi-dimensional integral especially when the integrand is expensive to evaluate, and alternative acquisition functions, such as the Posterior Variance Contribution (PVC) function, have been developed for adaptive experiment design of the integration points. However, those sequential design strategies also prevent BC from being implemented in a parallel scheme. Therefore, this paper aims at developing a parallelized adaptive BC method to further improve the computational efficiency.

By theoretically examining the multimodal behavior of the PVC function, it is concluded that the multiple local maxima all have important contribution to the integration accuracy as can be selected as design points, providing a practical way for parallelization of the adaptive BC. Inspired by the above finding, four multimodal optimization algorithms, including one newly developed in this work, are then introduced for finding multiple local maxima of the PVC function in one run, and further for parallel implementation of the adaptive BC.

The superiority of the parallel schemes and the performance of the four multimodal optimization algorithms are then demonstrated and compared with the k-means clustering method by using two numerical benchmarks and two engineering examples.

Multimodal behavior of acquisition function for BC is comprehensively investigated. All the local maxima of the acquisition function contribute to adaptive BC accuracy. Parallelization of adaptive BC is realized with four multimodal optimization methods.

This paper aims to develop a novel method and apply it to solve multiple definite integrals. The proposed method is constructed based on multiple sets of correlation extreme learning machines (MCELM).

The authors present a novel method for solving multiple definite integrals. By using an extreme learning machine (ELM) to learn the integrand function, the primitive function is analytically derived based on the functional expression of the trained ELM and expressed by another ELM, while the correlations between the two ELMs are established. Solutions of multiple definite integrals can be realized by applying this process repeatedly.

To verify the validity and effectiveness of the proposed method, various examples are selected and its numerical solutions are obtained by using the proposed method. The proposed method has high computational accuracy and efficiency, and the superiority is illustrated by comparing with some other existing methods.

MCELM method is proposed for solving multiple definite integrals. The method can be applied for solving multiple definite integrals appearing in applications, the strong applicability of the method in engineering problems is demonstrated in structural system reliability analysis of a cantilever beam.

This work compares the performance of the three boundary element techniques for solving Helmholtz problems: dual reciprocity, multiple reciprocity and direct interpolation. All techniques transform domain integrals into boundary integrals, despite using different principles to reach this purpose.

Comparisons here performed include the solution of eigenvalue and response by frequency scanning, analyzing many features that are not comprehensively discussed in the literature, as follows: the type of boundary conditions, suitable number of degrees of freedom, modal content, number of primitives in the multiple reciprocity method (MRM) and the requirement of internal interpolation points in techniques that use radial basis functions as dual reciprocity and direct interpolation.

Among the other aspects, this work can conclude that the solution of the eigenvalue and response problems confirmed the reasonable accuracy of the dual reciprocity boundary element method (DRBEM) only for the calculation of the first natural frequencies. Concerning the direct interpolation boundary element method (DIBEM), its interpolation characteristic allows more accessibility for solving more elaborate problems. Despite requiring a greater number of interpolating internal points, the DIBEM has presented higher-quality results for the eigenvalue and response problems. The MRM results were satisfactory in terms of accuracy just for the low range of frequencies; however, the neglected higher-order primitives impact the accuracy of the dynamic response as a whole.

There are safe alternatives for solving engineering stationary dynamic problems using the boundary element method (BEM), but there are no suitable comparisons between these different techniques. This paper presents the particularities and detailed comparisons approaching the accuracy of the three important BEM techniques, aiming at response and frequency evaluation, which are not found in the specialized literature.

In this paper, the author introduces a degenerate exponential integral function and further studies some of its analytical properties. The new function is a generalization of the classical exponential integral function and the properties established are analogous to those satisfied by the classical function.

The methods adopted in establishing the results are theoretical in nature.

A degenerate exponential integral function which is a generalization of the classical exponential integral function has been introduced and its properties investigated. Upon taking some limits, the established results reduce to results involving the classical exponential integral function.

The results obtained in this paper are new and have the potential of inspiring further research on the subject.

The purpose of this paper is to develop a multi-attribute group decision-making (MAGDM) method under the q-rung orthopair trapezoidal fuzzy environment, which calculates the interaction between the criteria depending on the proposed q-rung orthopair trapezoidal fuzzy aggregation Choquet integral (q-ROTrFACI) and employ TODIM (an acronym in Portuguese of Interactive and Multi-criteria Decision Making) to consider the risk psychology of decision-makers, to determine the optimal ranking of alternatives.

In MAGDM, q-rung orthopair trapezoidal fuzzy numbers (q-ROTrFNs) are efficient to indicate the quantitative vagueness of decision-makers. The q-ROTrFACI operator is defined and some properties are proved. Then, a novel similarity measure is developed by fusing the area and coordinates of the q-rung orthopair trapezoidal fuzzy function. Based on the above, a Choquet integral-based TODIM (CI-TODIM) method to consider the risk psychology of decision-makers is proposed and two cases are provided to prove superiority of the method.

The paper investigates q-ROTrFACI operator to productively solve problems with interdependent criteria. Then, an approach is proposed to determine the center point of q--ROTrFNs and a q-rung orthopair trapezoidal fuzzy similarity is constructed. Furthermore, CI-TODIM method is devised based on the proposed q-ROTrFACI operator and similarity in q-rung orthopair trapezoidal fuzzy context. The illustration example of business models' solutions and hypertension health management are given to demonstrate the effectiveness and superiority of proposed method.

The paper develops a novel CI-TODIM method that effectively solves the MAGDM problems under the premise of fully considering the priority of criteria and the risk preference of decision-makers, which provides guiding advantages for practical decision-making and enriches the application of decision-making theory.

The purpose of this paper is to study large deviations for the solution processes of a stochastic equation incorporated with the effects of nonlocal condition.

A weak convergence approach is adopted to establish the Laplace principle, which is same as the large deviation principle in a Polish space. The sufficient condition for any family of solutions to satisfy the Laplace principle formulated by Budhiraja and Dupuis is used in this work.

Freidlin–Wentzell type large deviation principle holds good for the solution processes of the stochastic functional integral equation with nonlocal condition.

The asymptotic exponential decay rate of the solution processes of the considered equation towards its deterministic counterpart can be estimated using the established results.

The purpose of this paper is to propose and validate a robust industrial control system. The aim is to design a Multivariable Proportional Integral controller that accommodates multiple responses while considering the process's control and noise parameters. In addition, this paper intended to develop a multidisciplinary approach by combining computational science, control engineering and statistical methodologies to ensure a resilient process with the best use of available resources.

Taguchi's robust design methodology and multi-response optimisation approaches are adopted to meet the research aims. Two-Input-Two-Output transfer function model of the distillation column system is investigated. In designing the control system, the Steady State Gain Matrix and process factors such as time constant (t) and time delay (?) are also used. The unique methodology is implemented and validated using the pilot plant's distillation column. To determine the robustness of the proposed control system, a simulation study, statistical analysis and real-time experimentation are conducted. In addition, the outcomes are compared to different control algorithms.

Research indicates that integral control parameters (K_i) affect outputs substantially more than proportional control parameters (K_p). The results of this paper show that control and noise parameters must be considered to make the control system robust. In addition, Taguchi's approach, in conjunction with multi-response optimisation, ensures robust controller design with optimal use of resources. Eventually, this research shows that the best outcomes for all the performance indices are achieved when Kp11 = 1.6859, Kp12 = −2.061, Kp21 = 3.1846, Kp22 = −1.2176, Ki11 = 1.0628, Ki12 = −1.2989, Ki21 = 2.454 and Ki22 = −0.7676.

This paper provides a step-by-step strategy for designing and validating a multi-response control system that accommodates controllable and uncontrollable parameters (noise parameters). The methodology can be used in any industrial Multi-Input-Multi-Output system to ensure process robustness. In addition, this paper proposes a multidisciplinary approach to industrial controller design that academics and industry can refine and improve.

The purpose of this paper is to apply the concept of equivalent initial flaw size (EIFS) to the anisotropic nickel-based single crystal (SX) material, and to predict the fatigue life on this basis. The crack propagation law of SX material at different temperatures and the weak correlation of EIFS values verification under different loading conditions are also investigated.

A three-parameter time to crack initial (TTCI) method with multiple reference crack lengths under different loading conditions is established, which include the TTCI backstepping method and EIFS fitting method. Subsequently, the optimized EIFS distribution is obtained based on the random crack propagation rate and maximum likelihood estimation of median fatigue life. Then, an effective driving force based on anisotropic and mixed crack propagation mode is proposed to describe the crack propagation rate in the small crack stage. Finally, the fatigue life of three different temperature ESE(T) standard specimens is predicted based on the EIFS values under different survival rates.

The optimized EIFS distribution based on EIFS fitting - maximum likelihood estimation (MLE) method has the highest accuracy in predicting the total fatigue life, with the range of EIFS values being about [0.0028, 0.0875] (mm), and the mean value of EIFS being 0.0506 mm. The error between the predicted fatigue life based on the crack propagation rate and EIFS distribution for survival rates ranges from 5% to 95% and the experimental life is within two times dispersion band.

This paper systematically proposes a new anisotropic material EIFS prediction method, establishing a framework for predicting the fatigue life of SX material at different temperatures using fracture mechanics to avoid inaccurate anisotropic constitutive models and fatigue damage accumulation theory.

The purpose of this paper is to develop the linguistic q-rung orthopair fuzzy set (LqROFS) information VIKOR method based on the bi-direction Choquet integral (BDCI), taking into account the correlation between information. The method can enrich the existing studies related to LqROFS information and better solve the problem of MAGDM problem.

Since applying Choquet integral (CI) depict information interaction is a common operation in MAGDM. However, the traditional CI has some limitations. The unidirectional alignment may affect the MAGDM results. Therefore, a LqROFS-VIKOR method based on BDCI is proposed, where BDCI is used to aggregate the decision matrix. Furthermore, it is not reasonable to apply exact numbers to express the similarity between two qualitative data. Then a new method of defining similarity using linguistics is proposed. The similarity is used to calculate attribute weights.

The validity and potential application of MAGMD method with linguistic q-rung orthopair fuzzy information based on BDCI are demonstrated in a numerical examples study.

According to the study of available literature, the current research on LqROFS is incomplete. The existing studies of both similarity and aggregate operators have certain shortcomings. The definition of similarity proposed in this paper is more in line with reality. And compared with the existing methods, the BDCI-based aggregate operator can describe the interaction between information more reasonably. Based on this VIKOR method based on BDCI under the LqROFS environment can better select the alternative.