TY - JOUR

T1 - Parallelization of adaptive Bayesian cubature using multimodal optimization algorithms

AU - Hong, Fangqi

AU - Wei, Pengfei

AU - Beer, Michael

N1 - This work is supported by the National Natural Science Foundation of China under grant number 72171194, and the Sino-German Mobility Programme under grant number M-0175 (2021–2024) and the National Science and Technology Major Project (Project No.: J2019-V-0016-0111).

PY - 2024/4/16

Y1 - 2024/4/16

N2 - Purpose: 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. Design/methodology/approach: 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. Findings: 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. Originality/value: 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.

AB - Purpose: 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. Design/methodology/approach: 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. Findings: 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. Originality/value: 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.

KW - Acquisition function

KW - Adaptive experiment design

KW - Bayesian cubature

KW - Multimodal optimization

KW - Parallel computation

UR - http://www.scopus.com/inward/record.url?scp=85189924722&partnerID=8YFLogxK

U2 - 10.1108/EC-12-2023-0957

DO - 10.1108/EC-12-2023-0957

M3 - Article

AN - SCOPUS:85189924722

VL - 41

SP - 413

EP - 437

JO - Engineering Computations (Swansea, Wales)

JF - Engineering Computations (Swansea, Wales)

SN - 0264-4401

IS - 2

ER -