Set Convergence Research Articles

An improved multi-objective artificial physics optimization algorithm based on R2 indicator and its application

Artificial Physics Optimization (APO) algorithms in solving constrained multi-objective problems often encounter challenges such as uneven population distribution and imbalances between global and local search capabilities. To address these issues, we propose the Constrained Rank Multi-Objective Artificial Physical Optimization based on the R2 indicator (R2-ICRMOAPO) algorithm. This algorithm integrates non-dominated sorting with the R2 indicator and updates the external storage set using the contribution value derived from the R2 indicator formula, ensuring both set distribution and convergence. It also dynamically adjusts inertia weights and gravitational factors to enhance its global and local search capabilities. To evaluate the performance of the R2-ICRMOAPO algorithm, we compared it with four other multi-objective optimization algorithms using standard test functions. The results indicate that it demonstrates superior distribution and optimization performance. Furthermore, we applied it to optimize the parameters of a hydro-pneumatic suspension system. The experimental results show that it can reduce the root mean square value of car body vertical acceleration by approximately 21.4% and the root mean square value of dynamic tire load by about 19.6%. This reduction effectively enhances vehicle smoothness within a reasonable range. Consequently, these results confirm the feasibility of the R2-ICRMOAPO algorithm to solve practical problems.

Quantifying convergence and consistency.

The reproducibility crisis highlights several unresolved issues in science, including the need to develop measures that gauge both the consistency and convergence of data sets. While existing meta-analytic methods quantify the consistency of evidence, they do not quantify its convergence: the extent to which different types of empirical methods have provided evidence to support a hypothesis. To address this gap in meta-analysis, we and colleagues developed a summary metric-the cumulative evidence index (CEI)-which uses Bayesian statistics to quantify the degree of both consistency and convergence of evidence regarding causal hypotheses between two phenomena. Here, we outline the CEI's underlying model, which quantifies the extent to which studies of four types-positive intervention, negative intervention, positive non-intervention and negative non-intervention-lend credence to any of three types of causal relations: excitatory, inhibitory or no-connection. Along with p-values and other measures, the CEI can provide a more holistic perspective on a set of evidence by quantitatively expressing epistemic principles that scientists regularly employ qualitatively. The CEI can thus address the reproducibility crisis by formally demonstrating how convergent evidence across multiple study types can yield progress toward scientific consensus, even when an individual type of study fails to yield reproducible results.

The Kuratowski convergence of medial axis and conflict sets

The Kuratowski convergence of medial axis and conflict sets

Exploring the convergence of Metaverse, Blockchain, Artificial Intelligence, and digital twin for pioneering the digitization in the envision smart grid 3.0

The ongoing evolution of the Metaverse, digital twin (DT), artificial intelligence (AI), and Blockchain technologies is fundamentally transforming the utilization of sustainable energy resources (SERs) within smart grids (SGs), ushering in the era of smart grid 3.0 (SG 3.0). This paradigm shift presents unprecedented opportunities to establish robust and sustainable energy trading frameworks between consumers and utilities within the SG 3.0 environment. The integration of DT, AI, and Blockchain technologies in the context of the Metaverse’s smart grids, facilitated by optimized communication protocols, represents a transformative approach. AI models play a pivotal role in predictive energy trading analysis, while DT assumes a crucial role in effectively managing diverse SERs within SGs. Simultaneously, Blockchain technology holds the potential to create a trusted and decentralized environment, laying the foundation for an energy trading system within the SG 3.0 universe of the Metaverse. This visionary convergence of DT and Blockchain sets the stage for a futuristic paradigm in SGs, establishing a robust energy trading and management foundation. This paper embarks on exploring uncharted research paths and unlocking new dimensions by surveying and delving into the convergence of the Metaverse, Blockchain, AI, and DT. Its primary objective is to drive the digitization of SERs within SGs, to revolutionize energy systems. The envisioned SG 3.0 represents a significant leap by amalgamating cutting-edge technologies in sustainable energy, paving the way for revolutionary advancements in SGs’ digitization, which aligns with and contributes to achieving the sustainable development goals outlined by the United Nations.

Role of Exact Exchange and Empirical Dispersion in Density Functional Theory-Based Three-Body Noncovalent Interactions.

Total and three-body interaction energies are calculated for a benchmark set of three-body systems using a range of different types of density functional theory (DFT) methods, with the results compared to CCSD(T)/CBS results from the benchmark reference [Phys. Chem. Chem. Phys. 2023, 25, 28621-28637]. Inclusion of Hartree-Fock exchange, via either a global or range-separated hybrid approach or inclusion of empirical dispersion corrections, increases accuracy for total and three-body interactions. Basis set convergence testing shows that the aug-cc-pVTZ basis set is well converged with little to no change seen when using quadruple-ζ basis sets. The accuracy of the DFT methods is similar when calculating interaction energies for both global and local minimum structures. Overall, the CAM-B3LYP-D3BJ, B97D3, and ωB97XD functionals are recommended for calculating three-body interactions.

New global minimum conformers for the Pt19\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{19}$$\\end{document} and Pt20\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{20}$$\\end{document} clusters: low symmetric species featuring different active sites

ContextThe study of platinum (Pt) clusters and nanoparticles is essential due to their extensive range of potential technological applications, particularly in catalysis. The electronic properties that yield optimal catalytic performance at the nanoscale are significantly influenced by the size and structure of Pt clusters. This research aimed to identify the lowest-energy conformers for Pt18\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{18}$$\\end{document}, Pt19\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{19}$$\\end{document}, and Pt20\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{20}$$\\end{document} species using Density Functional Theory (DFT). We discovered new low-symmetry conformers for Pt19\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{19}$$\\end{document} and Pt20\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{20}$$\\end{document}, which are 3.0 and 1.0 kcal/mol more stable, respectively, than previously reported structures. Our study highlights the importance of using density functional approximations that incorporate moderate levels of exact Hartree-Fock exchange, alongside basis sets of at least quadruple-zeta quality. The resulting structures are asymmetric with varying active sites, as evidenced by sigma hole analysis on the electrostatic potential surface. This suggests a potential correlation between electronic structure and catalytic properties, warranting further investigation.MethodsAn equivariant graph neural network interatomic potential (NequIP) within the Atomic Simulation Environment suite (ASE) was used to provide initial geometries of the aggregates under study. DFT calculations were performed with the ORCA 5 package, using functional approximations that included Generalized Gradient Approximation (PBE), meta-GGA (TPSS, M06-L), hybrid (PBE0, PBEh), meta-GGA hybrid (TPSSh), and range-separated hybrid (ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document}B97x) functionals. Def2-TZVP and Def2-QZVP as well as members of the cc-pwCVXZ-PP family to check basis set convergence were used. QTAIM calculations were performed using the AIMAll suite. Structures were visualized with the AVOGADRO code.

Multimodal multi-objective optimization algorithm based on hierarchical environment selection strategy.

The article proposes an optimization algorithm using a hierarchical environment selection strategyto solve the deficiencies of current multimodal multi-objective optimization algorithms in obtaining the completeness and convergence of Pareto optimal Sets (PSs). Firstly, the algorithm in this article is framed by a differential evolutionary algorithm (DE) and uses a special crowding distance to design a neighborhood-based individual variation strategy, which also ensures the diversity, and then special crowding distance is used to help populations with non-dominated sorting. In the stage of environmental selection, a strategy of hierarchical selection of individuals was designed, which selects sorted non-dominant ranked individual layer by layer according to the ratio, which allows potential individuals tobe explored. Finally, in the stage of evolution of individuals, the convergence and diversity of populations were investigated, anddifferent mutation strategies were selectedaccording to the characteristics of individuals. DE reproduction strategies are used for iteration, preventing individuals from avoiding premature convergence and ensuring the algorithm's searchability. These strategies help the algorithm to obtain more diverse and uniformly distributed PSs and Pareto Front (PF). The algorithm of this article compares with several other excellent algorithms on 13 test problems, and the test results show that all the algorithms of this article exhibit superior performance.

Investigating the Basis Set Convergence of Diagrammatically Decomposed Coupled-Cluster Correlation Energy Contributions for the Uniform Electron Gas.

We investigate the convergence of coupled-cluster (CC) correlation energies and related quantities with respect to the employed basis set size for the uniform electron gas (UEG) to gain a better understanding of the basis set incompleteness error (BSIE). To this end, coupled-cluster doubles (CCD) theory is applied to the three-dimensional UEG for a range of densities, basis set sizes, and electron numbers. We present a detailed analysis of individual diagrammatically decomposed contributions to the amplitudes at the level of CCD theory. In particular, we show that only two terms from the amplitude equations contribute to the asymptotic large-momentum behavior of the transition structure factor, corresponding to the cusp region at short interelectronic distances. However, due to the coupling present in the amplitude equations, all decomposed correlation energy contributions show the same asymptotic convergence behavior to the complete basis set limit. These findings provide an additional rationale for the success of a recently proposed correction to the BSIE of CC theory. Lastly, we examine the BSIE in the CCD plus perturbative triples [CCD(T)] method, as well as in the newly proposed CCD plus complete perturbative triples [CCD(cT)] method.

Strong convergence of split equality variational inequality, variational inclusion, and multiple sets fixed point problems in Hilbert spaces with application

This paper introduces an innovative inertial simultaneous cyclic iterative algorithm designed to address a range of mathematical problems within the realm of split equality variational inequalities. Specifically, the algorithm accommodates finite families of split equality variational inequality problems, infinite families of split equality variational inclusion problems, and multiple-sets split equality fixed point problems involving demicontractive operators in infinite-dimensional Hilbert spaces. The algorithm integrates well-established methods, including the cyclic method, the inertial method, the viscosity approximation method, and the projection method. We establish the strong convergence of this proposed algorithm, demonstrating its applicability in various scenarios and unifying disparate findings from existing literature. Additionally, a numerical example is presented to validate the primary convergence theorem.

On convergence of occupational measures sets of a discrete-time stochastic control system, with applications to averaging of hybrid systems

We establish that, under certain conditions, the set of random occupational measures generated by the state-control trajectories of a discrete-time stochastic system as well as the set of their mathematical expectations converge to a non-random, convex and compact set. We apply these results to the averaging a hybrid system with a slow continuous-time component and a fast discrete-time component. It is shown that the solutions of the hybrid system are approximated by the solutions of a differential inclusion. The novelty of our results is that we allow the state-control space of the fast component to be non-denumerable.

On the sensitivity of computed partial charges toward basis set and (exchange-)correlation treatment.

Partial charges are a central concept in general chemistry and chemical biology, yet dozens of different computational definitions exist. In prior work [Cho et al., ChemPhysChem 21, 688-696 (2020)], we showed that these can be reduced to at most three 'principal components of ionicity'. The present study addressed the dependence of computed partial charges on 1-particle basis set and (for WFT methods) -particle correlation treatment or (for DFT methods) exchange-correlation functional, for several representative partial charge definitions such as QTAIM, Hirshfeld, Hirshfeld-I, HLY (electrostatic), NPA, and GAPT. Our findings show that semi-empirical double hybrids can closely approach the CCSD(T) 'gold standard' for this property. In fact, owing to an error compensation in MP2, CCSD partial charges are further away from CCSD(T) than is MP2. The nonlocal correlation is important, especially when there is a substantial amount of nonlocal exchange. Employing range separation proves to be "mostly" not advantageous, while global hybrids perform optimally for 20%-30% Hartree-Fock exchange across all charge types. Basis set convergence analysis shows that an augmented triple-zeta heavy-aug-cc-pV(T+d)Z basis set or a partially augmented jun-cc-pV(T+d)Z basis set is sufficient for Hirshfeld, Hirshfeld-I, HLY, and GAPT charges. In contrast, QTAIM and NPA display slower basis set convergence. It is noteworthy that for both NPA and QTAIM, HF exhibits markedly slower basis set convergence than the correlation components of MP2 and CCSD. Triples corrections in CCSD(T), denoted as CCSD(T)-CCSD, exhibit even faster basis set convergence.

The p-block challenge: assessing quantum chemistry methods for inorganic heterocycle dimerizations.

The elements of the p-block of the periodic table are of high interest in various chemical and technical applications like frustrated Lewis-pairs (FLP) or opto-electronics. However, high-quality benchmark data to assess approximate density functional theory (DFT) for their theoretical description are sparse. In this work, we present a benchmark set of 604 dimerization energies of 302 "inorganic benzenes" composed of all non-carbon p-block elements of main groups III to VI up to polonium. This so-called IHD302 test set comprises two classes of structures formed by covalent bonding and by weaker donor-acceptor (WDA) interactions, respectively. Generating reliable reference data with ab initio methods is challenging due to large electron correlation contributions, core-valence correlation effects, and especially the slow basis set convergence. To compute reference values for these dimerization reactions, after thorough testing, we applied a computational protocol using state-of-the-art explicitly correlated local coupled cluster theory termed PNO-LCCSD(T)-F12/cc-VTZ-PP-F12(corr.). It includes a basis set correction at the PNO-LMP2-F12/aug-cc-pwCVTZ level. Based on these reference data, we assess 26 DFT methods in combination with three different dispersion corrections and the def2-QZVPP basis set, five composite DFT approaches, and five semi-empirical quantum mechanical methods. For the covalent dimerizations, the r2SCAN-D4 meta-GGA, the r2SCAN0-D4 and ωB97M-V hybrids, and the revDSD-PBEP86-D4 double-hybrid functional are found to be the best-performing methods among the evaluated functionals of the respective class. However, since def2 basis sets for the 4th period are not associated to relativistic pseudo-potentials, we obtained significant errors in the covalent dimerization energies (up to 6 kcal mol-1) for molecules containing p-block elements of the 4th period. Significant improvements were achieved for systems containing 4th row elements by using ECP10MDF pseudopotentials along with re-contracted aug-cc-pVQZ-PP-KS basis sets introduced in this work with the contraction coefficients taken from atomic DFT (PBE0) calculations. Overall, the IHD302 set represents a challenge to contemporary quantum chemical methods. This is due to a large number of spatially close p-element bonds which are underrepresented in other benchmark sets, and the partial covalent bonding character for the WDA interactions. The IHD302 set may be helpful to develop more robust and transferable approximate quantum chemical methods in the future.

Finite-time set stabilization of probabilistic Boolean control networks via output-feedback control

This paper investigates the finite-time output-feedback set stabilization problem for probabilistic Boolean control networks (PBCNs), a crucial aspect of set convergence. Firstly, an algorithm is presented to effectively identify the largest output-feedback control invariant subset in a given set. Next, a sufficient condition is put forward to bridge set stabilization and the question at hand via the minimum transient period. Additionally, sufficient and necessary criteria are proposed, addressing the solvability of finite-time output-feedback set stabilization, followed by two corresponding algorithms for effectively designing all feasible output-feedback stabilizers. Finally, a biological example is successfully explored to demonstrate our theoretical results.

Effects of aleatoric and epistemic errors in reference data on the learnability and quality of NN-based potential energy surfaces

The effect of noise in the input data for learning potential energy surfaces (PESs) based on neural networks for chemical applications is assessed. Noise in energies and forces can result from aleatoric and epistemic errors in the quantum chemical reference calculations. Statistical (aleatoric) noise arises for example due to the need to set convergence thresholds in the self consistent field (SCF) iterations whereas systematic (epistemic) noise is due to, inter alia, particular choices of basis sets in the calculations. The two molecules considered here as proxies are H2CO and HONO which are examples for single- and multi-reference problems, respectively, for geometries around the minimum energy structure. For H2CO it is found that adding noise to energies and forces with magnitudes representative of single-point calculations does not deteriorate the quality of the final PESs whereas increasing the noise level commensurate with electronic structure calculations for more complicated, e.g. metal-containing, systems is expected to have a more notable effect. On the other hand, for HONO which requires a multi-reference treatment, a clear correlation between model quality and the degree of multi-reference character as measured by the T1 amplitude is found. It is concluded that for chemically “simple” cases the effect of aleatoric and epistemic errors is manageable without evident deterioration of the trained model, but more care needs to be exercised for situations in which multi-reference effects are present.

Multi-objective optimization design of anti-roll torsion bar using improved beluga whale optimization algorithm

PurposeThis study aims to reduce the redundant weight of the anti-roll torsion bar brought by the traditional empirical design and improving its strength and stiffness.Design/methodology/approachBased on the finite element approach coupled with the improved beluga whale optimization (IBWO) algorithm, a collaborative optimization method is suggested to optimize the design of the anti-roll torsion bar structure and weight. The dimensions and material properties of the torsion bar were defined as random variables, and the torsion bar's mass and strength were investigated using finite elements. Then, chaotic mapping and differential evolution (DE) operators are introduced to improve the beluga whale optimization (BWO) algorithm and run case studies.FindingsThe findings demonstrate that the IBWO has superior solution set distribution uniformity, convergence speed, solution correctness and stability than the BWO. The IBWO algorithm is used to optimize the anti-roll torsion bar design. The error between the optimization and finite element simulation results was less than 1%. The weight of the optimized anti-roll torsion bar was lessened by 4%, the maximum stress was reduced by 35% and the stiffness was increased by 1.9%.Originality/valueThe study provides a methodological reference for the simulation optimization process of the lateral anti-roll torsion bar.

MaOEA/D with adaptive external population guided weight vector adjustment

In order to make multiobjective evolutionary algorithm based on decomposition (MOEA/D) have better performance in dealing with many-objective optimization problems (MaOPs) with different Pareto fronts (PFs), this paper proposes a MOEA/D with adaptive external population guided weight vector adjustment (MaOEA/D-AEW). The algorithm modifies the MOEA/D method of constructing mating pools by calculating the utility function value of individuals in the population to produce high-quality offspring. For the maintenance of non-dominated solutions in the external population (EP), solutions exceeding the capacity of the EP are removed by the method of minimum Euclidean distance combined with the aggregation function value to ensure the convergence and diversity of non-dominated solutions in the EP. For the addition of weight vectors, a new sparsity evaluating method (SL+) is proposed to determine the sparsity of weight vectors by multiplying parallel distances between individuals. For the deletion of the weight vector, the minimum Euclidean distance combined with the aggregation function value is used to delete the weight vector corresponding to invalid or crowded individuals in the population. Comparing MaOEA/D-AEW with six state-of-the-art algorithms on regular PF and irregular PF test problems respectively, simulation results show that the proposed algorithm can adapt well to MaOPs with different PFs and significantly outperforms other comparative algorithms in terms of convergence and diversity of solution sets.

Fundamental vibrational frequencies of pnictogen (Pn) containing linear triatomic anions: OCPn - and SCPn - where Pn = N, P, As and Sb

The fundamental vibrational frequencies of the isolated OCAs − and SCAs − ions, as well as their heavier antimony analogues, are reported for the first time. A thorough analysis of basis set convergence of the bond lengths and harmonic vibrational frequencies, as well as the examination of anharmonic corrections, is conducted using the MP2 and CCSD(T) ab initio methods with robust basis sets for the OC P n − and SC P n − anions moving down the pnictogen series from Pn = N to Sb. Second-order vibrational perturbation theory (VPT2) and vibrational configuration interaction (VCI) theory are used to compute the fundamental vibrational frequencies of each triatomic anion approaching the MP2 and CCSD(T) complete basis set (CBS) limits. While fundamental vibrational frequencies have been reported for OCN − and SCN − in the gas phase, only experimental vibrational frequencies of the salts for the heavier pnictogen analogues (P and As) have been reported. Experimental Raman vibrational frequencies for OCAs − and SCAs − salts were found to be coincidentally in good agreement with the CCSD(T) VCI frequencies near the CBS limit, differing by at most 13 cm − 1 from the former and 9 cm − 1 from the latter. These modest differences are likely due to the presence of counter-ions and other environmental effects in the solid state, and the overall agreement between the salts and gas-phase frequencies is quite similar to that observed for the OCP − and SCP − ions.

Third-harmonic scattering optical activity: QED theory, symmetry considerations, and quantum chemistry applications in the framework of response theory.

In this work, expressions for the third-harmonic scattering optical activity (THS-OA) spectroscopic responses are derived by combining molecular quantum electrodynamics (QED) and response theory, allowing their computational implementation. The QED theory of THS-OA presented here is meant to be an extension of a previous study by Andrews [Symmetry 12, 1466 (2020)]. In particular, the THS-OA phenomena are described within the Power-Zienau-Woolley multipolar Hamiltonian by including the electric-dipole, magnetic-dipole, and electric-quadrupole interactions for the absorption as well as the emission processes between the dynamic electromagnetic field (the photons) and matter. Moreover, we derive the expressions for the differential scattering ratios as a function of the scattering angle defined by the wavevectors of the incident and scattered photons. We show how the pure and mixed second hyperpolarizabilities can be obtained in the framework of response theory as specific cases of a generic cubic response function, thus enabling the computational implementation of THS-OA spectroscopy. We prove the origin-independence of the theory for exact wavefunctions. Preliminary computations on a prototype chiral molecule (methyloxirane) are considered together with an analysis of the basis set convergence and of the origin-dependence.

Convergence aspects for sets of measures with divergences and boundary conditions

In this paper we study set convergence aspects for Banach spaces of vector-valued measures with divergences (represented by measures or by functions) and applications. We consider a form of normal trace characterization to establish subspaces of measures that directionally vanish in parts of the boundary, and present examples constructed with binary trees. Subsequently we study convex sets with total variation bounds and their convergence properties together with applications to the stability of optimization problems.

Unveiling Brand Convergence in Consideration Sets: The Effects of Purchase for Self or Others and Product Nature

Abstract The creation of a consideration set, which constitutes a cluster of brands that a consumer contemplates buying from, plays a vital role in purchasing behavior. An understanding of its composition aids in the development of effective marketing strategies. However, existing research does not sufficiently explore brand convergence in the consideration set across consumers (BCCS). This study explored variations in BCCS according to whether the purchase is intended for self-use or as a gift, as well as the nature of the product (utilitarian or hedonic). A questionnaire survey involving 1219 participants provided data for empirical analysis. The findings revealed a higher BCCS for purchases made for self with respect to utilitarian products such as ballpoint pens, whereas hedonic products such as chocolates showed a higher BCCS when purchased for others than for self. These findings offer new insights into the underexplored area of BCCS; they suggest that desired assortment size should be tailored according to purchase intent and the nature of the targeted product (utilitarian or hedonic).