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In this paper, we study a discrete momentum consensus-based optimization (Momentum-CBO) algorithm which corresponds to a second-order generalization of the discrete first-order CBO [S.-Y. Ha, S. Jin and D. Kim, Convergence of a first-order consensus-based global optimization algorithm, Math. Models Methods Appl. Sci. 30 (2020) 2417–2444]. The proposed algorithm can be understood as the modification of ADAM-CBO, replacing the normalization term by unity. For the proposed Momentum-CBO, we provide a sufficient framework which guarantees the convergence of algorithm toward a global minimum of the objective function. Moreover, we present several experimental results showing that Momentum-CBO has an improved success rate of finding the global minimum compared to vanilla-CBO and show the stability of Momentum-CBO under different initialization schemes. We also show that Momentum-CBO can be used as the alternative of ADAM-CBO which does not have a proper convergence analysis. Finally, we give an application of Momentum-CBO for Lyapunov function approximation using symbolic regression techniques. [ABSTRACT FROM AUTHOR]

This editorial paper reviews the articles published in a special issue devoted to the application of active particle methods applied to the study of the collective dynamics of large systems of interacting entities in science and society. The applications presented in this special issue focus on the study of financial markets, cell dynamics in the context of cancer modeling, vehicle and crowd vehicle and crowd dynamics, and classical problems in the kinetic theory of active particles and swarm theory. A critical analysis is proposed to look forward to research. Perspectives with emphasis on the interaction of multiple social dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we analyze nonconforming virtual element methods on polytopal meshes with small faces for the second-order elliptic problem. We propose new stability forms for 2D and 3D nonconforming virtual element methods. For the 2D case, the stability form is defined by the sum of an inner product of approximate tangential derivatives and a weighed L 2 -inner product of certain projections on the mesh element boundaries. For the 3D case, the stability form is defined by a weighted L 2 -inner product on the mesh element boundaries. We prove the optimal convergence of the nonconforming virtual element methods equipped with such stability forms. Finally, several numerical experiments are presented to verify our analysis and compare the performance of the proposed stability forms with the standard stability form [B. Ayuso de Dios, K. Lipnikov and G. Manzini, The nonconforming virtual element method, ESAIM Math. Model. Numer. Anal. 50 (2016) 879–904]. [ABSTRACT FROM AUTHOR]