Consensus-based optimisation (CBO) is a versatile multi-particle metaheuristic optimisation method suitable for performing non-convex and non-smooth global optimisations in high dimensions. It has proven effective in various applications while at the same time being amenable to a theoretical convergence analysis. In this paper, we explore a variant of CBO, which incorporates truncated noise in order to enhance the well-behavedness of the statistics of the law of the dynamics. By introducing this additional truncation in the noise term of the CBO dynamics, we achieve that, in contrast to the original version, higher moments of the law of the particle system can be effectively bounded. As a result, our proposed variant exhibits enhanced convergence performance, allowing in particular for wider flexibility in choosing the noise parameter of the method as we confirm experimentally. By analysing the time evolution of the Wasserstein-$2$ distance between the empirical measure of the interacting particle system and the global minimiser of the objective function, we rigorously prove convergence in expectation of the proposed CBO variant requiring only minimal assumptions on the objective function and on the initialisation. Numerical evidences demonstrate the benefit of truncating the noise in CBO.

In stereology or applied geometric probability quantitative characterization of aggregates of particles from information on lower-dimensional sections plays a major role. Most stereological methods developed for particle aggregates are based on the assumption that the particles are of the same, known (simple) shape. Information on the volume-weighted distribution of particle size may, however, be obtained under fairly general assumptions about particle shape if particle volume is chosen as size parameter. In fact, there exists in this case an unbiased stereological estimator of the first moment under the sole assumption that the particles are convex. In the present paper, we consider a particle aggregate in ℝ and derive estimators of the q th moment of the volume-weighted distribution of particle volume, based on point-sampling of particles and measurements on q -flats through sampled particles. The estimators are valid for arbitrarily shaped particles but if the particles are non-convex it is necessary for the determination of the estimators to be able to identify the different separated parts on a q-flat through the particle aggregate which belong to the same particle. Explicit forms of the estimators are given for q = 1. For q = 2, an explicit form of one of the estimators is derived for an aggregate of triaxial ellipsoids in three-dimensional space.