Convergence rates for energies of interacting particles whose distribution spreads out as their number increases

IF 1.3 3区 数学 Q4 AUTOMATION & CONTROL SYSTEMS Esaim-Control Optimisation and Calculus of Variations Pub Date : 2022-04-06 DOI:10.1051/cocv/2022083
P. Meurs, Ken’ichiro Tanaka
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引用次数: 2

Abstract

We consider a class of particle systems which appear in various applications such as approximation theory, plasticity, potential theory and space-filling designs. The positions of the particles on the real line are described as a global minimum of an interaction energy, which consists of a nonlocal, repulsive interaction part and a confining part. Motivated by the applications, we cover non-standard scenarios in which the confining potential weakens as the number of particles increases. This results in a large area over which the particles spread out. Our aim is to approximate the particle interaction energy by a corresponding continuum interacting energy. Our main results are bounds on the corresponding energy difference and on the difference between the related potential values. We demonstrate that these bounds are useful to problems in approximation theory and plasticity. The proof of these bounds relies on convexity assumptions on the interaction and confining potentials. It combines recent advances in the literature with a new upper bound on the minimizer of the continuum interaction energy.
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相互作用粒子的能量的收敛率,其分布随着粒子数量的增加而分散
我们考虑了一类出现在各种应用中的粒子系统,如近似理论、塑性理论、势理论和空间填充设计。粒子在实线上的位置被描述为相互作用能量的全局最小值,它由非局部排斥相互作用部分和约束部分组成。在应用的激励下,我们涵盖了限制势随着粒子数量增加而减弱的非标准场景。这就产生了一个很大的区域,粒子在上面扩散。我们的目的是用相应的连续相互作用能来近似粒子相互作用能。我们的主要结果是对应的能量差和相关势值之间的差的界限。我们证明了这些边界对近似理论和塑性问题是有用的。这些边界的证明依赖于相互作用和约束势的凸性假设。它结合了文献的最新进展和连续体相互作用能最小值的新上界。
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来源期刊
Esaim-Control Optimisation and Calculus of Variations
Esaim-Control Optimisation and Calculus of Variations Mathematics-Computational Mathematics
自引率
7.10%
发文量
77
期刊介绍: ESAIM: COCV strives to publish rapidly and efficiently papers and surveys in the areas of Control, Optimisation and Calculus of Variations. Articles may be theoretical, computational, or both, and they will cover contemporary subjects with impact in forefront technology, biosciences, materials science, computer vision, continuum physics, decision sciences and other allied disciplines. Targeted topics include: in control: modeling, controllability, optimal control, stabilization, control design, hybrid control, robustness analysis, numerical and computational methods for control, stochastic or deterministic, continuous or discrete control systems, finite-dimensional or infinite-dimensional control systems, geometric control, quantum control, game theory; in optimisation: mathematical programming, large scale systems, stochastic optimisation, combinatorial optimisation, shape optimisation, convex or nonsmooth optimisation, inverse problems, interior point methods, duality methods, numerical methods, convergence and complexity, global optimisation, optimisation and dynamical systems, optimal transport, machine learning, image or signal analysis; in calculus of variations: variational methods for differential equations and Hamiltonian systems, variational inequalities; semicontinuity and convergence, existence and regularity of minimizers and critical points of functionals, relaxation; geometric problems and the use and development of geometric measure theory tools; problems involving randomness; viscosity solutions; numerical methods; homogenization, multiscale and singular perturbation problems.
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