scholarly journals A multi-step Lagrangian scheme for spatially inhomogeneous evolutionary games

2021 ◽  
Author(s):  
Stefano Almi ◽  
Marco Morandotti ◽  
Francesco Solombrino
Keyword(s):  
Mean Field ◽  
Evolutionary Games ◽  
Type Systems ◽  
Convergence Result ◽  
Performance Criterion ◽  
Mean Field Limit ◽  
Spatially Inhomogeneous ◽  
Convergence Results ◽  
Special Cases ◽  
Lagrangian Scheme

AbstractA multi-step Lagrangian scheme at discrete times is proposed for the approximation of a nonlinear continuity equation arising as a mean-field limit of spatially inhomogeneous evolutionary games, describing the evolution of a system of spatially distributed agents with strategies, or labels, whose payoff depends also on the current position of the agents. The scheme is Lagrangian, as it traces the evolution of position and labels along characteristics, and is a multi-step scheme, as it develops on the following two stages: First, the distribution of strategies or labels is updated according to a best performance criterion, and then, this is used by the agents to evolve their position. A general convergence result is provided in the space of probability measures. In the special cases of replicator-type systems and reversible Markov chains, variants of the scheme, where the explicit step in the evolution of the labels is replaced by an implicit one, are also considered and convergence results are provided.

scholarly journals A consensus-based model for global optimization and its mean-field limit

2017 ◽  
Vol 27 (01) ◽  
pp. 183-204 ◽  
Author(s):  
René Pinnau ◽  
Claudia Totzeck ◽  
Oliver Tse ◽  
Stephan Martin
Keyword(s):  
Global Optimization ◽  
Mean Field ◽  
Parabolic Partial Differential Equation ◽  
Field Limit ◽  
Mean Field Limit ◽  
Consensus Formation ◽  
Convergence Results ◽  
Multiple Dimensions ◽  
Si Model ◽  
Degenerate Parabolic

We introduce a novel first-order stochastic swarm intelligence (SI) model in the spirit of consensus formation models, namely a consensus-based optimization (CBO) algorithm, which may be used for the global optimization of a function in multiple dimensions. The CBO algorithm allows for passage to the mean-field limit, which results in a nonstandard, nonlocal, degenerate parabolic partial differential equation (PDE). Exploiting tools from PDE analysis we provide convergence results that help to understand the asymptotic behavior of the SI model. We further present numerical investigations underlining the feasibility of our approach.

scholarly journals The Microscopic Derivation and Well-Posedness of the Stochastic Keller–Segel Equation

2020 ◽  
Vol 31 (1) ◽  
Author(s):  
Hui Huang ◽  
Jinniao Qiu
Keyword(s):  
Existence Of Solutions ◽  
Particle System ◽  
Mean Field ◽  
Field Limit ◽  
Mean Field Limit ◽  
Unique Existence ◽  
Interacting Particle ◽  
Microscopic Derivation ◽  
The Mean ◽  
Limit Result

AbstractIn this paper, we propose and study a stochastic aggregation–diffusion equation of the Keller–Segel (KS) type for modeling the chemotaxis in dimensions $$d=2,3$$ d = 2 , 3 . Unlike the classical deterministic KS system, which only allows for idiosyncratic noises, the stochastic KS equation is derived from an interacting particle system subject to both idiosyncratic and common noises. Both the unique existence of solutions to the stochastic KS equation and the mean-field limit result are addressed.

scholarly journals Spatially Inhomogeneous Evolutionary Games

2021 ◽  
Vol 74 (7) ◽  
pp. 1353-1402
Author(s):  
Luigi Ambrosio ◽  
Massimo Fornasier ◽  
Marco Morandotti ◽  
Giuseppe Savaré
Keyword(s):  
Evolutionary Games ◽  
Spatially Inhomogeneous

scholarly journals Derivation of the Landau–Pekar Equations in a Many-Body Mean-Field Limit

2021 ◽  
Vol 240 (1) ◽  
pp. 383-417
Author(s):  
Nikolai Leopold ◽  
David Mitrouskas ◽  
Robert Seiringer
Keyword(s):  
Mean Field ◽  
Einstein Condensate ◽  
Back Reaction ◽  
Many Body ◽  
Field Limit ◽  
Mean Field Limit ◽  
Phonon Field ◽  
Bose Einstein Condensate ◽  
Weakly Couple ◽  
Bose Einstein

AbstractWe consider the Fröhlich Hamiltonian in a mean-field limit where many bosonic particles weakly couple to the quantized phonon field. For large particle numbers and a suitably small coupling, we show that the dynamics of the system is approximately described by the Landau–Pekar equations. These describe a Bose–Einstein condensate interacting with a classical polarization field, whose dynamics is effected by the condensate, i.e., the back-reaction of the phonons that are created by the particles during the time evolution is of leading order.

scholarly journals Mean Field Limit and Propagation of Chaos for a Pedestrian Flow Model

2016 ◽  
Vol 166 (2) ◽  
pp. 211-229 ◽  
Author(s):  
Li Chen ◽  
Simone Göttlich ◽  
Qitao Yin
Keyword(s):  
Flow Model ◽  
Mean Field ◽  
Propagation Of Chaos ◽  
Pedestrian Flow ◽  
Field Limit ◽  
Mean Field Limit

scholarly journals A Mean Field Limit for the Vlasov–Poisson System

2017 ◽  
Vol 225 (3) ◽  
pp. 1201-1231 ◽  
Author(s):  
Dustin Lazarovici ◽  
Peter Pickl
Keyword(s):  
Mean Field ◽  
Poisson System ◽  
Field Limit ◽  
Mean Field Limit

scholarly journals Limit Theorems and Fluctuations for Point Vortices of Generalized Euler Equations

2021 ◽  
Vol 182 (3) ◽  
Author(s):  
Carina Geldhauser ◽  
Marco Romito
Keyword(s):  
Euler Equations ◽  
Mean Field ◽  
Point Vortices ◽  
Positive Temperature ◽  
Field Limit ◽  
Mean Field Limit ◽  
Large Numbers ◽  
The Mean ◽  
Formal Solutions ◽  
2D Torus

AbstractWe prove a mean field limit, a law of large numbers and a central limit theorem for a system of point vortices on the 2D torus at equilibrium with positive temperature. The point vortices are formal solutions of a class of equations generalising the Euler equations, and are also known in the literature as generalised inviscid SQG. The mean-field limit is a steady solution of the equations, the CLT limit is a stationary distribution of the equations.

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