Mean-field control variate methods for kinetic equations with uncertainties and applications to socio-economic sciences

We examine mean field control problems on a finite state space, in continuous time and over a finite time horizon. We characterize the value function of the mean field control problem as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation in the simplex. In absence of any convexity assumption, we exploit this characterization to prove convergence, as $N$ grows, of the value functions of the centralized $N$-agent optimal control problem to the limit mean field control problem value function, with a convergence rate of order $\frac{1}{\sqrt{N}}$. Then, assuming convexity, we show that the limit value function is smooth and establish propagation of chaos, i.e. convergence of the $N$-agent optimal trajectories to the unique limiting optimal trajectory, with an explicit rate.

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Beyond the Cahn-Hilliard Equation: a Vacancy-Based Kinetic Theory

Solid State Phenomena ◽

10.4028/www.scientific.net/ssp.172-174.321 ◽

2011 ◽

Vol 172-174 ◽

pp. 321-330 ◽

Cited By ~ 3

Author(s):

Maylise Nastar

Keyword(s):

Kinetic Theory ◽

Structure Function ◽

Diffusion Mechanism ◽

Kinetic Equations ◽

Mean Field ◽

Early Time ◽

Present Theory ◽

Kinetic Coupling ◽

Amplification Rate ◽

Cahn Hilliard Equation

A Self-Consistent Mean Field (SCMF) kinetic theory including an explicit description ofthe vacancy diffusion mechanism is developed. The present theory goes beyond the usual local equi-librium hypothesis. It is applied to the study of the early time spinodal decomposition in alloys. Theresulting analytical expression of the structure function highlights the contribution of the vacancydiffusion mechanism. Instead of the single amplification rate of the Cahn-Hillard linear theory, thelinearized SCMF kinetic equations involve three constant rates, first one describing the vacancy re-laxation kinetics, second one related to the kinetic coupling between local concentrations and paircorrelations and the third one representing the spinodal amplification rate. Starting from the same va-cancy diffusion model, we perform kineticMonte Carlo simulations of a Body Centered Cubic (BCC)demixting alloy. The resulting spherically averaged structure function is compared to the SCMF pre-dictions. Both qualitative and quantitative agreements are satisfying.

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Collective Target Tracking Mean Field Control for Markovian Jump-Driven Models of Electric Water Heating Loads

Control of Complex Systems ◽

10.1016/b978-0-12-805246-4.00020-3 ◽

2016 ◽

pp. 559-584 ◽

Cited By ~ 9

Author(s):

A.C. Kizilkale ◽

R.P. Malhamé

Keyword(s):

Target Tracking ◽

Mean Field ◽

Markovian Jump ◽

Water Heating ◽

Field Control ◽

Heating Loads

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Approximation and Mean Field Control of Systems of Large Populations

10.1007/978-3-030-85325-9_7 ◽

2021 ◽

pp. 103-122

Author(s):

Carmen G. Higuera-Chan

Keyword(s):

Mean Field ◽

Field Control ◽

Large Populations

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Neutrino quantum kinetic equations

International Journal of Modern Physics E ◽

10.1142/s0218301315410098 ◽

2015 ◽

Vol 24 (09) ◽

pp. 1541009 ◽

Cited By ~ 33

Author(s):

Cristina Volpe

Keyword(s):

Evolution Equations ◽

Kinetic Equations ◽

Mean Field ◽

Boltzmann Equations ◽

The Mean ◽

Salient Features ◽

Neutrino Propagation

Neutrinos propagate in astrophysical and cosmological environments modifying their flavor in intriguing ways. The study of neutrino propagation in media is based on the mean-field, extended mean-field and Boltzmann equations. We summarize salient features of these evolution equations and the methods employed so far to derive them. We emphasize applications to situations of observational interest.

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