scholarly journals Existence of weak solutions to time-dependent mean-field games

2021 ◽  
Vol 212 ◽  
pp. 112470
Author(s):  
Rita Ferreira ◽  
Diogo Gomes ◽  
Teruo Tada
Keyword(s):  
Weak Solutions ◽  
Mean Field ◽  
Time Dependent ◽  
Mean Field Games ◽  
Existence Of Weak Solutions

scholarly journals On the implementation of a primal-dual algorithm for second order time-dependent Mean Field Games with local couplings

2019 ◽  
Vol 65 ◽  
pp. 330-348
Author(s):  
L. Briceño-Arias ◽  
D. Kalise ◽  
Z. Kobeissi ◽  
M. Laurière ◽  
Á. Mateos González ◽  
...  
Keyword(s):  
Mean Field ◽  
Iterative Algorithms ◽  
Variational Problems ◽  
Stationary Problem ◽  
Time Dependent ◽  
Mean Field Games ◽  
Dual Algorithm ◽  
System Solution ◽  
Primal Dual ◽  
Preconditioned Iterative

We study a numerical approximation of a time-dependent Mean Field Game (MFG) system with local couplings. The discretization we consider stems from a variational approach described in [14] for the stationary problem and leads to the finite difference scheme introduced by Achdou and Capuzzo-Dolcetta in [3]. In order to solve the finite dimensional variational problems, in [14] the authors implement the primal-dual algorithm introduced by Chambolle and Pock in [20], whose core consists in iteratively solving linear systems and applying a proximity operator. We apply that method to time-dependent MFG and, for large viscosity parameters, we improve the linear system solution by replacing the direct approach used in [14] by suitable preconditioned iterative algorithms.

scholarly journals A minimization approach to the wave equation on time-dependent domains

2020 ◽  
Vol 13 (4) ◽  
pp. 425-436 ◽  
Author(s):  
Gianni Dal Maso ◽  
Lucia De Luca
Keyword(s):  
Wave Equation ◽  
Weak Solutions ◽  
Space Time ◽  
Time Dependent ◽  
Existence Of Weak Solutions ◽  
Homogeneous Wave ◽  
Minimization Approach ◽  
Homogeneous Wave Equation

AbstractWe prove the existence of weak solutions to the homogeneous wave equation on a suitable class of time-dependent domains. Using the approach suggested by De Giorgi and developed by Serra and Tilli, such solutions are approximated by minimizers of suitable functionals in space-time.

scholarly journals Existence and non-existence for time-dependent mean field games with strong aggregation

2021 ◽  
Author(s):  
Marco Cirant ◽  
Daria Ghilli
Keyword(s):  
Population Density ◽  
State Space ◽  
Time Horizon ◽  
Existence Of Solutions ◽  
Mean Field ◽  
The State ◽  
Time Dependent ◽  
Classical Solutions ◽  
Mean Field Games ◽  
Quadratic Mean

AbstractWe investigate the existence of classical solutions to second-order quadratic Mean-Field Games systems with local and strongly decreasing couplings of the form $$-\sigma m^\alpha $$ - σ m α ,$$\alpha \ge 2/N$$ α ≥ 2 / N , where m is the population density and N is the dimension of the state space. We prove the existence of solutions under the assumption that $$\sigma $$ σ is small enough. For large $$\sigma $$ σ , we show that existence may fail whenever the time horizon T is large.

scholarly journals Time-Dependent Mean-Field Games with Logarithmic Nonlinearities

2015 ◽  
Vol 47 (5) ◽  
pp. 3798-3812 ◽  
Author(s):  
Diogo A. Gomes ◽  
Edgard Pimentel
Keyword(s):  
Mean Field ◽  
Time Dependent ◽  
Mean Field Games
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