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 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

scholarly journals Alternating the population and control neural networks to solve high-dimensional stochastic mean-field games

2021 ◽  
Vol 118 (31) ◽  
pp. e2024713118
Author(s):  
Alex Tong Lin ◽  
Samy Wu Fung ◽  
Wuchen Li ◽  
Levon Nurbekyan ◽  
Stanley J. Osher
Keyword(s):  
Neural Networks ◽  
Mean Field ◽  
Saddle Point Problem ◽  
High Dimensional ◽  
Mean Field Games ◽  
Point Problem ◽  
Generative Adversarial Network ◽  
Adversarial Network ◽  
Solution Methods ◽  
Primal Dual

We present APAC-Net, an alternating population and agent control neural network for solving stochastic mean-field games (MFGs). Our algorithm is geared toward high-dimensional instances of MFGs that are not approachable with existing solution methods. We achieve this in two steps. First, we take advantage of the underlying variational primal-dual structure that MFGs exhibit and phrase it as a convex–concave saddle-point problem. Second, we parameterize the value and density functions by two neural networks, respectively. By phrasing the problem in this manner, solving the MFG can be interpreted as a special case of training a generative adversarial network (GAN). We show the potential of our method on up to 100-dimensional MFG problems.

scholarly journals Time-Dependent Mean-Field Games in the Subquadratic Case

2014 ◽  
Vol 40 (1) ◽  
pp. 40-76 ◽  
Author(s):  
Diogo A. Gomes ◽  
Edgard A. Pimentel ◽  
Héctor Sánchez-Morgado
Keyword(s):  
Mean Field ◽  
Time Dependent ◽  
Mean Field Games

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

Approximation Algorithm for Resource Allocation Problems with Time Dependent Penalties

2017 ◽  
Vol 28 (07) ◽  
pp. 931-943
Author(s):  
Vaishali M. Wadhwa ◽  
Deepak Garg
Keyword(s):  
Resource Allocation ◽  
Approximation Algorithm ◽  
Facility Location ◽  
Threshold Value ◽  
Time Dependent ◽  
Elapse Time ◽  
Demand Point ◽  
Dual Algorithm ◽  
Allocation Problems ◽  
Primal Dual

The Resource Allocation Problem with Time Dependent Penalties (RAPTP) is a variant of uncapacitated resource allocation problems generally referred as uncapacitated facility allocation problems or uncapacitated facility location problem (UFLP). Work done in this paper is motivated by the work of Du, Lu and Xu [7] in which authors considered facility location problems with submodular penalties and presented a 3-approximation primal dual algorithm. This paper considers that each unallocated demand point adds to penalty that increases as time passes and is thus represented by function [Formula: see text] where [Formula: see text] and [Formula: see text] are elapse time and priority of demand point [Formula: see text]. As this problem has been considered for emergency service allocation, all demand points should be allocated to some facility or resource within some stipulated time limit beyond which it may lose its purpose. Thus penalty incurred by a demand point is considered till that threshold value only. Thus it is assumed that penalty contribution by a demand point remains constant after a specified threshold value. By exploiting the properties of time dependent penalties, a 4-approximation primal-dual algorithm is proposed which is based on LP framework, and is the first constant-factor approximation algorithm for RAPTP.

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