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The present work develops hybrid multigrid methods for high-order discontinuous Galerkin discretizations of elliptic problems. Fast matrix-free operator evaluation on tensor product elements is used to devise a computationally efficient PDE…
This paper considers the efficient exact computation of the counterpart of the Gittins index for a finite-horizon discrete-state bandit, which measures for each initial state the average productivity, given by the maximum ratio of expected…
We formulate a class of mean field games on a finite state space with variational principles resembling those in continuous-state mean field games. We construct a controlled continuity equation featuring a nonlinear activation function on…
We propose a locally conservative enriched Galerkin scheme that preserves the physical bounds for an elliptic problem. To this end, we use a substantial over-penalization of the discrete solution's jumps to obtain optimal convergence. To…
In this article, we propose two numerical methods, the Gaussian Process (GP) method and the Fourier Features (FF) algorithm, to solve mean field games (MFGs). The GP algorithm approximates the solution of a MFG with maximum a posteriori…
Mean-field games (MFGs) have shown strong modeling capabilities for large systems in various fields, driving growth in computational methods for mean-field game problems. However, high order methods have not been thoroughly investigated. In…
We consider a general class of mean field control problems described by stochastic delayed differential equations of McKean-Vlasov type. Two numerical algorithms are provided based on deep learning techniques, one is to directly…
We present a Reinforcement Learning (RL) algorithm to solve infinite horizon asymptotic Mean Field Game (MFG) and Mean Field Control (MFC) problems. Our approach can be described as a unified two-timescale Mean Field Q-learning: The…
In many scientific and engineering problems, noise and nonlinearity are unavoidable, which could induce interesting mathematical problem such as transition phenomena. This paper focuses on efficiently discovering the most probable…
Recent advances at the intersection of dense large graph limits and mean field games have begun to enable the scalable analysis of a broad class of dynamical sequential games with large numbers of agents. So far, results have been largely…
Multi-agent reinforcement learning methods have shown remarkable potential in solving complex multi-agent problems but mostly lack theoretical guarantees. Recently, mean field control and mean field games have been established as a…
We introduce a zero-sum game problem of mean-field type as an extension of the classical zero-sum Dynkin game problem to the case where the payoff processes might depend on the value of the game and its probability law. We establish…
We introduce a max-plus analogue of the Petrov-Galerkin finite element method, to solve finite horizon deterministic optimal control problems. The method relies on a max-plus variational formulation, and exploits the properties of…
Deep generative models (DGMs) are effective on learning multilayered representations of complex data and performing inference of input data by exploring the generative ability. However, little work has been done on examining or empowering…
Autonomous racing is gaining attention for its potential to advance autonomous vehicle technologies. Accurate race car dynamics modeling is essential for capturing and predicting future states like position, orientation, and velocity.…
Reinforcement learning algorithms such as the deep deterministic policy gradient algorithm (DDPG) has been widely used in continuous control tasks. However, the model-free DDPG algorithm suffers from high sample complexity. In this paper we…
Domain decomposition methods (DDMs) are popular solvers for discretized systems of partial differential equations (PDEs), with one-level and multilevel variants. These solvers rely on several algorithmic and mathematical parameters,…
These notes present preliminary results regarding two different approximations of linear infinite-horizon optimal control problems arising in model predictive control. Input and state trajectories are parametrized with basis functions and a…
This note discusses properties of parametric discrete-time Mixed-Integer Optimal Control Problems (MIOCPs) as they often arise in mixed-integer NMPC. We argue that in want for a handle on similarity properties of parametric MIOCPs the…
We present a finite-horizon optimization algorithm that extends the established concept of Dual Dynamic Programming (DDP) in two ways. First, in contrast to the linear costs, dynamics, and constraints of standard DDP, we consider problems…