We are addressing the analysis and numerical methods for the tractable simulation and the optimal control of dynamical systems which are modeling the behavior of a large number N of complex interacting agents described by a large amount of parameters (high-dimension). We are facing fundamental challenges:
- Random projections and recovery for high-dimensional dynamical systems: we shall explore how concepts of data compression via Johnson-Lindenstrauss random embeddings onto lower-dimensional spaces can be applied for tractable simulation of complex dynamical interactions. As a fundamental subtask for the recovery of high-dimensional trajectories from low-dimensional simulated ones, we will address the efficient recovery of point clouds defined on embedded manifolds from random projections.
-Mean field equations: for the limit of the number N of agents to infinity, we shall further explore how the concepts of compression can be generalized to work for associated mean field equations.
- Approximating functions in high-dimension: differently from purely physical problems, in the real life the ”social forces” which are ruling the dynamics are actually not known. Hence we will address the problem of automatic learning from collected data the fundamental functions governing the dynamics.
- Homogenization of multibody systems: while the emphasis of our modelling is on “social” dynamics, we will also investigate methods to recast multibody systems into our high-dimensional framework in order to achieve nonstandard homogenization by random projections.
- Sparse optimal control in high-dimension and mean field optimal control: while self-organization of such dynamical systems has been so far a mainstream, we will focus on their sparse optimal control in high-dimension. We will investigate L1-minimization to design sparse optimal controls. We will learn high-dimensional (sparse) controls by random projections to lower dimension spaces and their mean field limit.
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