Interacting Particle Systems: Fast algorithms and non-convex optimization
Speaker: Prof. Dr. Shi Jin
Affiliation: Shanghai Jiao Tong University
Abstract: We first develop random batch methods for classical and quantum interacting particle systems with large number of particles. These methods use small but random batches for particle interactions, thus the computational cost is reduced from O(N^2) per time step to O(N), for a system with N particles with binary interactions. For classical particles we give a particle number independent error estimate under some special interactions. For quantum N-body Schrodinger equation, we obtain, for pair-wise random interactions, a convergence estimate for the Wigner transform of the single-particle reduced density matrix of the particle system at time t that is uniform in N > 1 and independent of the Planck constant hbar. We then introduce a stochastic interacting particle consensus system for global optimization of high dimensional non-convex functions. This algorithm does not use gradient of the function thus is suitable for non-smooth functions. We prove that under dimension-independent conditions on the parameters and with suitable initial data the algorithms converge to the neighborhood of the global minimum almost surely.