Multi-scale problems on collective dynamics and image processing

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Multi-scale problems appear in many contexts. In this thesis, we study two dif- ferent subjects involving multi-scale problems: (i) collective dynamics, and (ii) image processing.

For collective dynamics, we concentrate on flocking models, in particular, Cucker-Smale and Motsch-Tadmor systems. These models characterize the emergent behaviors of self-organized dynamics. We study flocking systems in three different scales, from microscopic agent-based models, through mesoscopic kineitc discriptions, to macroscopic fluid systems. Global existence theories are developed for all three scales, with the proof of asymptotic flocking behaviors. In the macroscopic level, a critical threhold phenomenon is addressed to obtain global regularity. Similar idea is implemented to other fluid systems as well, like Euler-Poisson equations. In the kinetic level, a discontinuous Galerkin method is introduced to overcome the numerical difficulty due to the precence of δ -singularity.

For image processing, we apply the idea of multi-scale image representation to construct uniformly bounded solutions for div U = F. Despite the fact that the equation is simple and linear, it is suprisingly true that its bounded solution can not be constructed through a linear procedure. In particular, the Holmholtz solution is not always bounded. A hierarchical construction of the bounded solution of the equation is proposed, borrowing the idea from image processing. We also present a numerical implementation to deal with the highly nonlinear construction procedure. Solid numerical result verifies that the constructed solution is indeed uniformly bounded.