Multi-scale problems on collective dynamics and image processing
| dc.contributor.advisor | Tadmor, Eitan | en_US |
| dc.contributor.author | Tan, Changhui | en_US |
| dc.contributor.department | Applied Mathematics and Scientific Computation | en_US |
| dc.contributor.publisher | Digital Repository at the University of Maryland | en_US |
| dc.contributor.publisher | University of Maryland (College Park, Md.) | en_US |
| dc.date.accessioned | 2014-10-11T05:47:01Z | |
| dc.date.available | 2014-10-11T05:47:01Z | |
| dc.date.issued | 2014 | en_US |
| dc.description.abstract | 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. | en_US |
| dc.identifier | https://doi.org/10.13016/M2WG6T | |
| dc.identifier.uri | http://hdl.handle.net/1903/15757 | |
| dc.language.iso | en | en_US |
| dc.subject.pqcontrolled | Applied mathematics | en_US |
| dc.subject.pquncontrolled | collective dynamics | en_US |
| dc.subject.pquncontrolled | flocking | en_US |
| dc.subject.pquncontrolled | image processing | en_US |
| dc.title | Multi-scale problems on collective dynamics and image processing | en_US |
| dc.type | Dissertation | en_US |
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