This thesis contains two different parts. For part I, we develop a PDE perspective on a model problem from the machine learning literature. The problem involves ''prediction" via ''regret minimization". Our PDE approach identifies the optimal strategies and the associated outcomes in a scaling limit where the number of time steps tends to 1. While our PDE is very nonlinear, explicit solutions are available in many cases due to a surprising and convenient link to the linear heat equation.
For part II, We consider H–αgradient flow of the Modica-Mortola energy. The evolution law can be viewed as a nonlocal analogue of the Cahn-Hilliard equation for α > 0; the limit as α= 0 is (formally at least) the volume preserving Allen-Cahn flow. When α > ½ , we prove a time-averaged upper bound on the typical length scale, of order t < 1/( 1 +2α) We also argue that for α > ½ the true time scale of the evolution is the original time scale (and that the case α < ½ is different). Finally, we derive the sharp interface limit of the flow using heuristic arguments.
|Advisor:||Kohn, Robert V.|
|School:||New York University|
|School Location:||United States -- New York|
|Source:||DAI-B 76/01(E), Dissertation Abstracts International|
|Keywords:||Machine learning, Modica-Mortola energy, Partial differential equations|
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