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Title: Analysis of gradient descents in random energies and heat baths
Author: Sullivan, Timothy John
ISNI:       0000 0004 2680 5237
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2009
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This thesis concerns the mathematical analysis of random gradient descent evolutions as models for rate-independent dissipative systems under the influence of thermal effects. The basic notions of the theory of gradient descents (especially rate-independent evolutions) are reviewed in chapter 2. Chapters 3 and 4 focus on the scaling regime in which the microstructure dominates the thermal effects and comprise a rigorous justification of rateindependent processes in smooth, convex energies as scaling limits of ratedependent gradient descents in energies that have rapidly-oscillating random microstructure: chapter 3 treats the one-dimensional case with quite a broad class of random microstructures; chapter 4 treats a case in which the microstructure is modeled by a sum of “dent functions” that are scattered in Rn using a suitable point process. Chapters 5 and 6 focus on the opposite scaling regime: a gradient descent system (typically a rate-independent process) is placed in contact with a heat bath. The method used to “thermalize” a gradient descent is an interior-point regularization of the Moreau–Yosida incremental problem for the original gradient descent. Chapter 5 treats the heuristics and generalities; chapter 6 treats the case of 1-homogeneous dissipation (rate independence) and shows that the heat bath destroys the rate independence in a controlled and deterministic way, and that the effective dynamics are a gradient descent in the original energetic potential but with respect to a different and non-trivial effective dissipation potential. The appendices contain some auxiliary definitions and results, most of them standard in the literature, that are used in the main text.
Supervisor: Not available Sponsor: Engineering and Physical Sciences Research Council (Great Britain) (EPSRC) ; University of Warwick (UoW) ;
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics