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Graduate Exam Abstract


Yang Zhang

Ph.D. Preliminary

November 2, 2011, 10:00am-12:00noon

Gifford 312

Continuum Limits of Markov Chains with Application to Network Modeling


Abstract: In this research we investigate the continuum limits of a class of Markov chains. The investigation of such limits is motivated by the desire to model networks with a very large number of nodes. We show that a sequence of such Markov chains indexed by N, the number of components in the system that they model, converges in some sense to its continuum limit, which is the solution of a partial differential equation (PDE), as N goes to infinity. We provide sufficient conditions for the convergence and characterize the rate of convergence. Our approach directly approximates the state of Markov chains at specific temporal and spatial points by continuum limits that are deterministic functions, in contrast to approximation methods that study the proportion of components in certain states, and to those that study the distribution of continuum limits that are random processes. As an application we approximate Markov chains modeling large networks by PDEs, thus capturing the dynamic time-space characteristics of the networks. While traditional Monte Carlo simulation for very large networks is practically infeasible, PDEs can be solved with reasonable computation overhead using well- established mathematical tools.

Adviser: Edwin K. P. Chong
Co-Adviser: N/A
Non-ECE Member: Donald Estep
Member 3: J. Rockey Luo
Addional Members: Ali Pezeshki

Publications:
Y. Zhang, E. K. P. Chong, J. Hannig, and D. Estep, "On continuum limits of Markov chains and network modeling," in Decision and Control (CDC), 2010 49th IEEE Conference on, pp.6779-6784, 15-17 Dec. 2010.
Y. Zhang, E. K. P. Chong, J. Hannig, and D. Estep, "Continuum limits of Markov chains and network modeling," presented at The Mathematical Association of America Rocky Mountain Section Meeting, 2010.
Y. Zhang and E. K. P. Chong, "Continuum models of large networks," presented in the poster session of the Graduate Student Visit Day of the ECE Dept., Colorado State University, 2010.
Y. Zhang, E. K. P. Chong, J. Hannig, and D. J. Estep, "Continuum limits of Markov chains with application to network modeling," arXiv:1106.4288v1, 2011.


Program of Study:
ECE 514
ECE 516
ECE 520
ECE 752
ECE 799
MATH 545
STAT 720
N/A