Title page for ETD etd-12092009-143340


Type of Document Dissertation
Author Zhang, Jingwei
Author's Email Address jwzhang@vt.edu
URN etd-12092009-143340
Title Numerical Methods for the Chemical Master Equation
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Watson, Layne T. Committee Chair
Beattie, Christopher A. Committee Member
Herdman, Terry L. Committee Member
Lin, Tao Committee Member
Ribbens, Calvin J. Committee Member
Keywords
  • Collocation Method
  • Radial Basis Function
  • Shepard Algorithm
  • M-estimation
  • Uniformization/Randomization Method
  • Aggregation/Disaggregation
  • Uniformization/Randomization Method
  • Stochastic Simulation Algorithm
  • Parallel Computing
  • Chemical Master Equation
  • Radial Basis Function
  • Stochastic Simulation Algorithm
  • Chemical Master Equation
  • Aggregation/Disaggregation
  • Parallel Computing
  • Collocation Method
Date of Defense 2009-12-02
Availability unrestricted
Abstract
The chemical master equation, formulated on the Markov assumption of underlying chemical kinetics, offers an accurate stochastic description of general chemical reaction systems on the mesoscopic scale. The chemical master equation is especially useful when formulating mathematical models of gene regulatory networks and protein-protein interaction networks, where the numbers of molecules of most species are around tens or hundreds. However, solving the master equation directly suffers from the so called "curse of dimensionality" issue. This thesis first tries to study the numerical properties of the master equation using existing numerical methods and parallel machines. Next, approximation algorithms, namely the adaptive aggregation method and the radial basis function collocation method, are proposed as new paths to resolve the "curse of dimensionality". Several numerical results are presented to illustrate the promises and potential problems of these new algorithms. Comparisons with other numerical methods like Monte Carlo methods are also included. Development and analysis of the linear Shepard algorithm and its variants, all of which could be used for high dimensional scattered data interpolation problems, are also included here, as a candidate to help solve the master equation by building surrogate models in high dimensions.
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