Tao, Molei (2011) Multiscale geometric integration of deterministic and stochastic systems. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:05262011-171044915
In order to accelerate computations and improve long time accuracy of numerical simulations, this thesis develops multiscale geometric integrators. For general multiscale stiff ODEs, SDEs, and PDEs, FLow AVeraging integratORs (FLAVORs) have been proposed for the coarse time-stepping without any identification of the slow or the fast variables. In the special case of deterministic and stochastic mechanical systems, symplectic, multisymplectic, and quasi-symplectic multiscale integrators are easily obtained using this strategy. For highly oscillatory mechanical systems (with quasi-quadratic stiff potentials and possibly high-dimensional), a specialized symplectic method has been devised to provide improved efficiency and accuracy. This method is based on the introduction of two highly nontrivial matrix exponentiation algorithms, which are generic, efficient, and symplectic (if the exact exponential is symplectic). For multiscale systems with Dirac-distributed fast processes, a family of symplectic, linearly-implicit and stable integrators has been designed for coarse step simulations. An application is the fast and accurate integration of constrained dynamics. In addition, if one cares about statistical properties of an ensemble of trajectories, but not the numerical accuracy of a single trajectory, we suggest tuning friction and annealing temperature in a Langevin process to accelerate its convergence. Other works include variational integration of circuits, efficient simulation of a nonlinear wave, and finding optimal transition pathways in stochastic dynamical systems (with a demonstration of mass effects in molecular dynamics).
|Item Type:||Thesis (Dissertation (Ph.D.))|
|Subject Keywords:||multiscale integration of ODEs, SDEs, and PDEs; structure preserving simulation; symplecticity and multisymplecticity; F-convergence; efficient and symplectic matrix exponentiation; linearly-implicit symplectic integration; temperature and friction accelerated sampling; variational integration of multiscale and noisy circuits; optimal transition by optimizing Freidlin-Wentzell action|
|Degree Grantor:||California Institute of Technology|
|Division:||Engineering and Applied Science|
|Major Option:||Control and Dynamical Systems|
|Thesis Availability:||Public (worldwide access)|
|Defense Date:||16 May 2011|
|Default Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Molei Tao|
|Deposited On:||27 May 2011 22:18|
|Last Modified:||26 Dec 2012 04:36|
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