A Caltech Library Service

Multiscale Geometric Integration of Deterministic and Stochastic Systems


Tao, Molei (2011) Multiscale Geometric Integration of Deterministic and Stochastic Systems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/6J83-7C18.


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
Minor Option:Physics
Awards:The W.P. Carey and Co., Inc., Prize in Applied Mathematics, 2011.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Owhadi, Houman (advisor)
  • Marsden, Jerrold E. (co-advisor)
Thesis Committee:
  • Owhadi, Houman (chair)
  • Hou, Thomas Y.
  • Desbrun, Mathieu
  • Murray, Richard M.
  • Ortiz, Michael
Defense Date:16 May 2011
Record Number:CaltechTHESIS:05262011-171044915
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:6457
Deposited By: Molei Tao
Deposited On:27 May 2011 22:18
Last Modified:09 Oct 2019 17:10

Thesis Files

PDF - Final Version
See Usage Policy.


Repository Staff Only: item control page