@article{38746,
  keywords = {equations, geometric diffusions, harmonic-analysis, nonlinear dimensionality reduction, pattern-formation, structure definition, proper orthogonal decomposition, diffusion maps, approximate inertial manifolds, dimensionality reduction, eigenvectors, Alignment, Heat kernel, Local principal component   analysis},
  author = {Benjamin Sonday and Amit Singer and Ioannis Kevrekidis},
  title = {Noisy dynamic simulations in the presence of symmetry: Data alignment and model reduction},
  abstract = { <p>We process snapshots of trajectories of evolution equations with intrinsic symmetries, and demonstrate the use of recently developed eigenvector-based techniques to successfully quotient out the degrees of freedom associated with the symmetries in the presence of noise. Our illustrative examples include a one-dimensional evolutionary partial differential (the Kuramoto-Sivashinsky) equation with periodic boundary conditions, as well as a stochastic simulation of nematic liquid crystals which can be effectively modeled through a nonlinear Smoluchowski equation on the surface of a sphere. This is a useful first step towards data mining the symmetry-adjusted ensemble of snapshots in search of an accurate low-dimensional parametrization and the associated reduction of the original dynamical system. We also demonstrate a technique (Vector Diffusion Maps) that combines, in a single formulation, the symmetry removal step and the dimensionality reduction step. (C) 2013 Elsevier Ltd. All rights reserved.</p> },
  year = {2013},
  journal = {Computers \& Mathematics with Applications},
  volume = {65},
  pages = {1535-1557},
  month = {05/2013},
  isbn = {0898-1221},
  language = {English},
}