Research

Non-Intrusive Least Squares Shadowing (NILSS)

• The sensitivity can be reflected by comparing a base trajectory and its shadowing trajectory.
  The shadowing trajectory is one with a perturbed parameter, but the first order approximation of its difference with the base trajectory does not diverge.
  This is a classical result.
  
• Such shadowing trajectory can be found through a minimization on the L-2 norm of the distance between the base trajectory and the shadowing trajectory.
  This idea is due to Qiqi Wang.
  The corresponding numerical algorithm is called the Least Squares Shadowing (LSS).
  
• With a particular presentation, the minimization problem in LSS can be carried out only in the unstable directions.
  This greatly reduces the computational cost.
  NILSS is one of the first algorithms efficient enough to be applied on complicated problems like 3-D chaotic flows.
  

Adjoint Shadowing Direction and Non-Intrusive Least Squares Adjoint Shadowing (NILSAS)

• The shadowing direction can be viewed as the image of a linear operator, whose adjoint operator gives the adjoint shadowing direction.
  
• The adjoint shadowing direction has the following properties:
  1) an inhomogeneous adjoint solution;
  2) its L-2 inner-product with the trajectory direction is zero;
  3) its norm is bounded by a constant independent of the trajectory length.
  
• It turns out, if we can find a function with these properties, this function approximates the adjoint shadowing direction.
  This reverse engineering is the first step of deriving the NILSAS algorithm.
• Since the adjoint flow has the same Covariant Lyapunov Vector (CLV) structure as the tangent flow,
  we can find a 'non-intrusive' formulation of the least squares problem, which allows NILSAS
  1) be built using existing adjoint solvers;
  2) constrain the minimization to over only the unstable adjoint directions.