30 research outputs found
Probability density adjoint for sensitivity analysis of the Mean of Chaos
Sensitivity analysis, especially adjoint based sensitivity analysis, is a
powerful tool for engineering design which allows for the efficient computation
of sensitivities with respect to many parameters. However, these methods break
down when used to compute sensitivities of long-time averaged quantities in
chaotic dynamical systems.
The following paper presents a new method for sensitivity analysis of {\em
ergodic} chaotic dynamical systems, the density adjoint method. The method
involves solving the governing equations for the system's invariant measure and
its adjoint on the system's attractor manifold rather than in phase-space. This
new approach is derived for and demonstrated on one-dimensional chaotic maps
and the three-dimensional Lorenz system. It is found that the density adjoint
computes very finely detailed adjoint distributions and accurate sensitivities,
but suffers from large computational costs.Comment: 29 pages, 27 figure
Least Squares Shadowing Sensitivity Analysis of a Modified Kuramoto-Sivashinsky Equation
Computational methods for sensitivity analysis are invaluable tools for
scientists and engineers investigating a wide range of physical phenomena.
However, many of these methods fail when applied to chaotic systems, such as
the Kuramoto-Sivashinsky (K-S) equation, which models a number of different
chaotic systems found in nature. The following paper discusses the application
of a new sensitivity analysis method developed by the authors to a modified K-S
equation. We find that least squares shadowing sensitivity analysis computes
accurate gradients for solutions corresponding to a wide range of system
parameters.Comment: 23 pages, 14 figures. Submitted to Chaos, Solitons and Fractals, in
revie
Least Squares Shadowing sensitivity analysis of chaotic limit cycle oscillations
The adjoint method, among other sensitivity analysis methods, can fail in
chaotic dynamical systems. The result from these methods can be too large,
often by orders of magnitude, when the result is the derivative of a long time
averaged quantity. This failure is known to be caused by ill-conditioned
initial value problems. This paper overcomes this failure by replacing the
initial value problem with the well-conditioned "least squares shadowing (LSS)
problem". The LSS problem is then linearized in our sensitivity analysis
algorithm, which computes a derivative that converges to the derivative of the
infinitely long time average. We demonstrate our algorithm in several dynamical
systems exhibiting both periodic and chaotic oscillations.Comment: submitted to JCP in revised for
Least Squares Shadowing method for sensitivity analysis of differential equations
For a parameterized hyperbolic system the derivative
of the ergodic average to the parameter can be computed via
the Least Squares Shadowing algorithm (LSS). We assume that the sytem is
ergodic which means that depends only on (not on the
initial condition of the hyperbolic system). After discretizing this continuous
system using a fixed timestep, the algorithm solves a constrained least squares
problem and, from the solution to this problem, computes the desired derivative
. The purpose of this paper is to prove that the
value given by the LSS algorithm approaches the exact derivative when the
discretization timestep goes to and the timespan used to formulate the
least squares problem grows to infinity.Comment: 21 pages, this article complements arXiv:1304.3635 and analyzes LSS
for the case of continuous hyperbolic system
Multigrid-in-time for sensitivity analysis of chaotic dynamical systems
The following paper discusses the application of a multigrid-in-time scheme to Least Squares Shadowing (LSS), a novel sensitivity analysis method for chaotic dynamical systems. While traditional sensitivity analysis methods break down for chaotic dynamical systems, LSS is able to compute accurate gradients. Multigrid is used because LSS requires solving a very large Karush–Kuhn–Tucker system constructed from the solution of the dynamical system over the entire time interval of interest. Several different multigrid-in-time schemes are examined, and a number of factors were found to heavily influence the convergence rate of multigrid-in-time for LSS. These include the iterative method used for the smoother, how the coarse grid system is formed and how the least squares objective function at the center of LSS is weighted.United States. National Aeronautics and Space Administration (Award NNH11ZEA001N)American Society for Engineering Education. National Defense Science and Engineering Graduate Fellowshi