4,363 research outputs found
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
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 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
The prospect of using LES and DES in engineering design, and the research required to get there
In this paper we try to look into the future to divine how large eddy and
detached eddy simulations (LES and DES, respectively) will be used in the
engineering design process about 20-30 years from now. Some key challenges
specific to the engineering design process are identified, and some of the
critical outstanding problems and promising research directions are discussed.Comment: accepted for publication in the Royal Society Philosophical
Transactions
Periodic Shadowing Sensitivity Analysis of Chaotic Systems
The sensitivity of long-time averages of a hyperbolic chaotic system to
parameter perturbations can be determined using the shadowing direction, the
uniformly-bounded-in-time solution of the sensitivity equations. Although its
existence is formally guaranteed for certain systems, methods to determine it
are hardly available. One practical approach is the Least-Squares Shadowing
(LSS) algorithm (Q Wang, SIAM J Numer Anal 52, 156, 2014), whereby the
shadowing direction is approximated by the solution of the sensitivity
equations with the least square average norm. Here, we present an alternative,
potentially simpler shadowing-based algorithm, termed periodic shadowing. The
key idea is to obtain a bounded solution of the sensitivity equations by
complementing it with periodic boundary conditions in time. We show that this
is not only justifiable when the reference trajectory is itself periodic, but
also possible and effective for chaotic trajectories. Our error analysis shows
that periodic shadowing has the same convergence rates as LSS when the time
span is increased: the sensitivity error first decays as and then,
asymptotically as . We demonstrate the approach on the Lorenz
equations, and also show that, as tends to infinity, periodic shadowing
sensitivities converge to the same value obtained from long unstable periodic
orbits (D Lasagna, SIAM J Appl Dyn Syst 17, 1, 2018) for which there is no
shadowing error. Finally, finite-difference approximations of the sensitivity
are also examined, and we show that subtle non-hyperbolicity features of the
Lorenz system introduce a small, yet systematic, bias
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