68 research outputs found
Data-based stochastic model reduction for the Kuramoto--Sivashinsky equation
The problem of constructing data-based, predictive, reduced models for the
Kuramoto-Sivashinsky equation is considered, under circumstances where one has
observation data only for a small subset of the dynamical variables. Accurate
prediction is achieved by developing a discrete-time stochastic reduced system,
based on a NARMAX (Nonlinear Autoregressive Moving Average with eXogenous
input) representation. The practical issue, with the NARMAX representation as
with any other, is to identify an efficient structure, i.e., one with a small
number of terms and coefficients. This is accomplished here by estimating
coefficients for an approximate inertial form. The broader significance of the
results is discussed.Comment: 23 page, 7 figure
Stochastic Optimal Prediction with Application to Averaged Euler Equations
Optimal prediction (OP) methods compensate for a lack of resolution in the
numerical solution of complex problems through the use of an invariant measure
as a prior measure in the Bayesian sense. In first-order OP, unresolved
information is approximated by its conditional expectation with respect to the
invariant measure. In higher-order OP, unresolved information is approximated
by a stochastic estimator, leading to a system of random or stochastic
differential equations.
We explain the ideas through a simple example, and then apply them to the
solution of Averaged Euler equations in two space dimensions.Comment: 13 pages, 2 figure
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