Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Stability of Filters for the Navier-Stokes Equation   Data assimilation methodologies are designed to incorporate noisy
observations of a physical system into an underlying model in order to infer
the properties of the state of the system. Filters refer to a class of data
assimilation algorithms designed to update the estimation of the state in a
on-line fashion, as data is acquired sequentially. For linear problems subject
to Gaussian noise filtering can be performed exactly using the Kalman filter.
For nonlinear systems it can be approximated in a systematic way by particle
filters. However in high dimensions these particle filtering methods can break
down. Hence, for the large nonlinear systems arising in applications such as
weather forecasting, various ad hoc filters are used, mostly based on making
Gaussian approximations. The purpose of this work is to study the properties of
these ad hoc filters, working in the context of the 2D incompressible
Navier-Stokes equation. By working in this infinite dimensional setting we
provide an analysis which is useful for understanding high dimensional
filtering, and is robust to mesh-refinement. We describe theoretical results
showing that, in the small observational noise limit, the filters can be tuned
to accurately track the signal itself (filter stability), provided the system
is observed in a sufficiently large low dimensional space; roughly speaking
this space should be large enough to contain the unstable modes of the
linearized dynamics. Numerical results are given which illustrate the theory.
In a simplified scenario we also derive, and study numerically, a stochastic
PDE which determines filter stability in the limit of frequent observations,
subject to large observational noise. The positive results herein concerning
filter stability complement recent numerical studies which demonstrate that the
ad hoc filters perform poorly in reproducing statistical variation about the
true signal.

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The categories of this paper are: gr-qc
23
The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ph
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The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
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The categories of this paper are: hep-lat
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The categories of this paper are: cond-mat
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The categories of this paper are: cs
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The categories of this paper are: eess
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-th
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ex
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The categories of this paper are: adap-org
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The categories of this paper are: dg-ga