Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Complete Deterministic Dynamics and Spectral Decomposition of the
  Ensemble Kalman Inversion   The Ensemble Kalman inversion (EKI), proposed by Iglesias et al. for the
solution of Bayesian inverse problems of type $y=A u^\dagger +\varepsilon$,
with $u^\dagger$ being an unknown parameter and $y$ a given datum, is a
powerful tool usually derived from a sequential Monte Carlo point of view. It
describes the dynamics of an ensemble of particles $\{u^j(t)\}_{j=1}^J$, whose
initial empirical measure is sampled from the prior, evolving over an
artificial time $t$ towards an approximate solution of the inverse problem.
Using spectral techniques, we provide a complete description of the
deterministic dynamics of EKI and their asymptotic behavior in parameter space.
In particular, we analyze the dynamics of deterministic EKI and mean-field EKI.
We demonstrate that the Bayesian posterior can only be recovered with the
mean-field limit and not with finite sample sizes or deterministic EKI.
Furthermore, we show that -- even in the deterministic case -- residuals in
parameter space do not decrease monotonously in the Euclidean norm and suggest
a problem-adapted norm, where monotonicity can be proved. Finally, we derive a
system of ordinary differential equations governing the spectrum and
eigenvectors of the covariance matrix.

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The categories of this paper are: ao-sci
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The categories of this paper are: funct-an
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The categories of this paper are: supr-con
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The categories of this paper are: mtrl-th
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The categories of this paper are: physics
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The categories of this paper are: chem-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: quant-ph
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The categories of this paper are: math-ph
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The categories of this paper are: solv-int
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-th