727 research outputs found
Theoretical proof of the existence of characteristic diffuse light in natural waters
This paper develops a mathematical model for the radiance distribution of light penetrating a homogeneous hydrosol on the basis of the general theory of radiative transfer. It is proved that the radiance distribution approaches an asymptotic pattern at great depths. This is in accord with previous field measurements of the directional patterns in underwater light and with L. V. Whitney\u27s conjecture that there is at some depth in natural waters a characteristic diffuse light symmetrically distributed around the vertical. The angular form of this equilibrium light pattern is derived in terms of the mathematical model presented
Decomposing data sets into skewness modes
We derive the nonlinear equations satisfied by the coefficients of linear
combinations that maximize their skewness when their variance is constrained to
take a specific value. In order to numerically solve these nonlinear equations
we develop a gradient-type flow that preserves the constraint. In combination
with the Karhunen-Lo\`eve decomposition this leads to a set of orthogonal modes
with maximal skewness. For illustration purposes we apply these techniques to
atmospheric data; in this case the maximal-skewness modes correspond to
strongly localized atmospheric flows. We show how these ideas can be extended,
for example to maximal-flatness modes.Comment: Submitted for publication, 12 pages, 4 figure
On dimension reduction in Gaussian filters
A priori dimension reduction is a widely adopted technique for reducing the
computational complexity of stationary inverse problems. In this setting, the
solution of an inverse problem is parameterized by a low-dimensional basis that
is often obtained from the truncated Karhunen-Loeve expansion of the prior
distribution. For high-dimensional inverse problems equipped with smoothing
priors, this technique can lead to drastic reductions in parameter dimension
and significant computational savings.
In this paper, we extend the concept of a priori dimension reduction to
non-stationary inverse problems, in which the goal is to sequentially infer the
state of a dynamical system. Our approach proceeds in an offline-online
fashion. We first identify a low-dimensional subspace in the state space before
solving the inverse problem (the offline phase), using either the method of
"snapshots" or regularized covariance estimation. Then this subspace is used to
reduce the computational complexity of various filtering algorithms - including
the Kalman filter, extended Kalman filter, and ensemble Kalman filter - within
a novel subspace-constrained Bayesian prediction-and-update procedure (the
online phase). We demonstrate the performance of our new dimension reduction
approach on various numerical examples. In some test cases, our approach
reduces the dimensionality of the original problem by orders of magnitude and
yields up to two orders of magnitude in computational savings
Karhunen-Lo`eve Decomposition of Extensive Chaos
We show that the number of KLD (Karhunen-Lo`eve decomposition) modes D_KLD(f)
needed to capture a fraction f of the total variance of an extensively chaotic
state scales extensively with subsystem volume V. This allows a correlation
length xi_KLD(f) to be defined that is easily calculated from spatially
localized data. We show that xi_KLD(f) has a parametric dependence similar to
that of the dimension correlation length and demonstrate that this length can
be used to characterize high-dimensional inhomogeneous spatiotemporal chaos.Comment: 12 pages including 4 figures, uses REVTeX macros. To appear in Phys.
Rev. Let
Low-dimensional dynamical system model for observed coherent structures in ocean satellite data
The dynamics of coherent structures present in real-world environmental data
is analyzed. The method developed in this Paper combines the power of the
Proper Orthogonal Decomposition (POD) technique to identify these coherent
structures in experimental data sets, and its optimality in providing Galerkin
basis for projecting and reducing complex dynamical models. The POD basis used
is the one obtained from the experimental data. We apply the procedure to
analyze coherent structures in an oceanic setting, the ones arising from
instabilities of the Algerian current, in the western Mediterranean Sea. Data
are from satellite altimetry providing Sea Surface Height, and the model is a
two-layer quasigeostrophic system. A four-dimensional dynamical system is
obtained that correctly describe the observed coherent structures (moving
eddies). Finally, a bifurcation analysis is performed on the reduced model.Comment: 23 pages, 7 figure
Statistics of Atmospheric Correlations
For a large class of quantum systems the statistical properties of their
spectrum show remarkable agreement with random matrix predictions. Recent
advances show that the scope of random matrix theory is much wider. In this
work, we show that the random matrix approach can be beneficially applied to a
completely different classical domain, namely, to the empirical correlation
matrices obtained from the analysis of the basic atmospheric parameters that
characterise the state of atmosphere. We show that the spectrum of atmospheric
correlation matrices satisfy the random matrix prescription. In particular, the
eigenmodes of the atmospheric empirical correlation matrices that have physical
significance are marked by deviations from the eigenvector distribution.Comment: 8 pages, 9 figs, revtex; To appear in Phys. Rev.
Wishart and Anti-Wishart random matrices
We provide a compact exact representation for the distribution of the matrix
elements of the Wishart-type random matrices , for any finite
number of rows and columns of , without any large N approximations. In
particular we treat the case when the Wishart-type random matrix contains
redundant, non-random information, which is a new result. This representation
is of interest for a procedure of reconstructing the redundant information
hidden in Wishart matrices, with potential applications to numerous models
based on biological, social and artificial intelligence networks.Comment: 11 pages; v2: references updated + some clarifications added; v3:
version to appear in J. Phys. A, Special Issue on Random Matrix Theor
Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
We present a simple Coulomb gas method to calculate analytically the
probability of rare events where the maximum eigenvalue of a random matrix is
much larger than its typical value. The large deviation function that
characterizes this probability is computed explicitly for Wishart and Gaussian
ensembles. The method is quite general and applies to other related problems,
e.g. the joint large deviation function for large fluctuations of top
eigenvalues. Our results are relevant to widely employed data compression
techniques, namely the principal components analysis. Analytical predictions
are verified by extensive numerical simulations.Comment: 4 pages, 3 .eps figures include
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