740 research outputs found
Deterministic Mean-field Ensemble Kalman Filtering
The proof of convergence of the standard ensemble Kalman filter (EnKF) from
Legland etal. (2011) is extended to non-Gaussian state space models. A
density-based deterministic approximation of the mean-field limit EnKF
(DMFEnKF) is proposed, consisting of a PDE solver and a quadrature rule. Given
a certain minimal order of convergence between the two, this extends
to the deterministic filter approximation, which is therefore asymptotically
superior to standard EnKF when the dimension . The fidelity of
approximation of the true distribution is also established using an extension
of total variation metric to random measures. This is limited by a Gaussian
bias term arising from non-linearity/non-Gaussianity of the model, which exists
for both DMFEnKF and standard EnKF. Numerical results support and extend the
theory
Fast logarithmic Fourier-Laplace transform of nonintegrable functions
We present an efficient and very flexible numerical fast Fourier-Laplace
transform, that extends the logarithmic Fourier transform (LFT) introduced by
Haines and Jones [Geophys. J. Int. 92(1):171 (1988)] for functions varying over
many scales to nonintegrable functions. In particular, these include cases of
the asymptotic form and with
arbitrary real . Furthermore, we prove that the numerical transform
converges exponentially fast in the number of data points, provided that the
function is analytic in a cone with a finite
opening angle around the real axis and satisfies
as with a positive constant , which is
the case for the class of functions with power-law tails. Based on these
properties we derive ideal transformation parameters and discuss how the
logarithmic Fourier transform can be applied to convolutions. The ability of
the logarithmic Fourier transform to perform these operations on multiscale
(non-integrable) functions with power-law tails with exponentially small errors
makes it the method of choice for many physical applications, which we
demonstrate on typical examples. These include benchmarks against known
analytical results inaccessible to other numerical methods, as well as physical
models near criticality.Comment: 14 pages, 8 figure
Lorenz-Mie theory for 2D scattering and resonance calculations
This PhD tutorial is concerned with a description of the two-dimensional
generalized Lorenz-Mie theory (2D-GLMT), a well-established numerical method
used to compute the interaction of light with arrays of cylindrical scatterers.
This theory is based on the method of separation of variables and the
application of an addition theorem for cylindrical functions. The purpose of
this tutorial is to assemble the practical tools necessary to implement the
2D-GLMT method for the computation of scattering by passive scatterers or of
resonances in optically active media. The first part contains a derivation of
the vector and scalar Helmholtz equations for 2D geometries, starting from
Maxwell's equations. Optically active media are included in 2D-GLMT using a
recent stationary formulation of the Maxwell-Bloch equations called
steady-state ab initio laser theory (SALT), which introduces new classes of
solutions useful for resonance computations. Following these preliminaries, a
detailed description of 2D-GLMT is presented. The emphasis is placed on the
derivation of beam-shape coefficients for scattering computations, as well as
the computation of resonant modes using a combination of 2D-GLMT and SALT. The
final section contains several numerical examples illustrating the full
potential of 2D-GLMT for scattering and resonance computations. These examples,
drawn from the literature, include the design of integrated polarization
filters and the computation of optical modes of photonic crystal cavities and
random lasers.Comment: This is an author-created, un-copyedited version of an article
published in Journal of Optics. IOP Publishing Ltd is not responsible for any
errors or omissions in this version of the manuscript or any version derived
from i
A study of digital gyro compensation loops
The feasibility is discussed of replacing existing state-of-the-art analog gyro compensation loops with digital computations. This was accomplished by designing appropriate compensation loops for the dry turned TDF gyro, selecting appropriate data conversion and processing techniques and algorithms, and breadboarding the design for laboratory evaluation. A breadboard design was established in which one axis of a Teledyne turned-gimbal TDF gyro was caged digitally while the other was caged using conventional analog electronics. The digital loop was designed analytically to closely resemble the analog loop in performance. The breadboard was subjected to various static and dynamic tests in order to establish the relative stability characteristics and frequency responses of the digital and analog loops. Several variations of the digital loop configuration were evaluated. The results were favorable
Wavelet Estimators in Nonparametric Regression: A Comparative Simulation Study
Wavelet analysis has been found to be a powerful tool for the nonparametric estimation of spatially-variable objects. We discuss in detail wavelet methods in nonparametric regression, where the data are modelled as observations of a signal contaminated with additive Gaussian noise, and provide an extensive review of the vast literature of wavelet shrinkage and wavelet thresholding estimators developed to denoise such data. These estimators arise from a wide range of classical and empirical Bayes methods treating either individual or blocks of wavelet coefficients. We compare various estimators in an extensive simulation study on a variety of sample sizes, test functions, signal-to-noise ratios and wavelet filters. Because there is no single criterion that can adequately summarise the behaviour of an estimator, we use various criteria to measure performance in finite sample situations. Insight into the performance of these estimators is obtained from graphical outputs and numerical tables. In order to provide some hints of how these estimators should be used to analyse real data sets, a detailed practical step-by-step illustration of a wavelet denoising analysis on electrical consumption is provided. Matlab codes are provided so that all figures and tables in this paper can be reproduced
Data Assimilation: A Mathematical Introduction
These notes provide a systematic mathematical treatment of the subject of
data assimilation
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