29,076 research outputs found
The vector floor and ceiling model
This paper motivates and develops a nonlinear extension of the Vector Autoregressive model which we call the Vector Floor and Ceiling model. Bayesian and classical methods for estimation and testing are developed and compared in the context of an application involving U.S. macroeconomic data. In terms of statistical significance both classical and Bayesian methods indicate that the (Gaussian) linear model is inadequate. Using impulse response functions we investigate the economic significance of the statistical analysis. We find evidence of strong nonlinearities in the contemporaneous relationships between the variables and milder evidence of nonlinearity in the conditional mean
Quantum feedback control and classical control theory
We introduce and discuss the problem of quantum feedback control in the context of established formulations of classical control theory, examining conceptual analogies and essential differences. We describe the application of state-observer-based control laws, familiar in classical control theory, to quantum systems and apply our methods to the particular case of switching the state of a particle in a double-well potential
The Saffman-Taylor problem on a sphere
The Saffman-Taylor problem addresses the morphological instability of an
interface separating two immiscible, viscous fluids when they move in a narrow
gap between two flat parallel plates (Hele-Shaw cell). In this work, we extend
the classic Saffman-Taylor situation, by considering the flow between two
curved, closely spaced, concentric spheres (spherical Hele-Shaw cell). We
derive the mode-coupling differential equation for the interface perturbation
amplitudes and study both linear and nonlinear flow regimes. The effect of the
spherical cell (positive) spatial curvature on the shape of the interfacial
patterns is investigated. We show that stability properties of the fluid-fluid
interface are sensitive to the curvature of the surface. In particular, it is
found that positive spatial curvature inhibits finger tip-splitting. Hele-Shaw
flow on weakly negative, curved surfaces is briefly discussed.Comment: 26 pages, 4 figures, RevTex, accepted for publication in Phys. Rev.
Physics-informed Neural Networks for Solving Inverse Problems of Nonlinear Biot's Equations: Batch Training
In biomedical engineering, earthquake prediction, and underground energy
harvesting, it is crucial to indirectly estimate the physical properties of
porous media since the direct measurement of those are usually
impractical/prohibitive. Here we apply the physics-informed neural networks to
solve the inverse problem with regard to the nonlinear Biot's equations.
Specifically, we consider batch training and explore the effect of different
batch sizes. The results show that training with small batch sizes, i.e., a few
examples per batch, provides better approximations (lower percentage error) of
the physical parameters than using large batches or the full batch. The
increased accuracy of the physical parameters, comes at the cost of longer
training time. Specifically, we find the size should not be too small since a
very small batch size requires a very long training time without a
corresponding improvement in estimation accuracy. We find that a batch size of
8 or 32 is a good compromise, which is also robust to additive noise in the
data. The learning rate also plays an important role and should be used as a
hyperparameter.Comment: arXiv admin note: text overlap with arXiv:2002.0823
Cosmological Density and Power Spectrum from Peculiar Velocities: Nonlinear Corrections and PCA
We allow for nonlinear effects in the likelihood analysis of galaxy peculiar
velocities, and obtain ~35%-lower values for the cosmological density parameter
Om and the amplitude of mass-density fluctuations. The power spectrum in the
linear regime is assumed to be a flat LCDM model (h=0.65, n=1, COBE) with only
Om as a free parameter. Since the likelihood is driven by the nonlinear regime,
we "break" the power spectrum at k_b=0.2 h/Mpc and fit a power law at k>k_b.
This allows for independent matching of the nonlinear behavior and an unbiased
fit in the linear regime. The analysis assumes Gaussian fluctuations and
errors, and a linear relation between velocity and density. Tests using proper
mock catalogs demonstrate a reduced bias and a better fit. We find for the
Mark3 and SFI data Om_m=0.32+-0.06 and 0.37+-0.09 respectively, with
sigma_8*Om^0.6 = 0.49+-0.06 and 0.63+-0.08, in agreement with constraints from
other data. The quoted 90% errors include cosmic variance. The improvement in
likelihood due to the nonlinear correction is very significant for Mark3 and
moderately so for SFI. When allowing deviations from LCDM, we find an
indication for a wiggle in the power spectrum: an excess near k=0.05 and a
deficiency at k=0.1 (cold flow). This may be related to the wiggle seen in the
power spectrum from redshift surveys and the second peak in the CMB anisotropy.
A chi^2 test applied to modes of a Principal Component Analysis (PCA) shows
that the nonlinear procedure improves the goodness of fit and reduces a spatial
gradient of concern in the linear analysis. The PCA allows addressing spatial
features of the data and fine-tuning the theoretical and error models. It shows
that the models used are appropriate for the cosmological parameter estimation
performed. We address the potential for optimal data compression using PCA.Comment: 18 pages, LaTex, uses emulateapj.sty, ApJ in press (August 10, 2001),
improvements to text and figures, updated reference
Identifying Finite-Time Coherent Sets from Limited Quantities of Lagrangian Data
A data-driven procedure for identifying the dominant transport barriers in a
time-varying flow from limited quantities of Lagrangian data is presented. Our
approach partitions state space into pairs of coherent sets, which are sets of
initial conditions chosen to minimize the number of trajectories that "leak"
from one set to the other under the influence of a stochastic flow field during
a pre-specified interval in time. In practice, this partition is computed by
posing an optimization problem, which once solved, yields a pair of functions
whose signs determine set membership. From prior experience with synthetic,
"data rich" test problems and conceptually related methods based on
approximations of the Perron-Frobenius operator, we observe that the functions
of interest typically appear to be smooth. As a result, given a fixed amount of
data our approach, which can use sets of globally supported basis functions,
has the potential to more accurately approximate the desired functions than
other functions tailored to use compactly supported indicator functions. This
difference enables our approach to produce effective approximations of pairs of
coherent sets in problems with relatively limited quantities of Lagrangian
data, which is usually the case with real geophysical data. We apply this
method to three examples of increasing complexity: the first is the double
gyre, the second is the Bickley Jet, and the third is data from numerically
simulated drifters in the Sulu Sea.Comment: 14 pages, 7 figure
Radial fingering in a Hele-Shaw cell: a weakly nonlinear analysis
The Saffman-Taylor viscous fingering instability occurs when a less viscous
fluid displaces a more viscous one between narrowly spaced parallel plates in a
Hele-Shaw cell. Experiments in radial flow geometry form fan-like patterns, in
which fingers of different lengths compete, spread and split. Our weakly
nonlinear analysis of the instability predicts these phenomena, which are
beyond the scope of linear stability theory. Finger competition arises through
enhanced growth of sub-harmonic perturbations, while spreading and splitting
occur through the growth of harmonic modes. Nonlinear mode-coupling enhances
the growth of these perturbations with appropriate relative phases, as we
demonstrate through a symmetry analysis of the mode coupling equations. We
contrast mode coupling in radial flow with rectangular flow geometry.Comment: 36 pages, 5 figures, Latex, added references, to appear in Physica D
(1998
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