18,256 research outputs found
Observers for compressible Navier-Stokes equation
We consider a multi-dimensional model of a compressible fluid in a bounded
domain. We want to estimate the density and velocity of the fluid, based on the
observations for only velocity. We build an observer exploiting the symmetries
of the fluid dynamics laws. Our main result is that for the linearised system
with full observations of the velocity field, we can find an observer which
converges to the true state of the system at any desired convergence rate for
finitely many but arbitrarily large number of Fourier modes. Our
one-dimensional numerical results corroborate the results for the linearised,
fully observed system, and also show similar convergence for the full nonlinear
system and also for the case when the velocity field is observed only over a
subdomain
SO(3)-invariant asymptotic observers for dense depth field estimation based on visual data and known camera motion
In this paper, we use known camera motion associated to a video sequence of a
static scene in order to estimate and incrementally refine the surrounding
depth field. We exploit the SO(3)-invariance of brightness and depth fields
dynamics to customize standard image processing techniques. Inspired by the
Horn-Schunck method, we propose a SO(3)-invariant cost to estimate the depth
field. At each time step, this provides a diffusion equation on the unit
Riemannian sphere that is numerically solved to obtain a real time depth field
estimation of the entire field of view. Two asymptotic observers are derived
from the governing equations of dynamics, respectively based on optical flow
and depth estimations: implemented on noisy sequences of synthetic images as
well as on real data, they perform a more robust and accurate depth estimation.
This approach is complementary to most methods employing state observers for
range estimation, which uniquely concern single or isolated feature points.Comment: Submitte
Locally adaptive image denoising by a statistical multiresolution criterion
We demonstrate how one can choose the smoothing parameter in image denoising
by a statistical multiresolution criterion, both globally and locally. Using
inhomogeneous diffusion and total variation regularization as examples for
localized regularization schemes, we present an efficient method for locally
adaptive image denoising. As expected, the smoothing parameter serves as an
edge detector in this framework. Numerical examples illustrate the usefulness
of our approach. We also present an application in confocal microscopy
High-resolution simulations and modeling of reshocked single-mode Richtmyer-Meshkov instability: Comparison to experimental data and to amplitude growth model predictions
The reshocked single-mode Richtmyer-Meshkov instability is simulated in two spatial dimensions using the fifth- and ninth-order weighted essentially nonoscillatory shock-capturing method with uniform spatial resolution of 256 points per initial perturbation wavelength. The initial conditions and computational domain are modeled after the single-mode, Mach 1.21 air(acetone)/SF6 shock tube experiment of Collins and Jacobs [J. Fluid Mech. 464, 113 (2002)]. The simulation densities are shown to be in very good agreement with the corrected experimental planar laser-induced fluorescence images at selected times before reshock of the evolving interface. Analytical, semianalytical, and phenomenological linear and nonlinear, impulsive, perturbation, and potential flow models for single-mode Richtmyer-Meshkov unstable perturbation growth are summarized. The simulation amplitudes are shown to be in very good agreement with the experimental data and with the predictions of linear amplitude growth models for small times, and with those of nonlinear amplitude growth models at later times up to the time at which the driver-based expansion in the experiment (but not present in the simulations or models) expands the layer before reshock. The qualitative and quantitative differences between the fifth- and ninth-order simulation results are discussed. Using a local and global quantitative metric, the prediction of the Zhang and Sohn [Phys. Fluids 9, 1106 (1997)] nonlinear Padé model is shown to be in best overall agreement with the simulation amplitudes before reshock. The sensitivity of the amplitude growth model predictions to the initial growth rate from linear instability theory, the post-shock Atwood number and amplitude, and the velocity jump due to the passage of the shock through the interface is also investigated numerically
Numerical Methods for the Fractional Laplacian: a Finite Difference-quadrature Approach
The fractional Laplacian is a non-local operator which
depends on the parameter and recovers the usual Laplacian as . A numerical method for the fractional Laplacian is proposed, based on
the singular integral representation for the operator. The method combines
finite difference with numerical quadrature, to obtain a discrete convolution
operator with positive weights. The accuracy of the method is shown to be
. Convergence of the method is proven. The treatment of far
field boundary conditions using an asymptotic approximation to the integral is
used to obtain an accurate method. Numerical experiments on known exact
solutions validate the predicted convergence rates. Computational examples
include exponentially and algebraically decaying solution with varying
regularity. The generalization to nonlinear equations involving the operator is
discussed: the obstacle problem for the fractional Laplacian is computed.Comment: 29 pages, 9 figure
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