20,281 research outputs found
A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media
We present a fast method for numerically solving the inhomogeneous Helmholtz
equation. Our iterative method is based on the Born series, which we modified
to achieve convergence for scattering media of arbitrary size and scattering
strength. Compared to pseudospectral time-domain simulations, our modified Born
approach is two orders of magnitude faster and nine orders of magnitude more
accurate in benchmark tests in 1-dimensional and 2-dimensional systems
Shifted Laplacian multigrid for the elastic Helmholtz equation
The shifted Laplacian multigrid method is a well known approach for
preconditioning the indefinite linear system arising from the discretization of
the acoustic Helmholtz equation. This equation is used to model wave
propagation in the frequency domain. However, in some cases the acoustic
equation is not sufficient for modeling the physics of the wave propagation,
and one has to consider the elastic Helmholtz equation. Such a case arises in
geophysical seismic imaging applications, where the earth's subsurface is the
elastic medium. The elastic Helmholtz equation is much harder to solve than its
acoustic counterpart, partially because it is three times larger, and partially
because it models more complicated physics. Despite this, there are very few
solvers available for the elastic equation compared to the array of solvers
that are available for the acoustic one. In this work we extend the shifted
Laplacian approach to the elastic Helmholtz equation, by combining the complex
shift idea with approaches for linear elasticity. We demonstrate the efficiency
and properties of our solver using numerical experiments for problems with
heterogeneous media in two and three dimensions
On the indefinite Helmholtz equation: complex stretched absorbing boundary layers, iterative analysis, and preconditioning
This paper studies and analyzes a preconditioned Krylov solver for Helmholtz
problems that are formulated with absorbing boundary layers based on complex
coordinate stretching. The preconditioner problem is a Helmholtz problem where
not only the coordinates in the absorbing layer have an imaginary part, but
also the coordinates in the interior region. This results into a preconditioner
problem that is invertible with a multigrid cycle. We give a numerical analysis
based on the eigenvalues and evaluate the performance with several numerical
experiments. The method is an alternative to the complex shifted Laplacian and
it gives a comparable performance for the studied model problems
Shape and Trajectory Tracking of Moving Obstacles
This work presents new methods and algorithms for tracking the shape and
trajectory of moving reflecting obstacles with broken rays, or rays reflecting
at an obstacle. While in tomography the focus of the reconstruction method is
to recover the velocity structure of the domain, the shape and trajectory
reconstruction procedure directly finds the shape and trajectory of the
obstacle. The physical signal carrier for this innovative method are ultrasonic
beams. When the speed of sound is constant, the rays are straight line segments
and the shape and trajectory of moving objects will be reconstructed with
methods based on the travel time equation and ellipsoid geometry. For variable
speed of sound, we start with the eikonal equation and a system of differential
equations that has its origins in acoustics and seismology. In this case, the
rays are curves that are not necessarily straight line segments and we develop
algorithms for shape and trajectory tracking based on the numerical solution of
these equations. We present methods and algorithms for shape and trajectory
tracking of moving obstacles with reflected rays when the location of the
receiver of the reflected ray is not known in advance. The shape and trajectory
tracking method is very efficient because it is not necessary for the reflected
signal to traverse the whole domain or the same path back to the transmitter.
It could be received close to the point of reflection or far away from the
transmitter. This optimizes the energy spent by transmitters for tracking the
object, reduces signal attenuation and improves image resolution. It is a safe
and secure method. We also present algorithms for tracking the shape and
trajectory of absorbing obstacles. The new methods and algorithms for shape and
trajectory tracking enable new applications and an application to one-hop
Internet routing is presented.Comment: 22 pages, 2 figures, 2 table
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