42 research outputs found
A pseudodifferential equation with damping for one-way wave propagation in inhomogeneous acoustic media
A one-way wave equation is an evolution equation in one of the space
directions that describes (approximately) a wave field. The exact wave field is
approximated in a high frequency, microlocal sense. Here we derive the
pseudodifferential one-way wave equation for an inhomogeneous acoustic medium
using a known factorization argument. We give explicitly the two highest order
terms, that are necessary for approximating the solution. A wave front
(singularity) whose propagation velocity has non-zero component in the special
direction is correctly described. The equation can't describe singularities
propagating along turning rays, i.e. rays along which the velocity component in
the special direction changes sign. We show that incorrectly propagated
singularities are suppressed if a suitable dissipative term is added to the
equation.Comment: 15 page
A dispersion minimizing scheme for the 3-D Helmholtz equation based on ray theory
We develop a new dispersion minimizing compact finite difference scheme for
the Helmholtz equation in 2 and 3 dimensions. The scheme is based on a newly
developed ray theory for difference equations. A discrete Helmholtz operator
and a discrete operator to be applied to the source and the wavefields are
constructed. Their coefficients are piecewise polynomial functions of ,
chosen such that phase and amplitude errors are minimal. The phase errors of
the scheme are very small, approximately as small as those of the 2-D
quasi-stabilized FEM method and substantially smaller than those of
alternatives in 3-D, assuming the same number of gridpoints per wavelength is
used. In numerical experiments, accurate solutions are obtained in constant and
smoothly varying media using meshes with only five to six points per wavelength
and wave propagation over hundreds of wavelengths. When used as a coarse level
discretization in a multigrid method the scheme can even be used with downto
three points per wavelength. Tests on 3-D examples with up to degrees of
freedom show that with a recently developed hybrid solver, the use of coarser
meshes can lead to corresponding savings in computation time, resulting in good
simulation times compared to the literature.Comment: 33 pages, 12 figures, 6 table
Semiclassical analysis for the Kramers-Fokker-Planck equation
We study some accurate semiclassical resolvent estimates for operators that
are neither selfadjoint nor elliptic, and applications to the Cauchy problem.
In particular we get a precise description of the spectrum near the imaginary
axis and precise resolvent estimates inside the pseudo-spectrum. We apply our
results to the Kramers-Fokker-Planck operator
A mathematical framework for inverse wave problems in heterogeneous media
This paper provides a theoretical foundation for some common formulations of
inverse problems in wave propagation, based on hyperbolic systems of linear
integro-differential equations with bounded and measurable coefficients. The
coefficients of these time-dependent partial differential equations respresent
parametrically the spatially varying mechanical properties of materials. Rocks,
manufactured materials, and other wave propagation environments often exhibit
spatial heterogeneity in mechanical properties at a wide variety of scales, and
coefficient functions representing these properties must mimic this
heterogeneity. We show how to choose domains (classes of nonsmooth coefficient
functions) and data definitions (traces of weak solutions) so that optimization
formulations of inverse wave problems satisfy some of the prerequisites for
application of Newton's method and its relatives. These results follow from the
properties of a class of abstract first-order evolution systems, of which
various physical wave systems appear as concrete instances. Finite speed of
propagation for linear waves with bounded, measurable mechanical parameter
fields is one of the by-products of this theory