140 research outputs found
High-dimensional wave atoms and compression of seismic datasets
Wave atoms are a low-redundancy alternative to curvelets, suitable for high-dimensional seismic data processing. This abstract extends the wave atom orthobasis construction to 3D, 4D, and 5D Cartesian arrays, and parallelizes it in a shared-memory environment. An implementation of the algorithm for NVIDIA CUDA capable graphics processing units (GPU) is also developed to accelerate computation for 2D and 3D data. The new transforms are benchmarked against the Fourier transform for compression of data generated from synthetic 2D and 3D acoustic models.National Science Foundation (U.S.); Alfred P. Sloan Foundatio
Wavelets on the sphere : Implementation and approximations
We continue the analysis of the continuous wavelet transform on the 2-sphere, introduced in a previous paper. After a brief review of the transform, we define and discuss the notion of directional spherical wavelet, i.e., wavelets on the sphere that are sensitive to directions. Then we present a calculation method for data given on a regular spherical grid . This technique, which uses the FFT, is based on the invariance of under discrete rotations around the z axis preserving the sampling. Next, a numerical criterion is given for controlling the scale interval where the spherical wavelet transform makes sense, and examples are given, both academic and realistic. In a second part, we establish conditions under which the reconstruction formula holds in strong Lp sense, for 1p<. This opens the door to techniques for approximating functions on the sphere, by use of an approximate identity, obtained by a suitable dilation of the mother wavelet
Conditioning bounds for traveltime tomography in layered media
This paper revisits the problem of recovering a smooth, isotropic, layered
wave speed profile from surface traveltime information. While it is classic
knowledge that the diving (refracted) rays classically determine the wave speed
in a weakly well-posed fashion via the Abel transform, we show in this paper
that traveltimes of reflected rays do not contain enough information to recover
the medium in a well-posed manner, regardless of the discretization. The
counterpart of the Abel transform in the case of reflected rays is a Fredholm
kernel of the first kind which is shown to have singular values that decay at
least root-exponentially. Kinematically equivalent media are characterized in
terms of a sequence of matching moments. This severe conditioning issue comes
on top of the well-known rearrangement ambiguity due to low velocity zones.
Numerical experiments in an ideal scenario show that a waveform-based model
inversion code fits data accurately while converging to the wrong wave speed
profile
On the spectral properties of L_{+-} in three dimensions
This paper is part of the radial asymptotic stability analysis of the ground
state soliton for either the cubic nonlinear Schrodinger or Klein-Gordon
equations in three dimensions. We demonstrate by a rigorous method that the
linearized scalar operators which arise in this setting, traditionally denoted
by L_{+-}, satisfy the gap property, at least over the radial functions. This
means that the interval (0,1] does not contain any eigenvalues of L_{+-} and
that the threshold 1 is neither an eigenvalue nor a resonance. The gap property
is required in order to prove scattering to the ground states for solutions
starting on the center-stable manifold associated with these states. This paper
therefore provides the final installment in the proof of this scattering
property for the cubic Klein-Gordon and Schrodinger equations in the radial
case, see the recent theory of Nakanishi and the third author, as well as the
earlier work of the third author and Beceanu on NLS. The method developed here
is quite general, and applicable to other spectral problems which arise in the
theory of nonlinear equations
Spectral Analysis for Matrix Hamiltonian Operators
In this work, we study the spectral properties of matrix Hamiltonians
generated by linearizing the nonlinear Schr\"odinger equation about soliton
solutions. By a numerically assisted proof, we show that there are no embedded
eigenvalues for the three dimensional cubic equation. Though we focus on a
proof of the 3d cubic problem, this work presents a new algorithm for verifying
certain spectral properties needed to study soliton stability. Source code for
verification of our comptuations, and for further experimentation, are
available at http://www.math.toronto.edu/simpson/files/spec_prop_code.tgz.Comment: 57 pages, 22 figures, typos fixe
Scattering in flatland: Efficient representations via wave atoms
This paper presents a numerical compression strategy for the boundary
integral equation of acoustic scattering in two dimensions. These equations
have oscillatory kernels that we represent in a basis of wave atoms, and
compress by thresholding the small coefficients to zero. This phenomenon was
perhaps first observed in 1993 by Bradie, Coifman, and Grossman, in the context
of local Fourier bases \cite{BCG}. Their results have since then been extended
in various ways. The purpose of this paper is to bridge a theoretical gap and
prove that a well-chosen fixed expansion, the nonstandard wave atom form,
provides a compression of the acoustic single and double layer potentials with
wave number as -by- matrices with
nonnegligible entries, with a constant that depends on the relative
accuracy \eps in an acceptable way. The argument assumes smooth, separated,
and not necessarily convex scatterers in two dimensions. The essential features
of wave atoms that enable to write this result as a theorem is a sharp
time-frequency localization that wavelet packets do not obey, and a parabolic
scaling wavelength (essential diameter). Numerical experiments
support the estimate and show that this wave atom representation may be of
interest for applications where the same scattering problem needs to be solved
for many boundary conditions, for example, the computation of radar cross
sections.Comment: 39 page
Global dynamics above the ground state for the nonlinear Klein-Gordon equation without a radial assumption
We extend our previous result on the focusing cubic Klein-Gordon equation in
three dimensions to the non-radial case, giving a complete classification of
global dynamics of all solutions with energy at most slightly above that of the
ground state.Comment: 40 page
A Centre-Stable Manifold for the Focussing Cubic NLS in
Consider the focussing cubic nonlinear Schr\"odinger equation in : It admits special solutions of the form
, where is a Schwartz function and a positive
() solution of The space of
all such solutions, together with those obtained from them by rescaling and
applying phase and Galilean coordinate changes, called standing waves, is the
eight-dimensional manifold that consists of functions of the form . We prove that any solution starting
sufficiently close to a standing wave in the norm and situated on a certain codimension-one local
Lipschitz manifold exists globally in time and converges to a point on the
manifold of standing waves. Furthermore, we show that \mc N is invariant
under the Hamiltonian flow, locally in time, and is a centre-stable manifold in
the sense of Bates, Jones. The proof is based on the modulation method
introduced by Soffer and Weinstein for the -subcritical case and adapted
by Schlag to the -supercritical case. An important part of the proof is
the Keel-Tao endpoint Strichartz estimate in for the nonselfadjoint
Schr\"odinger operator obtained by linearizing around a standing wave solution.Comment: 56 page
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