100 research outputs found
Scaling Law for Recovering the Sparsest Element in a Subspace
We address the problem of recovering a sparse -vector within a given
subspace. This problem is a subtask of some approaches to dictionary learning
and sparse principal component analysis. Hence, if we can prove scaling laws
for recovery of sparse vectors, it will be easier to derive and prove recovery
results in these applications. In this paper, we present a scaling law for
recovering the sparse vector from a subspace that is spanned by the sparse
vector and random vectors. We prove that the sparse vector will be the
output to one of linear programs with high probability if its support size
satisfies . The scaling law still holds when
the desired vector is approximately sparse. To get a single estimate for the
sparse vector from the linear programs, we must select which output is the
sparsest. This selection process can be based on any proxy for sparsity, and
the specific proxy has the potential to improve or worsen the scaling law. If
sparsity is interpreted in an sense, then the scaling law
can not be better than . Computer simulations show that
selecting the sparsest output in the or thresholded-
senses can lead to a larger parameter range for successful recovery than that
given by the sense
Numerical verification of a gap condition for a linearized nonlinear Schrödinger equation
We make a detailed numerical study of the spectrum of two Schrödinger operators L± arising from the linearization of the supercritical nonlinear Schrödinger equation (NLS) about the standing wave, in three dimensions. This study was motivated by a recent result of the second author on the conditional asymptotic stability of solitary waves in the case of a cubic nonlinearity. Underlying the validity of this result is a spectral condition on the operators L±, namely that they have no eigenvalues nor resonances in the gap (a region of the positive real axis between zero and the continuous spectrum), which we call the gap property. The present numerical study verifies this spectral condition and shows further that the gap property holds for NLS exponents of the form 2 β + 1, as long as β* < β ≤ 1, where
Our strategy consists of rewriting the original eigenvalue problem via the Birman–Schwinger method. From a numerical analysis viewpoint, our main contribution is an efficient quadrature rule for the kernel 1/|x - y| in {\mathbb R}^3 , i.e. proved spectrally accurate. As a result, we are able to give similar accuracy estimates for all our eigenvalue computations. We also propose an improvement in Petviashvili's iteration for the computation of standing wave profiles which automatically chooses the radial solution.
All our numerical experiments are reproducible. The Matlab code can be downloaded from http://www.acm.caltech.edu/~demanet/NLS/
Full waveform inversion with extrapolated low frequency data
The availability of low frequency data is an important factor in the success
of full waveform inversion (FWI) in the acoustic regime. The low frequencies
help determine the kinematically relevant, low-wavenumber components of the
velocity model, which are in turn needed to avoid convergence of FWI to
spurious local minima. However, acquiring data below 2 or 3 Hz from the field
is a challenging and expensive task. In this paper we explore the possibility
of synthesizing the low frequencies computationally from high-frequency data,
and use the resulting prediction of the missing data to seed the frequency
sweep of FWI. As a signal processing problem, bandwidth extension is a very
nonlinear and delicate operation. It requires a high-level interpretation of
bandlimited seismic records into individual events, each of which is
extrapolable to a lower (or higher) frequency band from the non-dispersive
nature of the wave propagation model. We propose to use the phase tracking
method for the event separation task. The fidelity of the resulting
extrapolation method is typically higher in phase than in amplitude. To
demonstrate the reliability of bandwidth extension in the context of FWI, we
first use the low frequencies in the extrapolated band as data substitute, in
order to create the low-wavenumber background velocity model, and then switch
to recorded data in the available band for the rest of the iterations. The
resulting method, EFWI for short, demonstrates surprising robustness to the
inaccuracies in the extrapolated low frequency data. With two synthetic
examples calibrated so that regular FWI needs to be initialized at 1 Hz to
avoid local minima, we demonstrate that FWI based on an extrapolated [1, 5] Hz
band, itself generated from data available in the [5, 15] Hz band, can produce
reasonable estimations of the low wavenumber velocity models
Velocity estimation via registration-guided least-squares inversion
This paper introduces an iterative scheme for acoustic model inversion where
the notion of proximity of two traces is not the usual least-squares distance,
but instead involves registration as in image processing. Observed data are
matched to predicted waveforms via piecewise-polynomial warpings, obtained by
solving a nonconvex optimization problem in a multiscale fashion from low to
high frequencies. This multiscale process requires defining low-frequency
augmented signals in order to seed the frequency sweep at zero frequency.
Custom adjoint sources are then defined from the warped waveforms. The proposed
velocity updates are obtained as the migration of these adjoint sources, and
cannot be interpreted as the negative gradient of any given objective function.
The new method, referred to as RGLS, is successfully applied to a few scenarios
of model velocity estimation in the transmission setting. We show that the new
method can converge to the correct model in situations where conventional
least-squares inversion suffers from cycle-skipping and converges to a spurious
model.Comment: 20 pages, 13 figures, 1 tabl
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