37 research outputs found
Iterative algorithms for total variation-like reconstructions in seismic tomography
A qualitative comparison of total variation like penalties (total variation,
Huber variant of total variation, total generalized variation, ...) is made in
the context of global seismic tomography. Both penalized and constrained
formulations of seismic recovery problems are treated. A number of simple
iterative recovery algorithms applicable to these problems are described. The
convergence speed of these algorithms is compared numerically in this setting.
For the constrained formulation a new algorithm is proposed and its convergence
is proven.Comment: 28 pages, 8 figures. Corrected sign errors in formula (25
Nonlinear regularization techniques for seismic tomography
The effects of several nonlinear regularization techniques are discussed in
the framework of 3D seismic tomography. Traditional, linear, penalties
are compared to so-called sparsity promoting and penalties,
and a total variation penalty. Which of these algorithms is judged optimal
depends on the specific requirements of the scientific experiment. If the
correct reproduction of model amplitudes is important, classical damping
towards a smooth model using an norm works almost as well as
minimizing the total variation but is much more efficient. If gradients (edges
of anomalies) should be resolved with a minimum of distortion, we prefer
damping of Daubechies-4 wavelet coefficients. It has the additional
advantage of yielding a noiseless reconstruction, contrary to simple
minimization (`Tikhonov regularization') which should be avoided. In some of
our examples, the method produced notable artifacts. In addition we
show how nonlinear methods for finding sparse models can be
competitive in speed with the widely used methods, certainly under
noisy conditions, so that there is no need to shun penalizations.Comment: 23 pages, 7 figures. Typographical error corrected in accelerated
algorithms (14) and (20
Practical recipes for the model order reduction, dynamical simulation, and compressive sampling of large-scale open quantum systems
This article presents numerical recipes for simulating high-temperature and
non-equilibrium quantum spin systems that are continuously measured and
controlled. The notion of a spin system is broadly conceived, in order to
encompass macroscopic test masses as the limiting case of large-j spins. The
simulation technique has three stages: first the deliberate introduction of
noise into the simulation, then the conversion of that noise into an equivalent
continuous measurement and control process, and finally, projection of the
trajectory onto a state-space manifold having reduced dimensionality and
possessing a Kahler potential of multi-linear form. The resulting simulation
formalism is used to construct a positive P-representation for the thermal
density matrix. Single-spin detection by magnetic resonance force microscopy
(MRFM) is simulated, and the data statistics are shown to be those of a random
telegraph signal with additive white noise. Larger-scale spin-dust models are
simulated, having no spatial symmetry and no spatial ordering; the
high-fidelity projection of numerically computed quantum trajectories onto
low-dimensionality Kahler state-space manifolds is demonstrated. The
reconstruction of quantum trajectories from sparse random projections is
demonstrated, the onset of Donoho-Stodden breakdown at the Candes-Tao sparsity
limit is observed, a deterministic construction for sampling matrices is given,
and methods for quantum state optimization by Dantzig selection are given.Comment: 104 pages, 13 figures, 2 table