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![]() | New insights into one-norm solvers from the Pareto curve | ![]() |
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As a concrete example of the use of the Pareto curve in
the geophysical context, we study the problem of wavefield
reconstruction with sparsity-promoting inversion in the curvelet
domain (CRSI - Herrmann and Hennenfent, 2008). The simulated acquired data,
shown in Figure 4(a), corresponds to a shot record with 35%
of the traces missing. The interpolated result, shown in Figure
4(b), is obtained by solving BP
using SPG
.
This problem has more than half a million unknowns and forty-two
thousand data points.
The points in Figure 5 are samples of the corresponding Pareto curve. The regularity of these points strongly indicates that the underlying curve--which we know to be convex--is smooth and well behaved, and empirically supports our earlier claim. However problems of practical interest are often significantly larger, and it may be prohibitively expensive to compute a similarly fine sampling of the curve.
Because the curve is well behaved, we can leverage its smoothness and
use a small set of samples to obtain a good interpolation. The solid
line in Figure 5 shows an interpolation based only on
information from the circled samples. The interpolated curve closely
matches the samples that were not included in the interpolation. The
figure also plots the iterates taken by SPG
in order to obtain
the reconstruction shown in Figure 4(b). The plot shows
that the iterates remain to the Pareto curve and that they convergence
towards the BP
solution.
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data,interp
Figure 4. CRSI on synthetic data. (a) Input and (b) interpolated data using CRSI with SPG ![]() |
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res
Figure 5. Pareto curve and SPG ![]() ![]() ![]() |
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![]() | New insights into one-norm solvers from the Pareto curve | ![]() |
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