973 research outputs found
A new approach to nonlinear constrained Tikhonov regularization
We present a novel approach to nonlinear constrained Tikhonov regularization
from the viewpoint of optimization theory. A second-order sufficient optimality
condition is suggested as a nonlinearity condition to handle the nonlinearity
of the forward operator. The approach is exploited to derive convergence rates
results for a priori as well as a posteriori choice rules, e.g., discrepancy
principle and balancing principle, for selecting the regularization parameter.
The idea is further illustrated on a general class of parameter identification
problems, for which (new) source and nonlinearity conditions are derived and
the structural property of the nonlinearity term is revealed. A number of
examples including identifying distributed parameters in elliptic differential
equations are presented.Comment: 21 pages, to appear in Inverse Problem
A Two-stage Method for Inverse Medium Scattering
We present a novel numerical method to the time-harmonic inverse medium
scattering problem of recovering the refractive index from near-field scattered
data. The approach consists of two stages, one pruning step of detecting the
scatterer support, and one resolution enhancing step with mixed regularization.
The first step is strictly direct and of sampling type, and faithfully detects
the scatterer support. The second step is an innovative application of
nonsmooth mixed regularization, and it accurately resolves the scatterer sizes
as well as intensities. The model is efficiently solved by a semi-smooth
Newton-type method. Numerical results for two- and three-dimensional examples
indicate that the approach is accurate, computationally efficient, and robust
with respect to data noise.Comment: 18 pages, 5 figure
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