Nonsmooth, nonconvex regularizers applied to linear elctromagnetic
inverse problems
Abstract
Tikhonov’s regularization method is the standard technique applied to
obtain models of the subsurface conductivity dis- tribution from
electric or electromagnetic measurements by solving UT (m) =
||F(m) - d||^2 + P(m): The second
term correspond to the stabilizing functional, with P(m) =
||m||^2 the usual approach, and
the regularization parameter. Due to the roughness penalizer inclusion,
the model developed by Tikhonov’s algorithm tends to smear
discontinuities, a feature that may be undesirable. An important
requirement for the regularizer is to allow the recovery of edges, and
smooth the homogeneous parts. As is well known, Total Variation (TV) is
now the standard approach to meet this requirement. Recently, Wang
et.al. proved convergence for alternating direction method of
multipliers in nonconvex, nonsmooth optimization. In this work we
present a study of several algorithms for model recovering of
Geosounding data based on Inmal Convolution, and also on hybrid, TV and
second order TV and nonsmooth, nonconvex regularizers, observing their
performance on synthetic and real data. The algorithms are based on
Bregman iteration and Split Bregman method, and the geosounding method
is the low-induction numbers magnetic dipoles. Non-smooth regularizers
are considered using the Legendre-Fenchel transform.