22,666 research outputs found
Optimization Methods for Inverse Problems
Optimization plays an important role in solving many inverse problems.
Indeed, the task of inversion often either involves or is fully cast as a
solution of an optimization problem. In this light, the mere non-linear,
non-convex, and large-scale nature of many of these inversions gives rise to
some very challenging optimization problems. The inverse problem community has
long been developing various techniques for solving such optimization tasks.
However, other, seemingly disjoint communities, such as that of machine
learning, have developed, almost in parallel, interesting alternative methods
which might have stayed under the radar of the inverse problem community. In
this survey, we aim to change that. In doing so, we first discuss current
state-of-the-art optimization methods widely used in inverse problems. We then
survey recent related advances in addressing similar challenges in problems
faced by the machine learning community, and discuss their potential advantages
for solving inverse problems. By highlighting the similarities among the
optimization challenges faced by the inverse problem and the machine learning
communities, we hope that this survey can serve as a bridge in bringing
together these two communities and encourage cross fertilization of ideas.Comment: 13 page
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
Multi-frequency imaging of perfectly conducting cracks via boundary measurements
Imaging of perfectly conducting crack(s) in a 2-D homogeneous medium using
boundary data is studied. Based on the singular structure of the Multi-Static
Response (MSR) matrix whose elements are normalized by an adequate test
function at several frequencies, an imaging functional is introduced and
analyzed. A non-iterative imaging procedure is proposed. Numerical experiments
from noisy synthetic data show that acceptable images of single and multiple
cracks are obtained.Comment: 4 pages, 3 figure
GMRES-Accelerated ADMM for Quadratic Objectives
We consider the sequence acceleration problem for the alternating direction
method-of-multipliers (ADMM) applied to a class of equality-constrained
problems with strongly convex quadratic objectives, which frequently arise as
the Newton subproblem of interior-point methods. Within this context, the ADMM
update equations are linear, the iterates are confined within a Krylov
subspace, and the General Minimum RESidual (GMRES) algorithm is optimal in its
ability to accelerate convergence. The basic ADMM method solves a
-conditioned problem in iterations. We give
theoretical justification and numerical evidence that the GMRES-accelerated
variant consistently solves the same problem in iterations
for an order-of-magnitude reduction in iterations, despite a worst-case bound
of iterations. The method is shown to be competitive against
standard preconditioned Krylov subspace methods for saddle-point problems. The
method is embedded within SeDuMi, a popular open-source solver for conic
optimization written in MATLAB, and used to solve many large-scale semidefinite
programs with error that decreases like , instead of ,
where is the iteration index.Comment: 31 pages, 7 figures. Accepted for publication in SIAM Journal on
Optimization (SIOPT
EIT Reconstruction Algorithms: Pitfalls, Challenges and Recent Developments
We review developments, issues and challenges in Electrical Impedance
Tomography (EIT), for the 4th Workshop on Biomedical Applications of EIT,
Manchester 2003. We focus on the necessity for three dimensional data
collection and reconstruction, efficient solution of the forward problem and
present and future reconstruction algorithms. We also suggest common pitfalls
or ``inverse crimes'' to avoid.Comment: A review paper for the 4th Workshop on Biomedical Applications of
EIT, Manchester, UK, 200
Aggregated motion estimation for real-time MRI reconstruction
Real-time magnetic resonance imaging (MRI) methods generally shorten the
measuring time by acquiring less data than needed according to the sampling
theorem. In order to obtain a proper image from such undersampled data, the
reconstruction is commonly defined as the solution of an inverse problem, which
is regularized by a priori assumptions about the object. While practical
realizations have hitherto been surprisingly successful, strong assumptions
about the continuity of image features may affect the temporal fidelity of the
estimated images. Here we propose a novel approach for the reconstruction of
serial real-time MRI data which integrates the deformations between nearby
frames into the data consistency term. The method is not required to be affine
or rigid and does not need additional measurements. Moreover, it handles
multi-channel MRI data by simultaneously determining the image and its coil
sensitivity profiles in a nonlinear formulation which also adapts to
non-Cartesian (e.g., radial) sampling schemes. Experimental results of a motion
phantom with controlled speed and in vivo measurements of rapid tongue
movements demonstrate image improvements in preserving temporal fidelity and
removing residual artifacts.Comment: This is a preliminary technical report. A polished version is
published by Magnetic Resonance in Medicine. Magnetic Resonance in Medicine
201
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