56 research outputs found
High-order regularized regression in Electrical Impedance Tomography
We present a novel approach for the inverse problem in electrical impedance
tomography based on regularized quadratic regression. Our contribution
introduces a new formulation for the forward model in the form of a nonlinear
integral transform, that maps changes in the electrical properties of a domain
to their respective variations in boundary data. Using perturbation theory the
transform is approximated to yield a high-order misfit unction which is then
used to derive a regularized inverse problem. In particular, we consider the
nonlinear problem to second-order accuracy, hence our approximation method
improves upon the local linearization of the forward mapping. The inverse
problem is approached using Newton's iterative algorithm and results from
simulated experiments are presented. With a moderate increase in computational
complexity, the method yields superior results compared to those of regularized
linear regression and can be implemented to address the nonlinear inverse
problem
Fast-Convergent Dynamics for Distributed Resource Allocation Over Sparse Time-Varying Networks
In this paper, distributed dynamics are deployed to solve resource allocation
over time-varying multi-agent networks. The state of each agent represents the
amount of resources used/produced at that agent while the total amount of
resources is fixed. The idea is to optimally allocate the resources among the
group of agents by reducing the total cost functions subject to fixed amount of
total resources. The information of each agent is restricted to its own state
and cost function and those of its immediate neighbors. This is motivated by
distributed applications such as in mobile edge-computing, economic dispatch
over smart grids, and multi-agent coverage control. The non-Lipschitz dynamics
proposed in this work shows fast convergence as compared to the linear and some
nonlinear solutions in the literature. Further, the multi-agent network
connectivity is more relaxed in this paper. To be more specific, the proposed
dynamics even reaches optimal solution over time-varying disconnected
undirected networks as far as the union of these networks over some bounded
non-overlapping time-intervals includes a spanning-tree. The proposed
convergence analysis can be applied for similar 1st-order resource allocation
nonlinear dynamics. We provide simulations to verify our results
D-SVM over Networked Systems with Non-Ideal Linking Conditions
This paper considers distributed optimization algorithms, with application in
binary classification via distributed support-vector-machines (D-SVM) over
multi-agent networks subject to some link nonlinearities. The agents solve a
consensus-constraint distributed optimization cooperatively via continuous-time
dynamics, while the links are subject to strongly sign-preserving odd nonlinear
conditions. Logarithmic quantization and clipping (saturation) are two examples
of such nonlinearities. In contrast to existing literature that mostly
considers ideal links and perfect information exchange over linear channels, we
show how general sector-bounded models affect the convergence to the optimizer
(i.e., the SVM classifier) over dynamic balanced directed networks. In general,
any odd sector-bounded nonlinear mapping can be applied to our dynamics. The
main challenge is to show that the proposed system dynamics always have one
zero eigenvalue (associated with the consensus) and the other eigenvalues all
have negative real parts. This is done by recalling arguments from matrix
perturbation theory. Then, the solution is shown to converge to the agreement
state under certain conditions. For example, the gradient tracking (GT) step
size is tighter than the linear case by factors related to the upper/lower
sector bounds. To the best of our knowledge, no existing work in distributed
optimization and learning literature considers non-ideal link conditions
Parametric Level Set Methods for Inverse Problems
In this paper, a parametric level set method for reconstruction of obstacles
in general inverse problems is considered. General evolution equations for the
reconstruction of unknown obstacles are derived in terms of the underlying
level set parameters. We show that using the appropriate form of parameterizing
the level set function results a significantly lower dimensional problem, which
bypasses many difficulties with traditional level set methods, such as
regularization, re-initialization and use of signed distance function.
Moreover, we show that from a computational point of view, low order
representation of the problem paves the path for easier use of Newton and
quasi-Newton methods. Specifically for the purposes of this paper, we
parameterize the level set function in terms of adaptive compactly supported
radial basis functions, which used in the proposed manner provides flexibility
in presenting a larger class of shapes with fewer terms. Also they provide a
"narrow-banding" advantage which can further reduce the number of active
unknowns at each step of the evolution. The performance of the proposed
approach is examined in three examples of inverse problems, i.e., electrical
resistance tomography, X-ray computed tomography and diffuse optical
tomography
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