3,815 research outputs found
On a continuation approach in Tikhonov regularization and its application in piecewise-constant parameter identification
We present a new approach to convexification of the Tikhonov regularization
using a continuation method strategy. We embed the original minimization
problem into a one-parameter family of minimization problems. Both the penalty
term and the minimizer of the Tikhonov functional become dependent on a
continuation parameter.
In this way we can independently treat two main roles of the regularization
term, which are stabilization of the ill-posed problem and introduction of the
a priori knowledge. For zero continuation parameter we solve a relaxed
regularization problem, which stabilizes the ill-posed problem in a weaker
sense. The problem is recast to the original minimization by the continuation
method and so the a priori knowledge is enforced.
We apply this approach in the context of topology-to-shape geometry
identification, where it allows to avoid the convergence of gradient-based
methods to a local minima. We present illustrative results for magnetic
induction tomography which is an example of PDE constrained inverse problem
A mathematical and numerical framework for ultrasonically-induced Lorentz force electrical impedance tomography
We provide a mathematical analysis and a numerical framework for Lorentz
force electrical conductivity imaging. Ultrasonic vibration of a tissue in the
presence of a static magnetic field induces an electrical current by the
Lorentz force. This current can be detected by electrodes placed around the
tissue; it is proportional to the velocity of the ultrasonic pulse, but depends
nonlinearly on the conductivity distribution. The imaging problem is to
reconstruct the conductivity distribution from measurements of the induced
current. To solve this nonlinear inverse problem, we first make use of a
virtual potential to relate explicitly the current measurements to the
conductivity distribution and the velocity of the ultrasonic pulse. Then, by
applying a Wiener filter to the measured data, we reduce the problem to imaging
the conductivity from an internal electric current density. We first introduce
an optimal control method for solving such a problem. A new direct
reconstruction scheme involving a partial differential equation is then
proposed based on viscosity-type regularization to a transport equation
satisfied by the current density field. We prove that solving such an equation
yields the true conductivity distribution as the regularization parameter
approaches zero. We also test both schemes numerically in the presence of
measurement noise, quantify their stability and resolution, and compare their
performance
Magnetic Induction Tomography Using Magnetic Dipole and Lumped Parameter Model
This paper presents a novel approach to analyze the magnetic field of magnetic induction tomography (MIT) using magnetic dipoles and a lumped parameter model. The MIT is a next-generation medical imaging technique that can identify the conductivity of target objects and construct images. It is noninvasive and can be compact in design and, thus, used as a portable instrument. However, it still exhibits inferior performance due to the nonlinearity, low signal-to-noise ratio of the magnetic field, and ill-posed inverse problem. To overcome such difficulties, the magnetic field of the MIT system is first modeled using magnetic dipoles and a lumped parameter. In particular, the extended distributed multipole (eDMP) model is proposed to analyze the system, using magnetic dipoles. The method can dramatically reduce the computational efforts and improve the ill-posed condition. Hence, the forward and inverse problems of MIT are solved using the eDMP method. The modeling method can be validated by comparing with experiments, varying the modeling parameters. Finally, the image can be reconstructed, and then, the position and shape of the object can be characterized to develop the MIT
Acousto-electrical speckle pattern in Lorentz force electrical impedance tomography
Ultrasound speckle is a granular texture pattern appearing in ultrasound
imaging. It can be used to distinguish tissues and identify pathologies.
Lorentz force electrical impedance tomography is an ultrasound-based medical
imaging technique of the tissue electrical conductivity. It is based on the
application of an ultrasound wave in a medium placed in a magnetic field and on
the measurement of the induced electric current due to Lorentz force. Similarly
to ultrasound imaging, we hypothesized that a speckle could be observed with
Lorentz force electrical impedance tomography imaging. In this study, we first
assessed the theoretical similarity between the measured signals in Lorentz
force electrical impedance tomography and in ultrasound imaging modalities. We
then compared experimentally the signal measured in both methods using an
acoustic and electrical impedance interface. Finally, a bovine muscle sample
was imaged using the two methods. Similar speckle patterns were observed. This
indicates the existence of an "acousto-electrical speckle" in the Lorentz force
electrical impedance tomography with spatial characteristics driven by the
acoustic parameters but due to electrical impedance inhomogeneities instead of
acoustic ones as is the case of ultrasound imaging
A mathematical model and inversion procedure for Magneto-Acousto-Electric Tomography (MAET)
Magneto-Acousto-Electric Tomography (MAET), also known as the Lorentz force
or Hall effect tomography, is a novel hybrid modality designed to be a
high-resolution alternative to the unstable Electrical Impedance Tomography. In
the present paper we analyze existing mathematical models of this method, and
propose a general procedure for solving the inverse problem associated with
MAET. It consists in applying to the data one of the algorithms of
Thermo-Acoustic tomography, followed by solving the Neumann problem for the
Laplace equation and the Poisson equation.
For the particular case when the region of interest is a cube, we present an
explicit series solution resulting in a fast reconstruction algorithm. As we
show, both analytically and numerically, MAET is a stable technique yilelding
high-resolution images even in the presence of significant noise in the data
Stochastic modeling error reduction using Bayesian approach coupled with an adaptive Kriging based model
Purpose - Magnetic material properties of an electromagnetic device (EMD) can be recovered by solving a coupled experimental numerical inverse problem. In order to ensure the highest possible accuracy of the inverse problem solution, all physics of the EMD need to be perfectly modeled using a complex numerical model. However, these fine models demand a high computational time. Alternatively, less accurate coarse models can be used with a demerit of the high expected recovery errors. The purpose of this paper is to present an efficient methodology to reduce the effect of stochastic modeling errors in the inverse problem solution.
Design/methodology/approach - The recovery error in the electromagnetic inverse problem solution is reduced using the Bayesian approximation error approach coupled with an adaptive Kriging-based model. The accuracy of the forward model is assessed and adapted a priori using the cross-validation technique.
Findings - The adaptive Kriging-based model seems to be an efficient technique for modeling EMDs used in inverse problems. Moreover, using the proposed methodology, the recovery error in the electromagnetic inverse problem solution is largely reduced in a relatively small computational time and memory storage.
Originality/value - The proposed methodology is capable of not only improving the accuracy of the inverse problem solution, but also reducing the computational time as well as the memory storage. Furthermore, to the best of the authors knowledge, it is the first time to combine the adaptive Kriging-based model with the Bayesian approximation error approach for the stochastic modeling error reduction
A Bayesian approach for the stochastic modeling error reduction of magnetic material identification of an electromagnetic device
Magnetic material properties of an electromagnetic device can be recovered by solving an inverse problem where measurements are adequately interpreted by a mathematical forward model. The accuracy of these forward models dramatically affects the accuracy of the material properties recovered by the inverse problem. The more accurate the forward model is, the more accurate recovered data are. However, the more accurate âfineâ models demand a high computational time and memory storage. Alternatively, less accurate âcoarseâ models can be used with a demerit of the high expected recovery errors. This paper uses the Bayesian approximation error approach for improving the inverse problem results when coarse models are utilized. The proposed approach adapts the objective function to be minimized with the a priori misfit between fine and coarse forward model responses. In this paper, two different electromagnetic devices, namely a switched reluctance motor and an EI core inductor, are used as case studies. The proposed methodology is validated on both purely numerical and real experimental results. The results show a significant reduction in the recovery error within an acceptable computational time
A robust inversion method for quantitative 3D shape reconstruction from coaxial eddy-current measurements
This work is motivated by the monitoring of conductive clogging deposits in
steam generator at the level of support plates. One would like to use monoaxial
coils measurements to obtain estimates on the clogging volume. We propose a 3D
shape optimization technique based on simplified parametrization of the
geometry adapted to the measurement nature and resolution. The direct problem
is modeled by the eddy current approximation of time-harmonic Maxwell's
equations in the low frequency regime. A potential formulation is adopted in
order to easily handle the complex topology of the industrial problem setting.
We first characterize the shape derivatives of the deposit impedance signal
using an adjoint field technique. For the inversion procedure, the direct and
adjoint problems have to be solved for each coil vertical position which is
excessively time and memory consuming. To overcome this difficulty, we propose
and discuss a steepest descent method based on a fixed and invariant
triangulation. Numerical experiments are presented to illustrate the
convergence and the efficiency of the method
The Linearized Inverse Problem in Multifrequency Electrical Impedance Tomography
This paper provides an analysis of the linearized inverse problem in
multifrequency electrical impedance tomography. We consider an isotropic
conductivity distribution with a finite number of unknown inclusions with
different frequency dependence, as is often seen in biological tissues. We
discuss reconstruction methods for both fully known and partially known
spectral profiles, and demonstrate in the latter case the successful employment
of difference imaging. We also study the reconstruction with an imperfectly
known boundary, and show that the multifrequency approach can eliminate
modeling errors and recover almost all inclusions. In addition, we develop an
efficient group sparse recovery algorithm for the robust solution of related
linear inverse problems. Several numerical simulations are presented to
illustrate and validate the approach.Comment: 25 pp, 11 figure
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