A nonlinear inverse problem inspired by three-dimensional diffuse tomography

2001 ◽  
Vol 17 (6) ◽  
pp. 1907-1922 ◽  
Author(s):  
F Alberto Grünbaum
Keyword(s):  
Inverse Problem ◽  
Three Dimensional ◽  
Nonlinear Inverse Problem

Identification of nonlinear B–H curves based on magnetic field computations and multigrid methods for ill-posed problems

2003 ◽  
Vol 14 (1) ◽  
pp. 15-38 ◽  
Author(s):  
BARBARA KALTENBACHER ◽  
MANFRED KALTENBACHER ◽  
STEFAN REITZINGER
Keyword(s):  
Magnetic Field ◽  
Inverse Problem ◽  
Numerical Solutions ◽  
Iterative Scheme ◽  
Three Dimensional ◽  
Multigrid Methods ◽  
Stopping Criterion ◽  
Nonlinear Inverse Problem ◽  
Field Problem ◽  
Ill Posed

Our task is the identification of the reluctivity $\nu\,{=}\,\nu(B)$ in $\vec{H}\,{=}\,\nu(B) \vec{B}$, ($B\,{=}\,|\vec{B}|$) from measurements of the magnetic flux for different excitation currents in a driving coil, in the context of a nonuniform magnetic field distribution. This is a nonlinear inverse problem and ill-posed in the sense of unstable data dependence, whose solution is done numerically by a Newton type iterative scheme, regularized by an appropriate stopping criterion. The computational complexity of this method is determined by the number of necessary forward evaluations, i.e. the number of numerical solutions to the three-dimensional magnetic field problem. We keep the effort minimal by applying a special discretization strategy to the inverse problem, based on multigrid methods for ill-posed problems. Numerical results demonstrate the efficiency of the proposed method.

Noniterative three‐dimensional inversion of magnetic data

Geophysics ◽  
1990 ◽  
Vol 55 (6) ◽  
pp. 782-785 ◽  
Author(s):  
A. M. Pustisek
Keyword(s):  
Inverse Problem ◽  
Potential Field ◽  
Field Data ◽  
Three Dimensional ◽  
Magnetic Data ◽  
Aeromagnetic Data ◽  
Nonlinear Inverse Problem ◽  
Potential Field Data

The interpretation of magnetic or aeromagnetic data often requires the inverse problem’s solution of the structure of the magnetization interface. This nonlinear inverse problem of mapping the basement topography from potential field data was first discussed by Peters (1949).

An inversion formula for the transport equation in ℝ3 using complex analysis in several variables

2019 ◽  
Vol 27 (3) ◽  
pp. 341-352
Author(s):  
Seyed Majid Saberi Fathi
Keyword(s):  
Inverse Problem ◽  
Transport Equation ◽  
Inversion Formula ◽  
Complex Analysis ◽  
Three Dimensional ◽  
Analytical Continuation ◽  
Radon Transforms ◽  
Photon Transport ◽  
X Ray ◽  
Several Variables

Abstract In this paper, the stationary photon transport equation has been extended by analytical continuation from {\mathbb{R}^{3}} to {\mathbb{C}^{3}} . A solution to the inverse problem posed by this equation is obtained on a hyper-sphere and a hyper-cylinder as X-ray and Radon transforms, respectively. We show that these results can be transformed into each other, and they agree with known results. Numerical reconstructions of a three-dimensional Shepp–Logan head phantom using the obtained inverse formula illustrate the analytical results obtained in this manuscript.

Inversion of induced polarization data

Geophysics ◽  
1994 ◽  
Vol 59 (9) ◽  
pp. 1327-1341 ◽  
Author(s):  
Douglas W. Oldenburg ◽  
Yaoguo Li
Keyword(s):  
Inverse Problem ◽  
Objective Function ◽  
Effective Conductivity ◽  
Optimization Theory ◽  
Induced Polarization ◽  
Dc Resistivity ◽  
Earth Structure ◽  
Nonlinear Inverse Problem ◽  
Polarization Data ◽  
The Earth

We develop three methods to invert induced polarization (IP) data. The foundation for our algorithms is an assumption that the ultimate effect of chargeability is to alter the effective conductivity when current is applied. This assumption, which was first put forth by Siegel and has been routinely adopted in the literature, permits the IP responses to be numerically modeled by carrying out two forward modelings using a DC resistivity algorithm. The intimate connection between DC and IP data means that inversion of IP data is a two‐step process. First, the DC potentials are inverted to recover a background conductivity. The distribution of chargeability can then be found by using any one of the three following techniques: (1) linearizing the IP data equation and solving a linear inverse problem, (2) manipulating the conductivities obtained after performing two DC resistivity inversions, and (3) solving a nonlinear inverse problem. Our procedure for performing the inversion is to divide the earth into rectangular prisms and to assume that the conductivity σ and chargeability η are constant in each cell. To emulate complicated earth structure we allow many cells, usually far more than there are data. The inverse problem, which has many solutions, is then solved as a problem in optimization theory. A model objective function is designed, and a “model” (either the distribution of σ or η)is sought that minimizes the objective function subject to adequately fitting the data. Generalized subspace methodologies are used to solve both inverse problems, and positivity constraints are included. The IP inversion procedures we design are generic and can be applied to 1-D, 2-D, or 3-D earth models and with any configuration of current and potential electrodes. We illustrate our methods by inverting synthetic DC/IP data taken over a 2-D earth structure and by inverting dipole‐dipole data taken in Quebec.

scholarly journals The Localization of Activity Sources in a Three-Dimensional Model of the Human Brain Using the Joint Analysis of EEG / sMRI Data

2019 ◽  
Vol 17 (3) ◽  
pp. 18-28
Author(s):  
E. Bykova ◽  
A. Savostyanov
Keyword(s):  
Inverse Problem ◽  
Brain Activity ◽  
Three Dimensional ◽  
Human Head ◽  
Joint Analysis ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
Use Of Data ◽  
The Individual ◽  
The Brain

Despite the large number of existing methods of the diagnosis of the brain, brain remains the least studied part of the human body. Electroencephalography (EEG) is one of the most popular methods of studying of brain activity due to its relative cheapness, harmless, and mobility of equipment. While analyzing the EEG data of the brain, the problem of solving of the inverse problem of electroencephalography, the localization of the sources of electrical activity of the brain, arises. This problem can be formulated as follows: according to the signals recorded on the surface of the head, it is necessary to determine the location of sources of these signals in the brain. The purpose of my research is to develop a software system for localization of brain activity sources based on the joint analysis of EEG and sMRI data. There are various approaches to solving of the inverse problem of EEG. To obtain the most exact results, some of them involve the use of data on the individual anatomy of the human head – structural magnetic resonance imaging (sMRI data). In this paper, one of these approaches is supposed to be used – Electromagnetic Spatiotemporal Independent Component Analysis (EMSICA) proposed by A. Tsai. The article describes the main stages of the system, such as preprocessing of the initial data; the calculation of the special matrix of the EMSICA approach, the values of which show the level of activity of a certain part of the brain; visualization of brain activity sources on its three-dimensional model.

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