scholarly journals Novel framework for the three-dimensional NLTE inverse problem

2022 ◽  
Author(s):  
J. Stepan ◽  
T. del Pino Aleman ◽  
J. Trujillo Bueno
Keyword(s):  
Inverse Problem ◽  
Three Dimensional

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.

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.

Seismic Rays

2017 ◽  
Author(s):  
John A. Adam
Keyword(s):  
Inverse Problem ◽  
Seismic Tomography ◽  
Seismic Waves ◽  
Seismic Station ◽  
Three Dimensional ◽  
Arrival Times ◽  
The Earth ◽  
Straight Lines ◽  
Ray Path ◽  
Uniform Composition

This chapter focuses on the underlying mathematics of seismic rays. Seismic waves caused by earthquakes and explosions are used in seismic tomography to create computer-generated, three-dimensional images of Earth's interior. If the Earth had a uniform composition and density, seismic rays would travel in straight lines. However, it is broadly layered, causing seismic rays to be refracted and reflected across boundaries. In order to calculate the speed along the wave's ray path, the time it takes for a seismic wave to arrive at a seismic station from an earthquake needs to be determined. Arrival times of different seismic waves allow scientists to define slower or faster regions deep in the Earth. The chapter first presents the relevant equations for seismic rays before discussing how rays are propagated in a spherical Earth. The Wiechert-Herglotz inverse problem is considered, along with the properties of X in a horizontally stratified Earth.

A Remote Temperature Sensing Technique for Estimating the Cutting Interface Temperature Distribution

1986 ◽  
Vol 108 (4) ◽  
pp. 252-263 ◽  
Author(s):  
David Wei Yen ◽  
Paul K. Wright
Keyword(s):  
Inverse Problem ◽  
Temperature Distribution ◽  
Measurement Errors ◽  
Three Dimensional ◽  
Cutting Temperature ◽  
Transformation Matrix ◽  
Dimensional Case ◽  
Interface Temperature ◽  
Unknown Boundary ◽  
Ellipsoidal Coordinates

Cutting temperature is a major factor in controlling tool wear rate. Thus, sensing and control of cutting temperature is important in achieving a desired tool performance. This paper is concerned with estimating the cutting interface temperature distribution based on remote temperature measurements. This class of problems of estimating unknown boundary conditions from known interior quantities is called the inverse problem. The inverse problem of a square insert under steady state conditions is considered in this paper. The temperature distribution in a square insert is best described in Ellipsoidal Coordinates. The mapping functional in the one-dimensional case is solved analytically. The mapping functionals in general three-dimensional cases are solved numerically using the semianalytical finite element method. The mapping functional in a three-dimensional case is represented by a transformation matrix which maps one vector representing the cutting interface temperature distribution to another vector representing the remote temperatures. The transformation matrix is then used to solve the inverse problem of estimating the interface temperature distribution with redundant remote measurements. Measurement errors and transformation matrix errors are imposed in simulation studies. The sensitivity of inverse solutions to these errors is discussed.

Sign in / Sign up

Export Citation Format

Share Document