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dc.rights.licenseRestricted to current Rensselaer faculty, staff and students. Access inquiries may be directed to the Rensselaer Libraries.
dc.contributorIsaacson, David
dc.contributorHenshaw, William D.
dc.contributorLi, Fengyan
dc.contributorNewell, Jonathan C.
dc.contributorSaulnier, Gary J.
dc.contributorKao, Tzu-Jen
dc.contributor.authorChen, JiaMing
dc.date.accessioned2021-11-03T08:37:34Z
dc.date.available2021-11-03T08:37:34Z
dc.date.created2016-08-16T08:59:56Z
dc.date.issued2016-05
dc.identifier.urihttps://hdl.handle.net/20.500.13015/1709
dc.descriptionMay 2016
dc.descriptionSchool of Science
dc.description.abstractThe D-Bar method is based on solving for the complex geometrical optics (CGO) solutions to the Schrödinger equation, which leads to solving a series of DBar equations numerically. Previous numerical efforts to solve the D-Bar equation include an integral equation solver and a direct PDE solver that uses finite difference method. This thesis introduces and analyzes some alternative solving strategies under the finite element method (FEM) framework.
dc.description.abstractThe computation efficiency aspect of the solver is then discussed. A specially designed FEM mesh is proposed and the performance of the reconstruction routine is roughly compared with that of the finite difference solver.
dc.description.abstractFinally, a detailed ventilation-perfusion V / Q index analysis is carried out and V /Q index is calculated using reconstructed conductivities.
dc.description.abstractWe argue that the original instability of the classical FEM formulation comes from the inappropriate treatment of the boundary condition. One chapter is therefore specifically dedicated to analyzing this issue, where three alternative boundary conditions are considered. Some numerical simulations are provided to support our argument.
dc.description.abstractIn electrical impedance tomography (EIT), body conductivity is recovered from voltage and current measurements made on the boundary of the body. The major advantage of EIT over other imaging modalities such as CT and MRI is that it is inexpensive, non-invasive and fast. Mainstream reconstruction methods can be categorized into linearized and non-linear, indirect and direct methods. This thesis is focused on the numerical implementations of one of the non-linear direct methods: the D-Bar method.
dc.description.abstractTwo different FEM formulations of the D-Bar Equation will be discussed in detail. The first one is the classical Galerkin method, which proves to be unstable in general. We also notice that it shares a similar stability issue on a uniform grid with the finite difference solver developed in [13]. The stability issue is resolved by introducing the second formulation, the Least Square Finite Element Method (LSFEM). The solver is proven to have a unique solution and the corresponding FEM problem is second order convergent. Reconstruction results using LSFEM D-Bar solver are demonstrated.
dc.language.isoENG
dc.publisherRensselaer Polytechnic Institute, Troy, NY
dc.relation.ispartofRensselaer Theses and Dissertations Online Collection
dc.subjectMathematics
dc.titleElectrical impedance tomography and D-Bar equation
dc.typeElectronic thesis
dc.typeThesis
dc.digitool.pid177358
dc.digitool.pid177359
dc.digitool.pid177360
dc.rights.holderThis electronic version is a licensed copy owned by Rensselaer Polytechnic Institute, Troy, NY. Copyright of original work retained by author.
dc.description.degreePhD
dc.relation.departmentDept. of Mathematical Sciences


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