dc.creator | Villalobos Guillén, Cristóbal | |
dc.date.accessioned | 2020-08-22T00:04:43Z | |
dc.date.available | 2019-03-29 | |
dc.date.issued | 2019-03-29 | |
dc.identifier.uri | https://etd.library.vanderbilt.edu/etd-03262019-051041 | |
dc.identifier.uri | http://hdl.handle.net/1803/11475 | |
dc.description.abstract | This work is motivated by the problem of recovering the magnetization M of a rock sample from
a given set of measurements for the magnetic field it generates. Modeling the magnetization by an
R 3 -valued measure, we focus on the study of inverse problems for the Poisson equation with source
term the divergence of M; that is,
∆Φ = divM,
where Φ denotes the Magnetic Scalar Potential whose gradient is assumed to be known on a set
disjoint from the support of the measure M. We develop methods for recovering M based on total
variation regularization of measures. We provide sufficient conditions for the unique recovery of a
magnetization in cases where it is uni-directional or when the magnetization has a support which
is sparse in the sense that it is purely 1-unrectifiable.
In the last chapter we work on the ideal case where the magnetized sample is contained in a
subset of the horizontal plane. For this case we show that all magnetizations which do not generate
a magnetic field can be decomposed as a superposition of loops. The findings presented in this
chapter rely on the theory of functions of Bounded Variation and sets of finite perimeter and give
a characterization for magnetizations that do not generate a magnetic field.
Numerical examples are provided to illustrate the main theoretical results. | |
dc.format.mimetype | application/pdf | |
dc.subject | geometric measure theory | |
dc.subject | BV fucntions | |
dc.subject | Inverse problems | |
dc.subject | Inverse problems in electromagnetism | |
dc.title | A Measure Theoretic Approach for the Recovery of Remanent Magnetizations | |
dc.type | dissertation | |
dc.contributor.committeeMember | Marcelo M. Disconzi | |
dc.contributor.committeeMember | Guilherme A.R. Gualda | |
dc.contributor.committeeMember | Edward B. Saff | |
dc.contributor.committeeMember | Akram Aldroubi | |
dc.type.material | text | |
thesis.degree.name | PHD | |
thesis.degree.level | dissertation | |
thesis.degree.discipline | Mathematics | |
thesis.degree.grantor | Vanderbilt University | |
local.embargo.terms | 2019-03-29 | |
local.embargo.lift | 2019-03-29 | |
dc.contributor.committeeChair | Douglas P. Hardin | |