A Measure Theoretic Approach for the Recovery of Remanent Magnetizations
Villalobos Guillén, Cristóbal
:
2019-03-29
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.