Presentations
Talks
Website
The problem presented was the one of inverse scanning magnetic microscopy, which aims to recover magnetization distributions of thin rock samples. It is based on the Poisson equation, \[\Delta\Phi = \boldsymbol{\mu},\] where the vector field $\boldsymbol{\mu}$ is to be recovered from measurements of one or more components of $\nabla\Phi$. This inverse problem is ill-posed since the forward operator has a kernel so extra assumptions are needed to ensure uniqueness of solutions. We will start by exploring the theoretical limitation of the inverse problem given by this kernel. Then, we will focus on the planar case, where we have found two particular cases where we could theoretically recover the original magnetization. This is done by taking the group LASSO regularization technique letting the regularizing parameter to zero. Unfortunately, this method relies in zero noise, which is not the case when working with real data and we will show what the problems of the naive use of the group LASSO technique. Then, we will show different techniques to overcome this issues, including extension of the data or changing the how the measurements are taken, together with different machine leaning techniques.
The problem presented was the one of inverse scanning magnetic microscopy, which aims to recover magnetization distributions of thin rock samples. It is based on the Poisson equation, \[\Delta\Phi = \boldsymbol{\mu},\] where the vector field $\boldsymbol{\mu}$ is to be recovered from measurements of one or more components of $\nabla\Phi$. This inverse problem is ill-posed since the forward operator has a kernel so extra assumptions are needed to ensure uniqueness of solutions. We will start by exploring the theoretical limitation of the inverse problem given by this kernel. Then, we will focus on the planar case, where we have found two particular cases where we could theoretically recover the original magnetization. This is done by taking the group LASSO regularization technique letting the regularizing parameter to zero. Unfortunately, this method relies in zero noise, which is not the case when working with real data and we will show what the problems of the naive use of the group LASSO technique. Then, we will show different techniques to overcome this issues, including extension of the data or changing the how the measurements are taken, together with different machine leaning techniques.
Website || Short paper
We shall discuss an inverse problem where the underlying model is related to sources generated by currents on an anisotropic layer. This problem is a generalization of another motivated by the recovering of magnetization distribution in a rock sample from outer measurements of the generated static magnetic field. The original problem can be formulated as inverse source problem for the Laplace equation [1,2] with sources being the divergence of the magnetization whereas the generalization comes from taking the Helmholtz equation. Either inverse problem is non uniquely solvable with a kernel of infinite dimension. We shall present a decomposition of the space of sources that will allow us to discuss constraints that may restore uniqueness and propose regularization schemes adapted to these assumptions.
This was a collaborative work with L. Baratchart and H. Haddar.
[1] L. Baratchart, C. Gerhards, and A. Kegeles, Decomposition of L2-Vector Fields on Lipschitz Surfaces: Characterization via Null-Spaces of the Scalar Potential, SIAM J. Math. Anal. 53(4) (2021), pp. 4096–4117.
[2] L. Baratchart, C. Villalobos-Guillen, D. P. Hardin, M. C. Northington, E. B. Saff, Inverse Potential Problems for Divergence of Measures with Total Variation Regularization, Found. Comp. Math. 20 (2020), pp. 1273–1307.
We shall discuss an inverse problem where the underlying model is related to sources generated by currents on an anisotropic layer. This problem is a generalization of another motivated by the recovering of magnetization distribution in a rock sample from outer measurements of the generated static magnetic field. The original problem can be formulated as inverse source problem for the Laplace equation [1,2] with sources being the divergence of the magnetization whereas the generalization comes from taking the Helmholtz equation. Either inverse problem is non uniquely solvable with a kernel of infinite dimension. We shall present a decomposition of the space of sources that will allow us to discuss constraints that may restore uniqueness and propose regularization schemes adapted to these assumptions.
This was a collaborative work with L. Baratchart and H. Haddar.
[1] L. Baratchart, C. Gerhards, and A. Kegeles, Decomposition of L2-Vector Fields on Lipschitz Surfaces: Characterization via Null-Spaces of the Scalar Potential, SIAM J. Math. Anal. 53(4) (2021), pp. 4096–4117.
[2] L. Baratchart, C. Villalobos-Guillen, D. P. Hardin, M. C. Northington, E. B. Saff, Inverse Potential Problems for Divergence of Measures with Total Variation Regularization, Found. Comp. Math. 20 (2020), pp. 1273–1307.
Website
Inverse magnetization problems can be modeled with the Laplace equation with source term in divergence form: \[\Delta\Phi = \nabla \cdot \bs{\mu},\] where $\Phi$ is a locally integrable that represents the magnetic potential and $\bs{\mu}$, a $\R^3$-valued Borel measure that models the source magnetization. When the supp $\bs{\mu}$ is known to have a connected complement (for example when support is contained in a proper subset of the plane), the only magnetizations that generates a zero-field are the ones that are divergence-free. It is for this reason that we turn our attention to the study of divergence-free measure on the plane. In this talk, I will present results in that directions from a joint paper [1] with Laurent Baratchart from the project FACTAS, INRIA, and Douglas P. Hardin from Vanderbilt University. We found that it is possible to decompose such a measure as a continuous sum of unit tangent vector fields on rectifiable Jordan curves. Based such decomposition, I will talk about methods for recovering a magnetization $\bs{\mu}$ by penalizing the measure theoretic total variation norm $\|\bs{\mu}\|_{TV}$. In particular, if a magnetization is supported in a plane, then TV -regularization schemes always have a unique minimizer, even in the presence of noise. It is further shown that T V -norm minimization (among magnetizations generating the same field) uniquely recovers planar magnetizations in the following cases: when the magnetization is carried by a collection of sufficiently separated line segments and a set that is purely 1-unrectifiable, or when a superset of the support is tree-like. We note that such magnetizations can be recovered via TV-regularization schemes in the zero noise limit by taking the regularization parameter to zero. This suggests definitions of sparsity in the present infinite dimensional context, that generate results akin to compressed sensing.
[1] Baratchart, Laurent, Villalobos Guillén, Cristóbal, and Hardin, Douglas P. Inverse potential problems in divergence form for measures in the plane. ESAIM: COCV, 27:87, 2021.
Inverse magnetization problems can be modeled with the Laplace equation with source term in divergence form: \[\Delta\Phi = \nabla \cdot \bs{\mu},\] where $\Phi$ is a locally integrable that represents the magnetic potential and $\bs{\mu}$, a $\R^3$-valued Borel measure that models the source magnetization. When the supp $\bs{\mu}$ is known to have a connected complement (for example when support is contained in a proper subset of the plane), the only magnetizations that generates a zero-field are the ones that are divergence-free. It is for this reason that we turn our attention to the study of divergence-free measure on the plane. In this talk, I will present results in that directions from a joint paper [1] with Laurent Baratchart from the project FACTAS, INRIA, and Douglas P. Hardin from Vanderbilt University. We found that it is possible to decompose such a measure as a continuous sum of unit tangent vector fields on rectifiable Jordan curves. Based such decomposition, I will talk about methods for recovering a magnetization $\bs{\mu}$ by penalizing the measure theoretic total variation norm $\|\bs{\mu}\|_{TV}$. In particular, if a magnetization is supported in a plane, then TV -regularization schemes always have a unique minimizer, even in the presence of noise. It is further shown that T V -norm minimization (among magnetizations generating the same field) uniquely recovers planar magnetizations in the following cases: when the magnetization is carried by a collection of sufficiently separated line segments and a set that is purely 1-unrectifiable, or when a superset of the support is tree-like. We note that such magnetizations can be recovered via TV-regularization schemes in the zero noise limit by taking the regularization parameter to zero. This suggests definitions of sparsity in the present infinite dimensional context, that generate results akin to compressed sensing.
[1] Baratchart, Laurent, Villalobos Guillén, Cristóbal, and Hardin, Douglas P. Inverse potential problems in divergence form for measures in the plane. ESAIM: COCV, 27:87, 2021.
Website
Discussed an inverse problem motivated by the recovering of magnetization distribution in a rock sample from outer measurements of the generated static magnetic field. It can be formulated as inverse source problem for the Laplace equation with sources being the divergence of the magnetization. It can also be generalized to the Helmholtz equation where the underlying model is related to sources generated by currents on an anisotropic layer.
This was a collaborative work with L. Baratchart, H. Haddar and D. P. Hardin.
Discussed an inverse problem motivated by the recovering of magnetization distribution in a rock sample from outer measurements of the generated static magnetic field. It can be formulated as inverse source problem for the Laplace equation with sources being the divergence of the magnetization. It can also be generalized to the Helmholtz equation where the underlying model is related to sources generated by currents on an anisotropic layer.
This was a collaborative work with L. Baratchart, H. Haddar and D. P. Hardin.
Presented a collaborative work with M. Cetina. Based on the work of J. Leaños, we were able to show that the rectilinear crossing number of K30 (that is, the minimal number of edge crossing for the drawing on the plane of a complete graph with 30 vertices where each of the 435 edges is a line segment) actually reached its current upper bound, 9726.
Posters
Website || Poster
Here we will present the research found on [1, 2]. This work is concerned with inverse potential problems with source term in divergence form. Such issues typically arise in source identification from field measurements for Maxwell's equations, in the quasi-static regime. They occur for instance in geomagnetism and paleomagnetism, as well as in several non-destructive testing problems. A model problem of our particular interest and which we will take as focus in this poster, is inverse scanning magnetic microscopy, to recover magnetization distributions of thin rock samples. However the considerations here are of a more general and abstract nature.
[1] L. Baratchart, C. Villalobos Guillén, and D. P. Hardin. Inverse potential problems in divergence form for measures in the plane. ESAIM: COCV, 27:87, 2021.
[2] L. Baratchart, C. Villalobos Guillén, D. P. Hardin, M. C. Northington, and E. B. Saff. Inverse potential problems for divergence of measures with total variation regularization. Foundations of Computational Mathematics, Nov 2019.
Here we will present the research found on [1, 2]. This work is concerned with inverse potential problems with source term in divergence form. Such issues typically arise in source identification from field measurements for Maxwell's equations, in the quasi-static regime. They occur for instance in geomagnetism and paleomagnetism, as well as in several non-destructive testing problems. A model problem of our particular interest and which we will take as focus in this poster, is inverse scanning magnetic microscopy, to recover magnetization distributions of thin rock samples. However the considerations here are of a more general and abstract nature.
[1] L. Baratchart, C. Villalobos Guillén, and D. P. Hardin. Inverse potential problems in divergence form for measures in the plane. ESAIM: COCV, 27:87, 2021.
[2] L. Baratchart, C. Villalobos Guillén, D. P. Hardin, M. C. Northington, and E. B. Saff. Inverse potential problems for divergence of measures with total variation regularization. Foundations of Computational Mathematics, Nov 2019.