Publications
Journals
Publication || .pdf
We show that a divergence-free measure on the plane is a continuous sum of unit tangent vector fields on rectifiable Jordan curves. This loop decomposition is more precise than the general decomposition in terms of elementary solenoids given by S.K. Smirnov when applied to the planar case. The proof involves extending the Fleming-Rishel formula to homogeneous BV functions (in any dimension), and establishing for such functions approximate continuity of measure theoretic connected components of suplevel sets as functions of the level. We apply these results to inverse potential problems whose source term is the divergence of some unknown (vector-valued) measure. A prototypical case is that of inverse magnetization problems when magnetizations are modeled by $\R^3$-valued Borel measures. We investigate methods for recovering a magnetization $\bs{\mu}$ by penalizing the measure theoretic total variation norm $\|\bs{\mu}\|_{TV}$. In particular, we show that 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 $TV$-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.
We show that a divergence-free measure on the plane is a continuous sum of unit tangent vector fields on rectifiable Jordan curves. This loop decomposition is more precise than the general decomposition in terms of elementary solenoids given by S.K. Smirnov when applied to the planar case. The proof involves extending the Fleming-Rishel formula to homogeneous BV functions (in any dimension), and establishing for such functions approximate continuity of measure theoretic connected components of suplevel sets as functions of the level. We apply these results to inverse potential problems whose source term is the divergence of some unknown (vector-valued) measure. A prototypical case is that of inverse magnetization problems when magnetizations are modeled by $\R^3$-valued Borel measures. We investigate methods for recovering a magnetization $\bs{\mu}$ by penalizing the measure theoretic total variation norm $\|\bs{\mu}\|_{TV}$. In particular, we show that 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 $TV$-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.
Publication || .pdf (2018 preprint)
We study inverse problems for the Poisson equation with source term the divergence of an $\R^3$-valued measure, that is, the potential $\Phi$ satisfies \[\Delta\Phi=\nabla\cdot\bs{\mu},\] and $\bs{\mu}$ is to be reconstructed knowing (a component of) the field $\nabla\Phi$ on a set disjoint from the support of $\bs{\mu}$. Such problems arise in several electro-magnetic contexts in the quasi-static regime, for instance when recovering a remanent magnetization from measurements of its magnetic field. We investigate methods for recovering $\bs{\mu}$ by penalizing the measure theoretic total variation norm $\|\bs{\mu}\|_{TV}$. We provide sufficient conditions for the unique recovery of $\bs{\mu}$, asymptotically when the regularization parameter and the noise tend to zero in a combined fashion, when it is uni-directional or when the magnetization has a support which is sparse in the sense that it is purely 1-unrectifiable.
Numerical examples are provided to illustrate the main theoretical results.
We study inverse problems for the Poisson equation with source term the divergence of an $\R^3$-valued measure, that is, the potential $\Phi$ satisfies \[\Delta\Phi=\nabla\cdot\bs{\mu},\] and $\bs{\mu}$ is to be reconstructed knowing (a component of) the field $\nabla\Phi$ on a set disjoint from the support of $\bs{\mu}$. Such problems arise in several electro-magnetic contexts in the quasi-static regime, for instance when recovering a remanent magnetization from measurements of its magnetic field. We investigate methods for recovering $\bs{\mu}$ by penalizing the measure theoretic total variation norm $\|\bs{\mu}\|_{TV}$. We provide sufficient conditions for the unique recovery of $\bs{\mu}$, asymptotically when the regularization parameter and the noise tend to zero in a combined fashion, when it is uni-directional or when the magnetization has a support which is sparse in the sense that it is purely 1-unrectifiable.
Numerical examples are provided to illustrate the main theoretical results.
Publication || .pdf
Presentamos nuevas caracterizaciones de las álgebras de Lie semisimples y reductivas en términos de la invertibilidad de la imagen, bajo representaciones irreducibles, del operador de Casimir asociado. En otras palabras, sea $(g,B)$ un algebra de Lie cuadrática, $g$ es semisimple si, y sólo si, para toda representación irreducible la imagen bajo la representación del operador de Casimir asociado es invertible.
(We present new characterizations of semisimple and reductive Lie algebras in terms of the image invertability, under irreducible representations, of the associated Casimir operator. In other words, let $(g,B)$ be a quadratic Lie algebra, $g$ is semisimple if and only if, for every irreducible representation the image under the representation of the associated Casimir operator is invertible.)
Presentamos nuevas caracterizaciones de las álgebras de Lie semisimples y reductivas en términos de la invertibilidad de la imagen, bajo representaciones irreducibles, del operador de Casimir asociado. En otras palabras, sea $(g,B)$ un algebra de Lie cuadrática, $g$ es semisimple si, y sólo si, para toda representación irreducible la imagen bajo la representación del operador de Casimir asociado es invertible.
(We present new characterizations of semisimple and reductive Lie algebras in terms of the image invertability, under irreducible representations, of the associated Casimir operator. In other words, let $(g,B)$ be a quadratic Lie algebra, $g$ is semisimple if and only if, for every irreducible representation the image under the representation of the associated Casimir operator is invertible.)
Publication
There are two properties shared by all known crossing-minimizing geometric drawings of $K_n$, for $n$ a multiple of 3. First, the underlying $n$-point set of these drawings minimizes the number of $(\leq k)$-edges, that means, has exactly $3\binom{k+2}{2}$ $(\leq k)$-edges, for all $0\leq k \lt n/3$. Second, all such drawings have the nn points divided into three groups of equal size; this last property is captured under the concept of 3-decomposability. In this paper we show that these properties are closely related: every $n$-point set with exactly $3\binom{k+2}{2}$ $(\leq k)$-edges for all $0\lt k \lt n/3$, is $3$-decomposable. The converse, however, is easy to see that it is false. As an application, we prove that the rectilinear crossing number of $K_{30}$ is 9726.
There are two properties shared by all known crossing-minimizing geometric drawings of $K_n$, for $n$ a multiple of 3. First, the underlying $n$-point set of these drawings minimizes the number of $(\leq k)$-edges, that means, has exactly $3\binom{k+2}{2}$ $(\leq k)$-edges, for all $0\leq k \lt n/3$. Second, all such drawings have the nn points divided into three groups of equal size; this last property is captured under the concept of 3-decomposability. In this paper we show that these properties are closely related: every $n$-point set with exactly $3\binom{k+2}{2}$ $(\leq k)$-edges for all $0\lt k \lt n/3$, is $3$-decomposable. The converse, however, is easy to see that it is false. As an application, we prove that the rectilinear crossing number of $K_{30}$ is 9726.
Conference Papers
Publication || .pdf
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.
[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.
[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.
Publication || .pdf
We discuss recent results from [1] on sparse recovery for inverse potential problem with source term in divergence form. The notion of sparsity which is set forth is measure-theoretic, namely pure 1-unrectifiability of the support. The theory applies when a superset of the support is known to be slender, meaning it has measure zero and all connected components of its complement has infinite measure in $\R^3$. We also discuss open issues in the non-slender case.
[1] 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 discuss recent results from [1] on sparse recovery for inverse potential problem with source term in divergence form. The notion of sparsity which is set forth is measure-theoretic, namely pure 1-unrectifiability of the support. The theory applies when a superset of the support is known to be slender, meaning it has measure zero and all connected components of its complement has infinite measure in $\R^3$. We also discuss open issues in the non-slender case.
[1] 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.
Manuscripts
.pdf
Inverse source problems in divergence form consist in finding a vector field with prescribed support S, whose divergence is the Laplacian of some observed potential. In this paper, we assume the unknown vector field is a vector-valued measure, and we study the corresponding least square inversion problems, regularized by penalizing the total variation, without discretizing the criterion nor the unknown. We prove that this problem that this problem has a unique minimizer in the case where S is a so-called slender set; i.e., it has zero Lebesgue measure and each connected component of its complement has infinite Lebesgue measure. The proof dwells on Smirnov’s decomposition of divergence-free measures [1] that quickly turns the paper into a measure-geometric one.
[1] S. K. Smirnov. Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows. St. Petersburg Math. J., 5:841–867, 1994.
Inverse source problems in divergence form consist in finding a vector field with prescribed support S, whose divergence is the Laplacian of some observed potential. In this paper, we assume the unknown vector field is a vector-valued measure, and we study the corresponding least square inversion problems, regularized by penalizing the total variation, without discretizing the criterion nor the unknown. We prove that this problem that this problem has a unique minimizer in the case where S is a so-called slender set; i.e., it has zero Lebesgue measure and each connected component of its complement has infinite Lebesgue measure. The proof dwells on Smirnov’s decomposition of divergence-free measures [1] that quickly turns the paper into a measure-geometric one.
[1] S. K. Smirnov. Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows. St. Petersburg Math. J., 5:841–867, 1994.
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We study an inverse source problem with right hand side in divergence form for the Helmholtz equation, whose underlying model can be related to weak scattering from thin interfaces. This inverse problem is not uniquely solvable, as the forward operator has infinite-dimensional kernel. We present a decomposition of (not necessarily tangent) vector fields of $L^2$-class on a closed Lipschitz surface in $\mathbb{R}^3$, which allows one to discuss an ansatz for the solution and constraints that restore uniqueness. This work can be seen as a generalization of references [1, 2] dealing with the Laplace equation, but in the Helmholtz case new ties arise between the observations from each side of the surface. Our proof is based on properties of the Calderón projector on the boundary of Lipschitz domains, that we establish in a $H^{-1} \times L^2$ setting.
[1] L. Baratchart, C. Gerhards, and A. Kegeles. Decomposition of $L^2$-vector fields on lipschitz surfaces: characterization via null-spaces of the scalar potential. SIAM Journal on Mathematical Analysis, 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.
We study an inverse source problem with right hand side in divergence form for the Helmholtz equation, whose underlying model can be related to weak scattering from thin interfaces. This inverse problem is not uniquely solvable, as the forward operator has infinite-dimensional kernel. We present a decomposition of (not necessarily tangent) vector fields of $L^2$-class on a closed Lipschitz surface in $\mathbb{R}^3$, which allows one to discuss an ansatz for the solution and constraints that restore uniqueness. This work can be seen as a generalization of references [1, 2] dealing with the Laplace equation, but in the Helmholtz case new ties arise between the observations from each side of the surface. Our proof is based on properties of the Calderón projector on the boundary of Lipschitz domains, that we establish in a $H^{-1} \times L^2$ setting.
[1] L. Baratchart, C. Gerhards, and A. Kegeles. Decomposition of $L^2$-vector fields on lipschitz surfaces: characterization via null-spaces of the scalar potential. SIAM Journal on Mathematical Analysis, 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.