arXiv subjects

Tracey Balehowsky

Publications and source records attributed to Tracey Balehowsky.

4 recordsLinked to original sources

arXiv2022

An Inverse Problem for the Relativistic Boltzmann Equation

We consider an inverse problem for the Boltzmann equation on a globally hyperbolic Lorentzian spacetime $(M,g)$ with an unknown metric $g$. We consider measurements done in a neighbourhood $V\subset M$ of a timelike path $μ$ that connects a point $x^-$ to a point $x^+$. The measurements are modelled by a source-to-solution map, which maps a source supported in $V$ to the restriction of the solution to the Boltzmann equation to the set $V$. We show that the source-to-solution map uniquely determines the Lorentzian spacetime, up to an isometry, in the set $I^+(x^-)\cap I^-(x^+)\subset M$. The set $I^+(x^-)\cap I^-(x^+)$ is the intersection of the future of the point $x^-$ and the past of the point $x^+$, and hence is the maximal set to where causal signals sent from $x^-$ can propagate and return to the point $x^+$. The proof of the result is based on using the nonlinearity of the Boltzmann equation as a beneficial feature for solving the inverse problem.

math.AP↗

arXiv2022

Shape analysis via gradient flows on diffeomorphism groups

We study a gradient flow on Sobolev diffeomorphisms for the problem of image registration. The energy functional quantifies the effect of transforming a template to a target, while also penalizing deformation of the metric tensor. The main result is well-posedness of the flow. We also give a geometric description of the gradient in terms of the momentum map.

math.DG↗

arXiv2021

Transformation Optics for the Modelling of Waves in a Universe with Nontrivial Topology

We consider how transformation optics and invisibility cloaking can be used to construct models in subsets $\mathbb{R}^3$ with a varying metric, where the time-harmonic waves for a given angular wavenumber $k$, are equivalent to the waves in some closed orientable manifold. The obtained models could in principle be physically implemented using a device built from metamaterials. In particular the measurements in the metamaterial device given by the Helmholtz source-to-solution operator are equivalent to Helmholtz source-to-solution measurements in a universe given by $(\mathbb{R}_+\times M, -dt^2 +g)$, where $(M,g)$ is a closed, orientable, $C^\infty$-smooth, 3-dimensional Riemannian manifold. Thus the obtained construction could be used to simulate cosmological models using metamaterial devices.

math.AP↗

arXiv2018

Determining a Riemannian Metric from Minimal Areas

We prove that if $(M,g)$ is a topological 3-ball with a $C^4$-smooth Riemannian metric $g$, and mean-convex boundary $\partial M$ then knowledge of least areas circumscribed by simple closed curves $γ\subset \partial M$ uniquely determines the metric $g$, under some additional geometric assumptions. These are that $g$ is either a) $C^3$-close to Euclidean or b) satisfies much weaker geometric conditions which hold when the manifold is to a sufficient degree either thin, or straight. %sufficiently thin. In fact, the least area data that we require is for a much more restricted class of curves $γ\subset \partial M$. We also prove a corresponding local result: assuming only that $(M,g)$ has strictly mean convex boundary at a point $p\in\partial M$, we prove that knowledge of the least areas circumscribed by any simple closed curve $γ$ in a neighbourhood $U\subset \partial M$ of $p$ uniquely determines the metric near $p$. Additionally, we sketch the proof of a global result with no thin/straight or curvature condition, but assuming the metric admits minimal foliations "from all directions". The proofs rely on finding the metric along a continuous sweep-out of $M$ by area-minimizing surfaces; they bring together ideas from the 2D-Calderón inverse problem, minimal surface theory, and the careful analysis of a system of pseudo-differential equations.

math.DG↗