arXiv subjects

Trent DeGiovanni

Publications and source records attributed to Trent DeGiovanni.

4 recordsLinked to original sources

arXiv2025

A potential theory on weighted graphs

We present an analog to classic potential theory on weighted graphs. With nodes partitioned into exterior, boundary and interior nodes and an appropriate decomposition of the Laplacian, we define discrete analogues to the trace operators, the single and double layer potential operators, and the boundary layer operators. As in the continuum, these operators can represent exterior or interior harmonic functions with different boundary conditions. The formalism we introduce includes a discrete Calderón calculus and brings some well known results from potential theory to weighted graphs, e.g. on the spectrum of the Neumann-Poincaré operator. We illustrate the formalism with a cloaking strategy on weighted graphs which allows to hide an anomaly from the perspective of electrical measurements made away from the anomaly.

math.PR↗

arXiv2023

Imaging with thermal noise induced currents

We use thermal noise induced currents to image the real and imaginary parts of the conductivity of a body. Covariances of the thermal noise currents measured at a few electrodes are shown to be related to a deterministic problem. We use the covariances obtained while selectively heating the body to recover the real power density in the body under known boundary conditions and at a known frequency. The resulting inverse problem is related to acousto-electric tomography, but where the conductivity is complex and only the real power is measured. We study the local solvability of this problem by determining where its linearization is elliptic. Numerical experiments illustrating this inverse problem are included.

math.NA↗

arXiv2022

Active exterior cloaking for the 2D Helmholtz equation with complex wavenumbers and application to thermal cloaking

We design sources for the two-dimensional Helmholtz equation that can cloak an object by cancelling out the incident field in a region, without the sources completely surrounding the object to hide. As in previous work for real positive wavenumbers, the sources are also determined by the Green identities. The novelty is that we prove that the same approach works for complex wavenumbers which makes it applicable to a variety of media, including media with dispersion, loss and gain. Furthermore, by deriving bounds on Graf's addition formulas with complex arguments, we obtain new estimates that allow to quantify the quality of the cloaking effect. We illustrate our results by applying them to achieve active exterior cloaking for the heat equation.

math.AP↗

arXiv2020

Active Thermal Cloaking and Mimicking

We present an active cloaking method for the parabolic heat (and mass or light diffusion) equation that can hide both objects and sources. By active we mean that it relies on designing monopole and dipole heat source distributions on the boundary of the region to be cloaked. The same technique can be used to make a source or an object look like a different one to an observer outside the cloaked region, from the perspective of thermal measurements. Our results assume a homogeneous isotropic bulk medium and require knowledge of the source to cloak or mimic, but are in most cases independent of the object to cloak.

math.AP↗