arXiv subjects

Anamika Purohit

Publications and source records attributed to Anamika Purohit.

6 recordsLinked to original sources

arXiv2026

Recovery of time-dependent coefficients for the convection-diffusion equation on conformally transversally anisotropic manifolds from partial data

We study an inverse problem of recovering a time-dependent convection term and density coefficient of the convection-diffusion equation from partial data on a certain type of compact Riemannian manifold of dimension at least three. We prove that the knowledge of a partial input-output operator determines both coefficients uniquely up to a natural gauge. Our geometric setting is conformally transversally anisotropic manifolds, that is, compact Riemannian manifolds with boundary that are conformally embedded into a product of the Euclidean line and a transversal manifold. Additionally, we assume that the attenuated geodesic ray transforms of one-forms and functions are both injective on the transversal manifold.

math.AP↗

arXiv2026

An Inverse problem for a fourth order nonlinear Schrödinger equation (NLS)

We study an inverse problem for the time-dependent nonlinear fourth-order Schrödinger equation on both compact Euclidean domains and compact Riemannian manifolds, (say) $M$. This model arises in nonlinear fiber optics and the theory of optical solitons in gyrotropic media. Our main objective is the identification of unknown coefficients from the associated source-to-solution map, which assigns to each source term $f$, supported in $(0, T)\times Γ$, the corresponding solution $u$ restricted to the same set, where $Γ\subset M$ is a neighborhood of $\partial M$. We prove that the zeroth-order term, the second-order coefficient, and the nonlinear coefficient are uniquely determined by this map. Moreover, the recovery of the symmetric second-order tensor reduces to the inversion of a divergent beam transform.

math.AP↗

arXiv2025

Inverse boundary value problem for the Convection-Diffusion equation with local data

We study a local data inverse problem for the time-dependent Convection-Diffusion Equation (CDE) in a bounded domain where a part of the boundary is treated to be inaccessible. Up on assuming the inaccessible part to be flat, we seek for the unique determination of the time-dependent convection and the density terms from the knowledge of the boundary data measured outside the inaccessible part. In the process, we show that there is a natural gauge in the perturbations, and we prove that this is the only obstruction in the uniqueness result.

math.AP↗

arXiv2024

Tensor tomography using V-line transforms with vertices restricted to a circle

In this article, we study the problem of recovering symmetric $m$-tensor fields (including vector fields) supported in a unit disk $\mathbb{D}$ from a set of generalized V-line transforms, namely longitudinal, transverse, and mixed V-line transforms, and their integral moments. We work in a circular geometric setup, where the V-lines have vertices on a circle, and the axis of symmetry is orthogonal to the circle. We present two approaches to recover a symmetric $m$-tensor field from the combination of longitudinal, transverse, and mixed V-line transforms. With the help of these inversion results, we are able to give an explicit kernel description for these transforms. We also derive inversion algorithms to reconstruct a symmetric $m$-tensor field from its first $(m+1)$ moment longitudinal/transverse V-line transforms.

math.NA↗

arXiv2024

Inverse problem for a time-dependent Convection-diffusion equation in admissible geometries

We consider a partial data inverse problem for a time-dependent convection-diffusion equation on an admissible manifold. We prove that the time-dependent convection term and time-dependent density can be recovered uniquely modulo a known gauge invariance. There have been several works on inverse problems related to the steady state convection-diffusion operator in Euclidean as well as in Riemannian geometry settings; however, inverse problems related to time-dependent convection-diffusion equation on a manifold are not studied in the prior works, which is the main aim of this paper. In fact, to the best of our knowledge, the problem studied here is the first work related to a partial data inverse problem for recovering both first and zeroth-order time-dependent perturbations of evolution equations in the Riemannian geometry setting.

math.AP↗

arXiv2024

Determining time-dependent convection and density terms in convection-diffusion equation using partial data

In this article, we study an inverse boundary value problem for the time-dependent convection-diffusion equation. We use the nonlinear Carleman weight to recover the time-dependent convection term and time-dependent density coefficient uniquely. Nonlinear weight allows us to prove the uniqueness of the coefficients by making measurements on a possibly very small subset of the boundary. We proved that the convection term and the density coefficient can be recovered up to the natural gauge from the knowledge of the Dirichlet to Neumann map measured on a very small open subset of the boundary.

math.AP↗