arXiv subjects

Soumen Senapati

Publications and source records attributed to Soumen Senapati.

10 recordsLinked to original sources

arXiv2026

Localizing acoustic and electromagnetic waves in space and time

We study time-dependent acoustic and electromagnetic waves governed by the scalar wave equation or Maxwell's equations in a bounded three-dimensional domain. We establish the existence of time-dependent boundary excitations that can be prescribed on any open subset of the boundary of the domain such that the associated waves are strongly localized in space in the sense that they possess arbitrarily large norms in a given subdomain and on a given time-interval, while remaining arbitrarily small in any other given subdomain for all times. Similarly, we also show the existence of boundary data such that the associated waves are strongly localized in time in the sense that they possess arbitrarily large norms in a given subdomain and on a given time-interval, while remaining arbitrarily small on the same subdomain but on any other prescribed time-interval. In case that we have access to the possibly inhomogeneous coefficients in the wave equation or in the Maxwell system, we also give explicit constructions to obtain boundary data that generate these localized waves, and we comment on possible applications.

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arXiv2023

Recovering both the wave speed and the source function in a time-domain wave equation by injecting contrasting droplets

Dealing with the inverse source problem for the scalar wave equation, we have shown recently that we can reconstruct the space-time dependent source function from the measurement of the wave, collected at a single point $x$ for a large enough interval of time, generated by a small scaled droplets, enjoying large contrasts of its bulk modulus, injected inside the domain to image. Here, we extend this result to reconstruct not only the source function but also the variable wave speed. Indeed, from the measured waves, we first localize the internal values of the travel-time function by looking at the behavior of this collected wave in terms of time. Then from the Eikonal equation, we recover the wave speed. Second, we recover the internal values of the wave generated only by the background (in the absence of the small droplets) from the same measured data by inverting a Volterra integral operator of the second kind. From this reconstructed wave, we recover the source function at the expense of a numerical differentiation.

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arXiv2023

Minnaert Frequency and Simultaneous Reconstruction of the Density, Bulk and Source in the Time-Domain Wave Equation

We deal with the inverse problem of reconstructing acoustic material properties or/and external sources for the time-domain acoustic wave model. The traditional measurements consist of repeated active (or passive) interrogations, as the Dirichlet-Neumann map, or point sources with source points varying outside of the domain of interest. It is reported in the existing literature, that based on such measurements, one can recover some (but not all) of the three parameters: mass density, bulk modulus or the external source term. In this work, we first inject isolated small-scales bubbles into the region of interest and then measure the generated pressure field at a {\it{single point}} outside, or at the boundary, of this region. Then we repeat such measurements by moving the bubble to scan the region of interest. Using such measurements, we show that 1. If either the mass density or the bulk modulus is known then we can simultaneously reconstruct the other one and the source term. 2. If the source term is known at the initial time, precisely we assume to know its first non vanishing time-derivative, at the initial time, then we reconstruct simultaneously the three parameters, namely the mass density, the bulk modulus and the source function. Here, the source term is space-time dependent. It is worth mentioning that in terms of dimentionality, the induced inverse problem is not overdetermined since we use $4=3+1$ dimensions ($3$ in space and $1$ in time) to recover $2$ coefficients of $3$ spatial dimensions, i.e. the mass density and the bulk modulus, and the $4=3+1$ dimensional source function. In addition, the result is local, meaning that we do reconstruction in any subpart, of the domain of interest, we want.

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arXiv2023

The Calderón problem for space-time fractional parabolic operators with variable coefficients

We study an inverse problem for variable coefficient fractional parabolic operators of the form $(\partial_t -\operatorname{div}(A(x) \nabla_x)^s + q(x,t)$ for $s\in(0,1)$ and show the unique recovery of $q$ from exterior measured data. Similar to the fractional elliptic case, we use Runge type approximation argument which is obtained via a global weak unique continuation property. The proof of such a unique continuation result involves a new Carleman estimate for the associated variable coefficient extension operator. In the latter part of the work, we prove analogous unique determination results for fractional parabolic operators with drift.

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arXiv2022

Extension problem for the fractional parabolic Lamé operator and unique continuation

In this paper, we introduce and analyse an explicit formulation of fractional powers of the parabolic Lamé operator $\mathbb{H}$ and we then study the extension problem associated to such non-local operators. We also study the various regularity properties of solutions to such an extension problem via a transformation which reduces the extension problem for the parabolic Lamé operator to another system that resembles the extension problem of the fractional heat operator. Finally in the case when $s \geq 1/2$, by proving a conditional doubling property for solutions to the corresponding reduced system followed by a blowup argument, we establish a space-like strong unique continuation result for $\mathbb{H}^s \textbf{u}=V\textbf{u}$.

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arXiv2022

Point sources and stability for an inverse problem for a hyperbolic PDE with space and time dependent coefficients

We study stability aspects for the determination of space and time-dependent lower order perturbations of the wave operator in three space dimensions with point sources. The problems under consideration here are formally determined and we establish Lipschitz stability results for these problems. The main tool in our analysis is a modified version of Bukgheim-Klibanov method based on Carleman estimates.

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arXiv2021

Stability for a formally determined inverse problem for a hyperbolic PDE with space and time dependent coefficients

We prove stability for a formally determined inverse problem for a hyperbolic PDE where the coefficients depend on space and time variables. The hyperbolic operator has constant wave speed and we study the recovery of zeroth order and first order coefficients and the space dimension can be one or higher. We use a modification of the Bukhgeim-Klibanov method to obtain our results.

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arXiv2021

Stability estimate for a partial data inverse problem for the convection-diffusion equation

In this article, we study the stability in the inverse problem of determining the time-dependent convection term and density coefficient appearing in the convection-diffusion equation, from partial boundary measurements. For dimension $n\geq 2$, we show the convection term (modulo the gauge term) admits log-log stability, whereas log-log-log stability estimate is obtained for the density coefficient.

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arXiv2020

Stability estimates for relativistic Schrödinger equation from partial boundary data

In this article we study stability aspects for the determination of time-dependent vector and scalar potentials in relativistic Schrödinger equation from partial knowledge of boundary measurements. For space dimensions strictly greater than 2 we obtain log-log stability estimates for the determination of vector potentials (modulo gauge equivalence) and log-log-log stability estimates for the determination of scalar potentials from partial boundary data assuming suitable a-priori bounds on these potentials.

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arXiv2020

A uniqueness result for light ray transform on symmetric 2-tensor fields

We study light ray transform of symmetric 2-tensor fields defined on a bounded time-space domain in $\mathbb{R}^{1+n}$ for $n\geq 3$. We prove a uniqueness result for such light ray transforms. More precisely, we characterize the kernel of the light ray transform vanishing near a fixed direction at each point in the time-space domain.

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