I work on Inverse Problems related to Partial Differential Equations (PDEs). Inverse problems are just the opposite of Direct Problems. Loosely speaking, a Direct Problem calculates the output or solves a physical system, provided all its parameters and initial or input boundary data are given. On the other hand, an Inverse Problem attempts to determine the parameters of a system, provided there is enough information about the input and output measurements. For a simple introduction to this research area please watch: Nuutti Hyvönen: "Inverse problems" and Roy Pike "What is an inverse problem?"
Since a large number of physical systems (wave propagation, heat flow, fluid flow, electromagnetic, elasticity, ...) are governed by PDEs, it is worth investigating inverse problems for PDEs. Some physical applications of inverse problems for PDEs can be found in Medical Imaging or Tomography such as: Ultrasonography, CT scan, MRI, EIT, etc. Some key words related to my research area are: Analysis and PDEs, Integral Geometry, Riemannian Geometry, Pseudodifferential Calculus, Fourier Integral Operators, Microlocal Analysis, Semiclassical Analysis etc.
In this area, we consider wave operators and the system of elastic waves modelling acoustic or seismic wave propagations. The coefficients that appear in the systems are related to several physical parameters such as the density of the medium, elasticity of the medium, wave speeds, etc. We measure the displacement and traction on a part of the boundary and determine the coefficients locally near the measured portions.
An added difficulty arises when we consider the medium is non-smooth, i.e. having layers with high contrast in the density and other parameters. This setup is useful to model Earth (PREM), Brain Imaging, Ultrasound Imaging, etc. The jump in the coefficients results in the scattering of the waves, which severely complicates the determination process.
In this area, we consider a harmonic or a polyharmonic operator with lower-order perturbations denoting various parameters in the domain. We consider the Dirichlet-Neumann map corresponding to the operators measured on the boundary and determine the coefficients in the interior of the domain. An example of such a problem for a second-order operator is the famous Calderón problem, the mathematical formulation of Electrical Impedance Tomography. In general, we study polyharmonic operators (of order 2m) with several lower-order tensorial perturbations and determine the perturbations from the boundary measurements.
Furthermore, it is practically relevant to measure data only on a part of the boundary to determine the coefficients. The lack of access to the full boundary creates several problems in determining the coefficients in the whole domain along with severe instability in the determination process.
Uploaded to arXiv
Direct and inverse problem for bi-wave equation with time-dependent coefficients from partial data
Jointly with Pranav Kumar. (ArXiv)
Density results of biharmonic functions on symmetric tensor fields and their applications to inverse problems
Jointly with Divyansh Agrawal and Pranav Kumar. (ArXiv)
Published
Recovery of piecewise smooth parameters in an acoustic-gravitational system of equations from exterior Cauchy data
Jointly with Maarten de Hoop, Vitaly Katsnelson, (2025). Inverse Problems 41 075013 DOI 10.1088/1361-6420/adec14 (ArXiv)
Inverse Problems For Third-Order Nonlinear Perturbations Of Biharmonic Operators
Jointly with Katya Krupchyk, Suman Kumar Sahoo, Gunther Uhlmann (2025). Communications in Partial Differential Equations, 1–34. https://doi.org/10.1080/03605302.2024.2444972. (ArXiv)
Local data inverse problem for the polyharmonic operator with anisotropic perturbations.
Jointly with Pranav Kumar (2024), Inverse Problems 40, no. 5, 055004. (ArXiv)
Momentum Ray Transforms and a Partial Data Inverse Problem for a Polyharmonic Operator.
Jointly with Venkateswaran P. Krishnan and Suman Kumar Sahoo (2023), SIAM Journal of Mathematical Analysis. (ArXiv)
Recovery of Piecewise Smooth Density and Lamé Parameters from High Frequency Exterior Cauchy Data.
Jointly with Maarten V-de Hoop, Vitaly Katsnelson and Gunther Uhlmann (2022),
SIAM Journal on Imaging Sciences 15 (4), 1910-1943. (ArXiv)
Recovery of wave speeds and density of mass across a heterogeneous smooth interface
from acoustic and elastic wave reflection operators.
Jointly with Maarten V-de Hoop, Vitaly Katsnelson and Gunther Uhlmann (2022)
GEM-International Journal on Geomathematics 13 (1), 9. (ArXiv)
An inverse problem on determining second order symmetric tensor for perturbed biharmonic operator.
Jointly with Tuhin Ghosh (2021), Mathematische Annalen. (ArXiv)
Inverse Problem for Fractional-Laplacian with Lower Order Non-local Perturbations.
Jointly with Tuhin Ghosh and Gunther Uhlmann, Transactions of American Mathematical Society (2020). (ArXiv)
Inverse boundary value problem of determining up to second order tensors appear
in the lower order perturbations of the polyharmonic operator.
Jointly with Tuhin Ghosh, Journal of Fourier Analysis and Applications 25 (2019), no. 3, 661--683. (ArXiv)
Local uniqueness of the density from partial boundary data for isotropic elastodynamics.
Inverse Problems 34 (2018), no. 12, 125001, 10 pp. (ArXiv)
Optimal stability estimate of the elliptic periodic coefficient problem by partial measurements.
Jointly with Cătălin I. Cârstea, Journal of Mathematical Analysis and Applications 466 (2018), no. 1, 642--654. (ArXiv)
An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data.
Inverse Problems and Imaging 12 (2018), no. 3, 801--830. (ArXiv)