Over the last few years, I have been working on Formally Determined Inverse Problems For Hyperbolic PDEs. Specifically, I have been working on the Fixed Angle Scattering Problem and the Backscattering Problem. I study the uniqueness, continuous dependence, inversion, and related theoretical questions, for such inverse problems.
Earlier I did considerable work on the Inversion Of The Spherical Mean Value Operator. This has strong connections with certain inverse problems for the constant speed wave equation.
Early in my career, I did considerable work on One-dimensional Inverse Problems For Hyperbolic PDEs and on Overdetermined Inverse Problems For Hyperbolic PDEs.
I also have some publications on Dynamical Systems, Geometry, Probability and Economics.
Fixed Angle Scattering for velocities, Riemannian metrics and Lorentzian metrics; UC Irvine online seminar 2024 (video, slides).
Unsolved Formally Determined Inverse Problems For The Wave Equation; AIP Grenoble 2019 talk. (slides)
Introduction to Inverse Problems For PDEs; three presentations at TIFR (Bangalore) Inverse Problems School 2014. (slides)
Rigidity in the Lorentzian Calder\'on problem with formally determined data, (with Lauri Oksanen and Mikko Salo), (2024).
Rigidity for fixed angle inverse scattering for Riemannian metrics, (with Lauri Oksanen and Mikko Salo), (2024).
Point sources and stability for an inverse problem for a hyperbolic PDE with space and time dependent coefficients, (with Venky Krishnan and Soumen Senapati), (2023) Journal of Differential Equations 342, 622-665.
Stability for a formally determined inverse problem for a hyperbolic PDE with space and time dependent coefficients (with Venky Krishnan and Soumen Senapati), (2021) SIAM J. Math. Anal. 53, no. 6, 6822–6846.
Fixed angle inverse scattering for almost symmetric or controlled perturbations (with Mikko Salo), SIAM J. Math. Anal., (2020) 52(6), 5467- 5499.
The fixed angle scattering problem and wave equation inverse problems with two measurements (with Mikko Salo), (2020) Inverse Problems 36 035005.
The point source inverse back-scattering problem (with Gunther Uhlmann), Contemporary Mathematics, Volume 644, (2015).
Uniqueness for the inverse back-scattering problem for angularly controlled potentials (with Gunther Uhlmann) Inverse Problems 30 065005 (2014).
Uniqueness for a hyperbolic inverse problem with angular control on the coefficients (with Paul Sacks) Journal Inverse Ill-Posed Problems 19 (2011), no. 1, 107-126.
A linearised inverse problem for the wave equation, Communications in PDE, 13, 573-601 (1988).
Recovering initial values from light cone traces of solutions of the wave equation (with Tao Yuan), Inverse Problems 34 (2018) 075003.
Spherical means with centers on a hyperplane in even dimensions (with E K Narayanan), Inverse Problems 26 (2010) 035014 (12pp).
Recovering a function from its spherical mean values in two and three dimensions (with D Finch), in Photoacoustic imaging and spectroscopy, edited by Lihong Wang, CRC Press (2009).
The spherical mean value operator with centers on a sphere (with D Finch), Inverse Problems Vol 23 No 6, S37-S49 (2007).
Inversion of spherical means and the wave equation in even dimensions (with D Finch and M Haltmeier), SIAM J Appl Math Vol 68 Issue 2, pages 392-412 (2007).
The range of the spherical mean value operator (with David Finch), Inverse Problems 22 (2006) 923-938.
Trace identities for solutions of the wave equation with initial data supported in a ball (with D. Finch), Math Meth Appl Sciences, Volume 28, Issue 16(2005),1897-1917.
Determining a function from its mean values over a family of spheres (with D. Finch and S. Patch), SIAM J Math Analysis, Vol 35, No 5, pp 1213-1240 (2004).
Determining the twist in an optical fiber (with Jiahua Tang and Andrew Lacey), Inverse Problems, Volume 32, No. 3 (2016).
Stability for an inverse problem for a two-speed hyperbolic PDE in one space dimension (with P Sacks), Inverse Problems 26 (2010) 025005 (20pp).
Inverse problems for the wave equation with a single coincident source-receiver pair, Inverse Problems 24, 015012 (2008).
An inverse problem for a layered medium with a point source, Inverse Problems 19, 497-506, (2003).
A One Dimensional Inverse Problem For A Hyperbolic System With Complex Coefficients, Inverse Problems, 17, 1401-1417 (2001).
Characterization Of Transmission Data For Webster's Horn Equation, Inverse Problems, 16, L9-L24 (2000).
Inversion Of Spherically Symmetric Potentials from Boundary Data For The Wave Equation, Inverse Problems, 14, 999-1007 (1998).
Potential Inversion From Transmission Data For The One Dimensional Wave Equation, Wave Motion, 25, 319-329 (1997).
Impedance Inversion From Transmission Data For the Wave Equation (with Paul Sacks), Wave Motion, 24, 263-274 (1996).
An inverse impedance transmission problem for the wave equation, Communications in PDE, 18, (1993), 583-600.
An inverse problem for the wave equation in the half plane, Inverse Problems, 9, 433-441 (1993).
Property C and an inverse problem for a hyperbolic equation (with A.G. Ramm), Journal of Mathematical Analysis and Applications, Vol. 156, No. 1, (1991), 209-219. Here is an earlier version of this article which may be of interest too.
Reconstruction for an inverse problem for the wave equation, Inverse Problems, 6 (1990), 91-98.
Uniqueness for an inverse problem for the wave equation (with W.W. Symes), Communications in PDE, 13, 87-96 (1988).
Input-to-state stability for hybrid periodic orbits of systems with impulse effects via Poincare analysis, (with Sushant Veer and Ioannis Poulakakis), IEEE Transactions on Automatic Control (2019) , Vol. 64, No. 11, November 2019.
A remark on unique continuation along and across lower dimensional planes for the wave equation, Math. Meth. Applied Sciences, Vol 36, 246-252 (2009).
The d-th linear polarization constant of R^d (with YJ Leung and Wenbo Li), Journal of Functional Analysis, Vol 255, Issue 10, 2861-2871 (2008).
Spectral analysis of Brownian motion with jump boundary (with YJ Leung and Wenbo Li), Proceedings of the AMS, Vol 136, 4427-4437 (2008).
Optimal ellipsoids and decomposition of positive definite matrices (with YJ Leung and W Li), Journal of Mathematical Analysis and Applications, Volume 331, Issue 2, 1452-1466 (2007).
Numerical solution of a timelike Cauchy problem for the wave equation (with M. Klibanov), Mathematical Methods in the Applied Sciences, 15, (1992), 559-570.
Dynamic Pricing and Ordering Decisions by a Monopolist (with A. Rajan and R. Steinberg), Management Science, Vol. 38, No. 2, (1992), 240 - 262.