Vitaly Katsnelson’s research focuses partial differential equations—specifically, microlocal analysis and inverse problems. He received his Ph.D. in Mathematics from Stanford University, where he used microlocal analysis to study elastic wave propagation on singular manifolds (elastic objects with edges and corners) to better understand how waves experienced during an earthquake interact with sharp objects.
Katsnelson continued his research through postdoctoral work at Rice University under the supervision of Professor Maarten V. de Hoop. The focus of his study was on inverse problems where they attempt to reconstruct material properties of the Earth using seismic surface data or by conducting seismic experiments on the surface itself. This process is not unlike medical imaging where scientists try to detect cancer cells or obtain an image of the body noninvasively by sending certain waves and then taking measurements of these scattered waves outside the body, akin to a CT scan.
In 2017, Katsnelson and his collaborators obtained a fundamental uniqueness result for the inverse boundary value problem associated with the acoustic wave equation with piecewise smooth wave speeds, based on the scattering control series they developed. The scattering control allowed for disentanglement of internal multiple scattering without knowledge of the wave speed or interfaces, which was the key to recovering the full wave speed without linearization. This research was then carried into the realm of the elastic wave equation in developing a scattering-control-based approach to the recovery of P-and S-wave speeds, and showing the extent that internal multiples may be suppressed and eliminated.
Currently, Vitaly’s research looks at the frequencies of the Earth’s free oscillations during an earthquake and how much structure of the planet’s interior one can reconstruct from such frequencies that are measured in practice.
Recent Projects/Research
- Spectral rigidity for spherically symmetric manifolds with discontinuities
- Rayleigh waves in the presence of edges and corners
- Microlocal analysis of FWI
Publications
- J. William Helton, Vitaly Katsnelson, and Igor Klep. “Sign patterns for chemical reaction networks.” J. Math. Chem. 47(1):403-429, 2010.
- Vitaly Katsnelson. “Diffraction of elastic waves by edges.” Annales de l'Institut Fourier. 2016. arXiv:1611.06994.
- Vitaly Katsnelson, Linh V. Nguyen. “Time reversal method for thermoacoustic tomography in elastic media: convergence with sharp observation time.” Appl. Math. Lett. (2017).
- Peter Caday, Maarten V. de Hoop, Vitaly Katsnelson, Gunther Uhlmann. “Scattering control for the wave equation with unknown wave speed.” Arch. Ration. Mech. Anal. 2017. arXiv:1701.01070.
- Peter Caday, Maarten V. de Hoop, Vitaly Katsnelson, Gunther Uhlmann. “Reconstruction of piecewise smooth wave speeds using multiple scattering.” Trans. Amer. Math. Soc. 2018. arXiv:1801.03144.
Courses Taught at New York Tech
- Math 151
- Math 170