NUMERICAL RELATIVITY
I am currently working on the design of numerical methods for the solution of initial-boundary value problems for the Einstein field equations. I am interested in methods for formulating equations and specifying boundary conditions in the way that achieves a stable evolution. I am studying ways to solve systems with differential constraints numerically. In particular, I am working on the development of the methods for prescribing well-posed boundary conditions in a manner consistent with the constraint equations and methods for constraint enforcement during the simulations such as constraint-damping, incorporating constraints into the evolution and design of the constraint-free discretization techniques.

FINITE ELEMENTS / DISCONTINUOUS GALERKIN METHODS
I am interested in the development of higher order discontinuous Galerkin methods for solving hyperbolic problems encountered in gas kinetics such as Boltzman-BGK equation. I am also working with applications of DG methods for second-order hyperbolic equations and model problems of constraint evolution.

DIFFUSION WEIGHTED MRI
The diffusion weighted MRI can measure rate of diffusion in a sample medium at any point in any direction. This information can be used to study orientation of the tissue fibers in the medium, since the fastest diffusion is assumed to occur along the fibers in the sample. By combining diffusion information, structure of the entire object can be revealed. I work with a CSUN student, John Sikora, on a research projects related to the fiber structure reconstruction in human brain based on the diffusion weighted MRI.

INVERSE PROBLEMS
My Ph.D. thesis was devoted to applications of optimal control theory to inverse problems of electromagnetic wave propagation. Problems of electromagnetic wave propagation arise, e.g., in non-invasive techniques of medical imaging, non-destructing testing of materials in engineering, radar sensing and geophysical prospecting. In these applications the information about the object can only be acquired on the surface of the object or on its outside layers. The problem then consists in determining the material parameters inside the object from the data measured on the surface. This translates into the mathematical (inverse) problem of reconstructing the coefficients (or the right side) in the governing differential equation from the information known on the boundary of the domain.

PAPERS AND PREPRINTS

Constraint-preserving boundary conditions for the linearized BSSN formulation. Abstract and Applied Analysis, Vol. 2008, Article ID 742040 (2008), http://www.hindawi.com/getarticle.aspx?doi=10.1155/2008/742040


Well-posed initial-boundary value problem for a constrained evolution system and radiation-controlling constraint-preserving boundary conditions. Journal of Hyperbolic Differential Equations. Vol.4 (2007), No.4, pp.587--612. http://www.worldscinet.com/jhde/04/0404/S02198916070404.html http://arXiv.org/abs/gr-qc/0611011


New first-order formulation for the Einstein equations, Physical Review D, vol.68 (2003), 064013. http://arXiv.org/abs/gr-qc/0210071


Numerical algorithms for identification problem in magnetoencephalography, Journal of Inverse and Ill-Posed Problems, vol.7 (1999), No.5, pp.387-408


Uniqueness of the stationary point for the one-dimensional inverse problem in two-dimensional space, Journal of Inverse and Ill-Posed Problems, vol.6 (1998), No.2, pp.95-114

Radiation-controlling boundary conditions for a problem of constrained evolution. Abstract for The International Conference on "Inverse and Ill-posed problems of mathematical physics", August 21-24, 2007, Novosibirsk, Russia. Click here for the posters


Well-posed initial-boundary value constrained evolution problems. abstract for the AMS Joint Meetings in New Orleans, January 2007. (2007)


Two examples of constraint evolution: 3-1=2 and 6-3=5. research notes, California State University, Northridge (2006)


A new symmetric hyperbolic formulation for the ADM system, abstract for the AMS Joint Meetings in Baltimore, January 2003. (2003)


Hyperbolic formulations for the linearized ADM system. research notes, University of Minnesota (2002)


A new symmetric hyperbolic formulation of ADM system. poster presentation for ``Hot Topics'' workshop on Numerical Relativity at IMA, UMN in June 2002.


Hyperbolic Formulations in Linearized Gravity, abstract for The International Conference in Inverse and Ill-Posed Problems, Novosibirsk, August 5-9, 2002. click here for the posters