Dimitar Trenev

Research Interests

My primary research interest is in the field of Numerical Analysis and in particular in the various methods for approximation of solutions to partial differential equations. I am also interested in computational mathematics and in the technology advancements for large scale computing.

Current Research

I have worked on the development of a numerical algorithm for approximately solving the elastic wave scattering problem. As this is a problem posed on an infinite domain a perfectly matched layer technique was used to truncate the problem. Full details on this work are available in the paper published in Mathematics of Computation (reference below).

Additionally, in my thesis I proposed and analyzed a change of coordinates approach to modify certain elliptic PDE with decaying solutions in such a way, that the solutions of the new equations decay much more rapidly and good approximations can be found by truncating the problems to reasonably small computational domains.
As a picture is worth a thousand words, here is an approximation, using exponential scaling of the Cartesian coordinates, of a solution to the Laplace equation on the exterior of a rectangle, approximated close to the rectangle.

Most recently, during my year at ExxonMobil Research and Engineering, I worked on the development and application of a perfectly matched layer absorbing boundary to seismic wave propagation, and investigated its effect on the seismic waveform inversion problem. A summary of the results is currently in preparation.

Further detail on my research interests can be found in my research statement.

Publications and Preprints

• J. H. Bramble, J. E. Pasciak and D. Trenev, Analysis of a finite PML approximation to the three dimensional elastic wave scattering problem, Math. Comp., 79 (2010).
• D. Trenev, Spatial scaling for the numerical approximation of problems on unbounded domains, thesis, Texas A&M University (available online).