Research

Applications of hyperboloidal foliations to relativistic perturbations of stars and black holes.

The problem of outer boundary conditions for partial differential equations on unbounded domains has a long history that goes back to Sommerfeld. There have been significant developments in the treatment of artificial outer boundaries, particularly since the 1970s, with the rise and prominence of numerical methods.

Wave propagation problems on unbounded domains are ubiquitous in computational mathematics, engineering, and technology, appearing in various domains such as ultrasonics, seismics, underwater acoustics, and electrodynamics. As a concrete example, consider the numerical computation of the radar cross-section of aircraft.

One of the most important open problems with the hyperboloidal method is its application to the nonlinear Einstein equations for gravitational wave astronomy. The problem is that the compactification introduces formally singular terms that are numerically unstable.

The remarkable success of the hyperboloidal approach in representing propagating fields in the exterior of isolated systems suggests that it may be a useful tool for the study of quantum fields.

Swarm-based optimization for non-convex functions