Department of Mathematics
"An Accurate and Fast Solver for High-Frequency Wave Propagation"
In science and engineering, the fast and accurate solution of high-frequency wave propagation problems in highly heterogeneous media is of extremely high interest. For example, these problems are a crucial part of inverse problems for oil exploration. In this context, it is important to solve the problem not only accurately but also fast. Recently, there have been many advances in the development of fast solvers (linear complexity with respect to the number of degrees of freedom). While most of those methods only scale optimally in the context of low-order discretizations and smooth wave speed distributions, the Method of Polarized Traces has been shown to be the only method that scales optimally for high-order discretizations and highly heterogeneous (and even discontinuous) wave speed distributions.
In this work, we show that the combination of Hybridizable Discontinuous Galerkin (HDG) Methods and the Method of Polarized Traces results in an accurate and fast solver for high frequency wave propagation in highly heterogeneous media. In particular, we show that on a uniform mesh, HDG methods are second order accurate, can be made pollution free, and can be solved in O(N) time where N is the number of total degrees of freedom. We concentrate on the aspects of discretization and its use in the context of the Method of Polarized Traces. However, our approach also shows a lot of inherent parallelism which can be used for further speed-up.
We motivate and introduce the new method and corroborate our claims using geophysical numerical examples.