Numerical Analysis: Fast methods for high frequency wave propagation problems

We design methods for wave propagation, whose cost grows slowly with the frequency for a fixed tolerance.

In this project we consider high-frequency wave propagation problems. For these problems the computational cost of standard numerical methods grows algebraically with the frequency, for a fixed accuracy. We want to design methods where the computational cost is independent of, or only grows slowly with, the frequency. In one step we propose and analyze a fast method for computing the solution of the high frequency Helmholtz equation in a bounded domain with a variable wave speed function. For this case we can show rigorously that in one dimension the algorithm is convergent and that indeed the computational cost only grows slowly with the frequency.