Optimal reduced-order solution to the 3D deterministic parabolic wave equation

Abstract

In weakly scattering media, the parabolic wave equation is often utilized to develop approximate solutions to the Helmholtz equation. However, in three dimensions, at high frequencies, and when the computational domain becomes very large, computational cost grows quickly. We propose using a reduced-order modeling approach for the deterministic 3D parabolic wave equation via the dynamical low-rank approximation. This approach is similar in spirit to the normal-mode technique; however, instead of using fixed modes for the entire computational domain, we instantaneously optimally evolve the modes in range according to the PDE dynamics. This dynamic order reduction provides a more accurate low-rank approximation to the parabolic wave equation. We demonstrate the efficacy of this technique on realistic ocean acoustic test cases.