Collaborative Research: Nonoscillatory Phase Methods for the Variable Coefficient Helmholtz Equation in the High-Frequency Regime

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The importance of the numerical simulation of physical phenomena by computers cannot be overstated. Such computations have become essential tools both in scientific research and in industrial research and development. This project concerns the numerical simulation of the scattering of waves. Such simulations have applications to sonar and radar, as well as in medical imaging, geophysics, and many other applications. Wave phenomena become more complicated to model as the frequency of the wave increases, and our current ability to accurately model high-frequency waves is quite limited. This project seeks to develop new methods for modeling high-frequency waves efficiently and to high accuracy. The project provides research training opportunities for graduate students. The numerical simulation of the scattering of waves from inhomogeneous media has important applications in radar and sonar, medical imaging, geophysics, and a host of other scientific applications. In many cases of interest, such simulations can be performed by solving the variable coefficient Helmholtz equation. The solutions of this equation are oscillatory, and the difficulty of calculating them using conventional approaches grows quickly with the frequency of the oscillations. Recently, one of the investigators developed a new class of solvers for the variable coefficient Helmholtz equation that achieve extremely high accuracy and have run times that scale much more slowly with increasing frequency than conventional solvers. They operate by solving the nonlinear Riccati equation that is satisfied by the logarithms of solutions of the Helmholtz equation. Currently, these solvers only apply in special cases. This project aims to extend them to the general case to develop a method for the variable coefficient Helmholtz equation that is significantly faster than current techniques.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.