#### PI: Laurent Demanet, Department of Mathematics, MIT

### Abstract

The proposed research project aims at designing new methods of model reduction for computational wave propagation, in order to deal with currently out-of-reach situations where the unknowns number in the billions. To bypass the limitations of all-purpose methods, this project will explore nonlinear optimization ideas under the name of phase tracking. The set of tools provided by model reduction is needed for making progress with the resolution of seismic imaging, and for providing measures of uncertainty in the results.

### Report

- Project Title: Model Reduction for High-frequency Wave Scattering
- Principal Investigator: Laurent Demanet, Department of Mathematics, MIT
- Grant Period: February 2017 – January 2018

**Participants in the project**:

At least:

Laurent Demanet (PI)

Dmitry Batenkov (postdoc)

Matt Li (graduate student)

**Summary**

The project’s goal is to develop computational methods for wave propagation, in order to address some of the fundamental challenges that arise in the high-frequency regime. In turn, the availability of such methods is key for enabling high-resolution imaging from waves, such as seismic imaging (with oil and gas applications), radar imaging (with defense applications), and ultrasound imaging (with medical applications).

This project’s approach is to design novel methods of model reduction, in order to reach parallel runtime complexities that scales more slowly than the total number of unknowns in the simulation. Preliminary results from our team indicate that sublinear scalings are indeed possible, which was quite a surprise in view of the common wisdom in numerical analysis. Specifically, on a sufficiently parallel machine (a large cluster), under reasonable assumptions, we have shown that it is possible to compute the solution in a number of steps independent of the number of grid points needed to represent the highly oscillatory solution. The approach is original because it applies “model reduction” in a nonstandard way at the level of the phases of the wavefields, rather than the wavefields themselves.