At the interface between semiclassical analysis and numerical analysis of wave propagation problems

Our understanding of wave propagation underpins several important technologies in everyday use. For example, WiFi and mobile phones use electromagnetic waves to transmit information, and the technologies in seismic and medical imaging use acoustic, elastic, and electromagnetic waves to obtain information about the ocean floor and the human body.

Over the last few years, the investigator has worked with, on the one hand, researchers interested in wave phenomena from a purely mathematical point of view (without any thought of applications), and, on the other hand, researchers interested primarily in improving how wave phenomena are simulated in applications (with this understanding then feeding into new technologies). This experience has uncovered huge untapped potential in the relationship between the theoretical aspects of wave propagation and the more-practical aspects, and this fellowship seeks to exploit this.

The overall goals are to (i) prove fundamental theoretical results about wave propagation, motivated by applications, and (ii) use these theoretical results to prove fundamental results about how wave propagation is simulated using computers, addressing long-standing open problems and developing new numerical methods that have the potential to change the technologies used in the huge variety of practical applications of wave propagation.

Our understanding of acoustic, electromagnetic and elastic wave propagation is used in a plethora of technologies upon which our society depends, for example: radar, sonar, mobile phone technology, ultrasound, noise barriers on motorways, optical fibres, seismic imaging. The results of this Fellowship will enhance our understanding of wave phenomena, and thus ultimately contribute to improved technologies which will benefit the general public. In the short term, there are potential pathways to certain industrial applications (listed below) from several results this Fellowship seeks to obtain. In the longer term, results from this Fellowship will form the basis of future investigations into wave propagation problems.

Examples of short-term potential industrial impacts of this Fellowship include the following.

- Medical imaging: ultrasound. Boundary integral equations are currently used in the simulation of high-frequency acoustic waves used in ultrasound. The new integral-equation formulations of the Fellowship therefore have the potential to make these high-frequency ultrasound simulations faster and more reliable.

- Medical imaging: microwaves. A particular example of the use of electromagnetic waves in medical imaging is the use of microwaves to diagnose strokes. The new methods of the Fellowship for the fast numerical solution of the electromagnetic waves will be able to simulate microwaves travelling through the brain, and so they have the potential to make this diagnosis process faster and more reliable.

-Seismic imaging. The seismic imaging community currently use time-domain solvers to simulate the elastic waves used in imaging technologies, but there is an ongoing "quest" for an optimal frequency-domain solver. There is strong precedent for extending Helmholtz solvers to frequency-domain elastic waves, and thus the methods of the Fellowship for the Helmholtz equation with variable wave speed will generate huge interest in this area.