Inverse problems for hyperbolic partial differential equations

A typical inverse boundary value problem arises when an object is probed by sending waves to it in order to determine some properties of the object from the measured responses. The proposed research focuses on the case where the measurements are modelled using the acoustic wave equation. The acoustic case can be seen as a prototype case for several geophysical imaging problems. These problems can be local, for example, when the goal is to detect geological deposits, or global when the goal is to chart the internal structure of the Earth.

The aim of the proposed research is to significantly extend the mathematical theory of inverse boundary value problems and to bridge the gap between this theory and practical applications in geophysical imaging. The main theoretical objective is to show that the inverse boundary value problem for the wave equation has a unique solution in the fully coordinate invariant context.

The main practical objective is to develop a new computational method for inverse boundary value problems. The presently used computational methods in geophysical imaging are not based on the mathematical theory of inverse boundary value problems. Bridging the gap between the theory and applications has a potential for a high impact on the latter, and the resulting physical understanding will also guide further mathematical study.

The research brings together ideas from analysis of partial differential equations, differential geometry, and numerical analysis.

The proposed research has transformative potential since solving the inverse boundary value problem for the wave equation in a coordinate invariant manner will guide solving other inverse boundary value problems of geometric nature, since it explores a promising new paradigm to solve unique continuation problems, and since it builds bridges between mathematics and geophysical imaging. Moreover, the methods to be developed are based on geometric techniques that have applications beyond inverse boundary value problems, for example, in cosmology.

The fellow has experience in communicating with the researchers in geometry and analysis. To disseminate the results to numerical analysis and geophysical imaging communities the connections of the collaborators will be used. Drawing from the fellow's experience in inter-disciplinary discussions with the Photoacoustic Imaging Group at University College London, in the context of his EPSRC 1st grant project (EP/L026473/1), a plan to communicate the results to geophysicists has been based on construction of computational examples.