Research teams in the FAME Flagship have made substantial progress in the mathematical theory of wave imaging. This theory forms the basis for various applications in seismic and medical imaging, thus providing a solid foundation for our impact in these fields.
A basic model for wave imaging problems is the Lorentzian Calderón problem, where the objective is to determine a possibly time dependent sound speed in a medium from measurements of waves at the boundary. This problem represents the wave analogue of the fundamental anisotropic Calderón problem of electrical impedance tomography and has long stood as a major open question in the field. The basic challenge has been the absence of techniques capable of dealing with smooth time-dependent sound speeds that do not satisfy real-analyticity assumptions.
However, Professor Lauri Oksanen from the University of Helsinki has demonstrate that an unknown potential can be determined from boundary measurements of waves. This breakthrough was achieved by extending a version of the boundary control method to the case of certain smooth Lorentzian metrics by relying on a new optimal unique continuation result in the exterior of double null cones for metrics satisfying suitable curvature conditions.
To address the problem of determining a time-dependent sound speed, the research teams of Oksanen and Professor Mikko Salo (University of Jyväskylä) introduced a novel method that utilises distorted plane wave measurements and a combination of geometric, topological and unique continuation arguments. Remarkably, the new method requires much smaller amount of measurements compared to previous approaches, thus solving formally determined versions of such inverse problems. This advancement paves the way for tackling the full Lorentzian Calderón problem for smooth time-dependent sound speeds.
There has also been considerable progress in inverse problems for nonlinear wave equations. The higher order linearization method introduced by the University of Helsinki team in 2018 has led to numerous results for which nonlinearity acts as an asset rather than a hindrance. While most of these advancements have focused on boundary measurements, which excludes critical scattering applications such as radar imaging, recent efforts by Professor Matti Lassas (University of Helsinki) and Assistant Professor Teemu Tyni (University of Oulu) have addressed this shortcoming. They generalized these methods for a scattering model by introducing novel scattering functionals that render the problem meaningful even in cases where global well-posedness may not hold. This progress opens possibilities for addressing various inverse scattering problems with nonlinear models.
