Inverse problems have been a very active field of mathematical and numerical research over the last decades, driven by many applications of important economic and societal impact.They are intrinsically difficult to solve: this fact is due in part to their very mathematical structure and to the effect that generally only partial data is available. In the last decade, new and exciting directions of research have emerged. On the one hand, the crossing of ideas from neighboring topics such as control theory, shape optimization or geometry has allowed substantial progress in analyzing and solving inverse problems. On the other hand, in an effort to increase the amount of information available for reconstruction, new imaging methodologies have been invented, that use multi-physics modalities and apriori knowledge of the possible multiscale character of a medium. Our project focuses on multiwave inverse problems which is one of these promising new directions, where our team is at the forefront of international research.

A variety of multiwave imaging modalities have been introduced in the last decade. The term multiwave imaging refers to the fact that two or more types of physical waves are used to probe the medium under study. Usually, the first type of waves «W 1 (p)» is sensitive to the contrast of the parameter «p» to be reconstructed, while the other types «W 2 (I)» can carry the internal information «I» revealed by the first type of waves to the boundary of the medium, where measurements are taken. By cleverly combining the interaction of different types of waves, one may hope to obtain at the same time sensitivity and resolution. Several mechanisms of interaction are possible: the interaction of the first type of waves with the medium may generate waves of the second type; a wave of the first type, which carries information about the contrast of the desired parameter, can be locally modulated by a wave of the second type that has better spatial resolution; a fast propagating wave can be used to acquire a spatio-temporal sequence of the propagation of a slower transient wave. Due to the large discrepancy in time scales and free paths between the two waves, usually the first wave is of diffusion type, while the second one is of hyperbolic type. In terms of reconstruction, one typically proceeds in two steps (1). Information from waves of the second type «W 2 » is measured at the boundary of the medium. A first step, which usually takes the form of a well-posed linear inverse source problem, provides internal data «I» for the contrast-sensitive waves of the first type «W 1 (p)». The second step consists in recovering the values of the desired parameter «p» from that internal data.