Uncertainty Quantification for Source Localization and Passive Imaging in Random

This project addresses Inverse problems; these problems arise in applications where inaccessible quantities of interest are determined from indirect measurements. Examples include tomographic imaging, optical characterization of nanostructured materials, and image deblurring. Two types of inverse problems are considered, namely the so‐called inverse source and inverse medium problems. The Inverse source and inverse medium problems are already used in fields such as seismology, medical imaging, acoustic design, nondestructive testing of materials, and ocean bottom exploration. However, most often conventional models concerning inverse problems disregards or pay little attention to uncertainties or naturally occurring random fluctuations. This is where we aim, through the project, to expand by allowing for said uncertainties and let either the medium of wave propagation or sources behave randomly by modelling these quantities as random fields. Furthermore, in addition to allowing for randomness, only passive measurements are assumed at hand. That is, measurements obtained from a sensor array where the sensors lack the capability of emitting an impulse and rely on signals emitted by ambient noise sources. The combination mentioned above of passive measurements and inverse problems are known as passive imaging. These extensions and modifications applied to the inverse source and medium problems add the necessity for new mathematical results and numerical algorithms to solve these problems. Therefore the main research topics of the project will be:

• uniqueness results and stability estimates for solutions of inverse source problems and passive medium imaging problems with random background media
• convergence estimates for relevant numerical inverse solvers
• uncertainty quantification of the effects of noise in measurements and elsewhere
• incomplete time‐domain measurements, and windowed frequency measurements, implementation and testing of suitable numerical inverse solvers, including the use of real‐world measurement data.