Many parameters of physical interest cannot be studied directly. Examples include: imaging the interior of the body or locating buried objects; determining the location and size of cracks within solid objects; reconstructing material parameters such as the conductivity of interior regions. When these problems are translated into mathematical terms they take the form of partial differential equations. However, since we have additional unknowns in the model, these introduce unknown parameters in the equations that we must additionally resolve by means of further measurements. A central theme of the proposal is the question of when a unique determination can be made (what is the minimal amount of data needed) as well as the design of algorithms for the efficient numerical recovery of the unknowns. This proposal considers the practical and computational aspects of this from a mathematical perspective. Specific problems addressed include the recovery of the location, shape, and material properties of interior objects from surface measurements. In such inverse problems two things must always be understood. First, the reconstructions will be extremely sensitive to small changes in the data, this is inherent in the underlying physics; in mathematical terms these are highly ill-conditioned problems containing both analytical and computational complexity. Second, the available data is always subject to error. However, we may know a model for the data error such as, for example, its mean and variance. This proposal seeks a formulation that will allow us to provide similar information on the geometry of the obstacle - namely a quantitative assessment of the ranges of reconstructions one could expect with a given level of data error. This would allow us to assign a probability that a particular feature would be identifiable or that, say, the volume of the object is greater than a given value.
The proposal has a range of broader impacts.These include not only the breadth of applications to science and engineering covered by these inverse problems, but there is an important training aspect involved. Specifically, many of the problems have simplified versions where both the experimental apparatus needed as well as some of the corresponding reconstruction algorithms are within reach of advanced undergraduates. This will enable a wider audience to gain an understanding of both the challenges and possible solutions to these ubiquitous but complex problems.
