Layer Potential Techniques

The anomaly detection algorithms described in this book rely on asymptotic expansions of the fields when the medium contains anomalies of small volume. Such asymptotics will be investigated in the case of the conduction equation, the Helmholtz equation, the operator of elasticity, and the Stokes system. As it will be shown in the subsequent chapters, a remarkable feature of these imaging techniques, is that they allow a stable and accurate reconstruction of the location and of the geometric features of the anomalies, even for moderately noisy data.

We prepare the way in this chapter by reviewing a number of basic facts on the layer potentials for these equations which are very useful for anomaly detection. The most important results in this chapter are what we call decomposition theorems for transmission problems. For such problems, we prove that the solution is the sum of two functions, one solving the homogeneous problem, the other inheriting geometric properties of the anomaly. These results have many applications. They have been used to prove global uniqueness results for anomaly detection problems [75, 77]. In this book, we will use them to provide asymptotic expansions of the solution perturbations due to presence of small volume anomalies.