Mathematics Colloquia and Seminars

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Recently, the level set method for describing propagating fronts has become quite popular in the application of medical or geophysical tomography. The goal in these applications is to reconstruct unknown objects inside a given domain from a finite set of boundary data. Mathematically, these problems define so-called nonlinear inverse problems, where usually iterative solution strategies are required. Starting from some initial guess for the unknown obstacles, successive corrections to this initial shape are calculated such that the so evolving shapes eventually converge to a shape which satifies the collected data. Since the hidden objects can have a complicated topological structure which is not known a priori, the shapes usually undergo several topology changes during this evolution before converging to the final solution. Therefore, a powerful and flexible tool for the numerical description of these propagating shapes is essential for the success of the inversion method of choice. In the talk, we present a recently developed two-step shape reconstruction method which uses a level set representation of the shapes for this purpose. Numerical results will be presented for three different practically relevant examples: cross-borehole electromagnetic tomography using a 2D Helmholtz model, surface to borehole 3D electromagnetic induction tomography (EMIT) using a model based on the full 3D system of Maxwell's equations, and diffuse optical tomography (DOT) for medical imaging using a model based on the radiative transfer equation in 2D.