Shape optimization and level set method applied to inversion problems - Albuquerque, Yuri Flores (2021)
This thesis is about efficient shape optimization methods for the reconstruction of sharp interfaces in the problems of electrical impedance tomography (EIT) and geophysical exploration by full waveform inversion (FWI). These two problems differ experimentally. In summary, electrical impedance tomography is a method of imaging the internal structure of an unknown physical body by applying electrical currents near the surface of the body. Seismic tomography, on the other hand, is a technique to determine the material structure of the earths subsoil based on recorded seismic signals produced by waves excited by some source and propagated through the ground. The objective here is to contribute with theoretical and practical aspects of shape optimization, electrical impedance tomography, and full waveform inversion. In particular, we study the averaged adjoint method, which can handle additional constraints, such as admitting discontinuities in material parameters, depending on the specific configurations of the function spaces. For that, results about the differentiability of the operator with respect to the shape are established, and a shape optimization algorithm that uses the average adjoint method is proposed. Furthermore, we use some ideas from classical optimization theory and employ a linear search procedure to improve the reconstruction of shape interfaces. Additionally, we use our version of the level set method to efficiently calculate evolving interfaces. Numerical results are presented for the inverse problems in electrical impedance tomography for point measurements in discontinuous boundary, and in geophysical exploration on reservoir scale in the solid domain, containing salt bodies for reconstruction of sharp interfaces in time-domain full waveform inversion. As part of this thesis, two implementations of shape optimization that depend on the calculation of the shape derivative of the cost function, used in the level set method, were developed.