Inverse problems across scales

Abstract

Predicting the shape of the shadow generated by a 3D object is a straightforward task. However, it can be challenging to solve the inverse problem of finding the object’s shape solely from its projected shadow. Analogous situations appear in many areas of Physics, either to detect gas reservoirs or to design a new quantum material. In this regard, inverse problems typically involve extracting physical properties from a noisy and limited information scenario. The typical length scale and the interactions between physical parameters introduce distinct challenges for each inverse problem. This motivated us to cross scales in this thesis by analyzing four problems and their inverse counterparts. Guided by a cross-fertilization scheme and following a multiscale approach, we started studying what mass density a hanging cable must possess to assume a specific shape. Then, we solve the Bayesian full waveform inversion problem for single and multiple timedependent seismic surveys. To address this inverse problem, we employed the Hamiltonian Monte Carlo method with a proposed mass matrix design. After that, we migrate to the forward problem of determining self-assembled magnetic particles’ ordered structures and their collapse conditions. Finally, we investigate the quantum inverse problem of finding the spatial position of randomly distributed impurities in a one-dimensional electronic device. These problems provide valuable insights into solving inverse problems and have applications in various fields of Physics.