Solving highly underdetermined inverse problems by an inexact penalty method and learned distances with an application to seismic imaging

Inverse problems are often solved by minimizing problem-specific distance functionals. In many applications, such as full waveform inversion (FWI), the leading technology in seismic imaging, standard functionals are non-convex and thus hard to optimize. Consequently, many convex replacements have been proposed, including learned distance functionals. However, even if one can find a minimizer of such a functional, it is likely not unique, particularly when the inverse problem is underdetermined, as is often the case in practical settings. To overcome this challenge, we propose and theoretically backup an inexact penalty method, that is computationally realizable and finds a solution to the inverse problem with physically preferred properties, that are described by a given regularization term. To validate this method, we apply it to FWI in the acoustic regime using the Marmousi model, a proven geophysical benchmark problem. We observe a significant enhancement in the reconstruction quality compared to standard methods.