An information field theory approach to engineering inverse problems

Inverse problems in infinite dimensions are ubiquitously encountered across the scien- tific disciplines. These problems are defined by the need to reconstruct continuous fields from incomplete, noisy measurements, which oftentimes leads to ill-posed problems. Almost universally, the solutions to these problems are constructed in a Bayesian framework. How- ever, in the infinite-dimensional setting, the theory is largely restricted to the Gaussian case, and the treatment of prior physical knowledge is lacking. We develop a new framework for Bayesian reconstruction of infinite-dimensional fields which encodes our physical knowledge directly into the prior, while remaining in the continuous setting. We then prove various characteristics of the method, including situations in which the problems we study have unique solutions under our framework. Finally, we develop numerical sampling schemes to characterize the various objects involved.