Bayesian inverse problems in all-at-onc...

Inverse problems find application in wide range of real-world problems reaching from medical imaging to geophysics. Thus, they form an important field inside applied mathematics.

However, often these problems are characterized through ill-posedness, which means the solution is not straight forward. Regularization techniques have to be devised and implemented to stabilize the inversion process or even facilitate it. In the broad spectrum of regularization techniques available, there is a distinction between deterministic and stochastic types. This thesis focuses on particular stochastic method of regularization, namely the Bayesian regularization. This tool has become quite popular as it allows for the specification of a priori information through probability distributions. It makes use of Bayes’ theorem to combine knowledge from prior distribution with the measured data.

In this thesis, not only the Bayesian setting for inverse problems is discussed, in addition, it is combined with the all-at-once approach to formulate inverse problems. This new type of combination is promising as it allows for the additional usage of a prior distribution for the state variable, not only for the parameter. The approach is analysed with the help of two prototypical examples, namely an inverse source problem for the Poisson equation and the backward heat equation, i.e., a stationary and a time-dependent problem. Appropriate function spaces, adjoint operators, and eigenvalues are investigated, joint priors are designed, and numerical reconstructions are accomplished.

Further, a new time-stepping scheme for the reconstruction of inverse problems involving fractionally damped wave equations is presented. These inverse problems are of high relevance in photoacoustic or thermoacoustic tomography. In this thesis, the inverse problem of identifying the initial data in a fractionally damped wave equation from time trace measurements on a surface is considered. A novel time-stepping method based on the Newmark scheme is derived. The time-stepping method with two different discretization schemes for the fractional derivative is evaluated numerically, and an error analysis is performed. In addition, the adjoint problem is established to derive an adjoint-based method for the gradient computation to obtain reconstructions efficiently. Finally, this gradient computation is used to perform numerical reconstructions in two space dimensions.