Uncertainty quantification for data-limited inverse problems via efficient probabilistic ML methods

Detecting (stochastic) inclusions via measurements at a boundary of the space-time cylinder in a setting modeled by a partial differential equation has various applications in the sciences. This project seeks the detection of stochastic (discontinuous) inclusions in a Bayesian inversion setting. The low regularity of the stochastic random field describing the inclusions limits the use of standard approaches. After providing a general formulation for a variable stochastic setting, adaptive numerical approaches are developed. Detecting\reconstructing the inclusion may be tackled via a point estimator like a maximum a-posteriori estimate. A machine-learning-based method is employed to lower the computational complexity the high-dimensionality that the problem inherently brings.