An Efficient Bayesian Framework for Uncertainty Quantification in Nonlinear Imaging Inverse Problems
Bayesian methods provide a natural framework for estimating a parameter in non-linear inverse problems and quantifying uncertainty in the estimation. However, when the forward model for such non-linear inverse problems is given by some Partial Differential Equation (PDE), Bayesian inference is typically carried out by resorting to MCMC methods. Since each MCMC iteration requires solving a PDE, these methods become computationally expensive and are often impractical for large-scale imaging problems. In this work, we develop a computationally efficient Bayesian framework for two such nonlinear imaging inverse problems: Quantitative Photoacoustic Tomography (QPAT) and Electrical Impedance Tomography (EIT). Building on a recently proposed two-stage pushforward methodology, we first formulate a Bayesian regression problem for an auxiliary variable whose posterior is available in closed form. This posterior is then pushed forward through a deterministic reconstruction map to obtain a posterior on the unknown parameter, avoiding MCMC sampling. We give a rigorous measure-theoretic justification to interpret the induced posterior as a Bayesian posterior and derive posterior contraction rates for both QPAT and EIT. Numerical results show that the proposed method provides accurate reconstructions and reliable uncertainty estimates at a arguably lower computational cost than standard Bayesian approaches.