A variational framework for Bayesian inversion in moving boundary Darcy flow
We develop an infinite-dimensional Bayesian framework for recovering spatially varying log-permeability from pressure observations in single-phase Darcy flow with a moving boundary, motivated by resin transfer moulding. The main contribution is an observation-wise representer formulation linking sensitivities of the parameter-to-observable map to the moving-boundary state and adjoint systems. This yields a reduced Levenberg--Marquardt method for maximum a posteriori (MAP) estimation on the Cameron--Martin space of a Gaussian prior and an explicit finite-rank covariance for the linearised MAP (LMAP) approximation. Once the representers are computed, the LM update and covariance assembly reduce to linear algebra in observation space, independently of the discretisation dimension. In 1D, the explicit moving-boundary solution is used to establish posterior well-posedness and existence of MAP estimators, and to derive closed-form Fréchet derivatives and representers. In 2D, we introduce a very weak mixed formulation coupling pressure, interface velocity and domain evolution. A formal one-sided directional shape linearisation yields the corresponding linearised state and adjoint systems, with observation-wise representers computed through independent, parallel adjoint problems. Numerical experiments show that the 1D LMAP approximation closely matches a reference MCMC posterior while requiring \(\mathcal{O}(10^1)\) rather than \(\mathcal{O}(10^5\text{--}10^6)\) forward simulations. In 2D, the method produces reconstructions and uncertainty estimates comparable to ensemble Kalman inversion with large ensembles, at substantially lower computational cost. The workflow is demonstrated across several geometries and experimental configurations without reformulating the inversion methodology.