A variable-separation ensemble transport method for Bayesian inference of parametric PDEs

This paper presents a variable-separation ensemble transport method (VSET) for inferring parameters in large-scale parametric partial differential equations (PDEs). Traditional Bayesian or ensemble-based approaches are often computationally prohibitive, as accurate inference typically requires many expensive partial differential equation (PDE) solves. To this end, we develop a coupled input--output reduction framework that constructs a low-rank approximation of the parameter-to-observable (PtO) map. The proposed method employs a shared stochastic basis, derived from the original system, to simultaneously span reduced subspaces for both the parameter field and the corresponding observables. The {variable-separation (VS)} expansion is precomputed offline using prior information, providing a finite-dimensional representation of the unknown parameters. Within a Kalman filtering framework, the forecast step employs the reduced VS model to approximate the forward PDE solution, while the analysis step updates the coefficient ensemble by assimilating observational data. Furthermore, to enhance the reusability and generalization of the stochastic basis functions, the VS representation, constructed from a single steady-state model, is subsequently applied to a family of steady-state and time-dependent models with varying boundary conditions, source terms, or diffusion coefficients. We also establish a Hellinger error estimate for the VS-truncated posterior measure. Finally, numerical experiments demonstrate that the proposed VSET method achieves a reliable and computationally tractable approximation of the estimated parameters for complex parametric PDEs.