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Reduced-order model tracking and interpolation to solve PDE-based Bayesian inverse problems

This work presents a computationally efficient probabilistic framework that enables the identification of model parameters from noisy measurements of the response. We consider transient PDE-based models, where the parameters correspond to physical properties. An efficient and reliable procedure for estimation of those unknown parameters is pursued. The proposed framework uses a Bayesian approach, an efficient sequential Monte Carlo sampling scheme, and adaptive reduced-order models (ROMs). The Bayesian approach has several advantages including the ability to provide not only point estimates of the quantities of interest, but also measures of credibility and robustness concerning those estimates. The associated sequential Monte Carlo method sampling scheme is embarrassingly parallelizable, as well as efficient in terms of the number of calls to the forward solver (e.g. a finite element code) used in evaluating the likelihood function. We propose to use a ROM adaptation procedure where projection-based ROMs are seen as points on a certain Riemannian manifold and are ‘tracked’ and interpolated during the sampling process using a database of precomputed ROMs. This approach ensures that an appropriate ROM is used for the likelihood evaluation in every region of the parameter space, thus leading to computational savings while conserving sufficient accuracy. Using numerical examples, we illustrate the capabilities of the proposed framework, and show that it leads to quality estimates with a quantified predictive uncertainty.