Data-free likelihood-informed dimension reduction of Bayesian inverse problems

Identifying a low-dimensional informed parameter subspace offers a viable path to alleviating the dimensionality challenge in the sampled-based solution to large-scale Bayesian inverse problems. This paper introduces a novel gradientbased dimension reduction method in which the informed subspace does not depend on the data. This permits online-offline computational strategy where the expensive low-dimensional structure of the problem is detected in an offline phase, meaning before observing the data. This strategy is particularly relevant for multiple inversion problems as the same informed subspace can be reused. The proposed approach allows to control the approximation error (in expectation over the data) of the posterior distribution. We also present sampling strategies which exploit the informed subspace to draw efficiently samples from the exact posterior distribution. The method is successfully illustrated on two numerical examples: a PDE-based inverse problem with a Gaussian process prior and a tomography problem with Poisson data and a Besov-B-11(2) prior.

Automated summary

Identifying a low-dimensional informed parameter subspace is a viable way to address the dimensionality challenge in sampled-based solutions to large-scale Bayesian inverse problems. The introduced gradient-based method allows for offline detection of the expensive low-dimensional structure before observing the data, enabling efficient control over the approximation error of the posterior distribution. Sampling strategies are presented to draw samples from the exact posterior distribution using the informed subspace.