Title: Ensemble sampler for infinite-dimensional inverse problems
Authors: Jeremie Coullon - Lancaster University (United Kingdom) [presenting]
Robert Webber - New York University (United States)
Abstract: A Markov chain Monte Carlo (MCMC) sampler for infinite-dimensional inverse problems is introduced. The proposed MCMC sampler is more efficient than preconditioned Crank-Nicolson, yet it is easy to implement and it requires no gradient information or posterior covariance information. The new sampler involves truncating the Karhunen-Loeve expansion of the function to decompose the space into a low-dimensional subspace that is mainly influenced by the likelihood, and the complementary subspace which approximately follows the prior distribution. The affine invariant ensemble sampler is used for proposals in the low-dimensional space while pCN is used for the complementary subspace. As a result, the new sampler is able to handle strongly correlated posteriors, is self-tuning, and is easy to parallelise. We use two numerical experiments to compare the performance of the new sampler to pCN and an existing gradient-free sampler in the literature. The first example involves inferring the wave speed and initial condition of the advection equation, and the second example involves parameter estimation for Langevin dynamics where we infer the two scalar parameters along with the posterior position path $X_t$. We find that the new sampler is more robust and outperforms both methods. We conclude with a discussion of the limitations of the method and avenues for future research.