2019/12/05 by Garbuno-Inigo, Alfredo, Nüsken, Nikolas, Reich, Sebastian · 3 citations
#62F15 #65C30 #65N21 #65N75 #90C56 #Dynamical Systems (math.DS) #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.1912.02859
We propose a computational method (with acronym ALDI) for sampling from a given target distribution based on first-order (overdamped) Langevin dynamics which satisfies the property of affine invariance. The central idea of ALDI is to run an ensemble of particles with their empirical covariance serving as a preconditioner for their underlying Langevin dynamics. ALDI does not require taking the inverse or square root of the empirical covariance matrix, which enables application to high-dimensional sampling problems. The theoretical properties of ALDI are studied in terms of non-degeneracy and ergodicity. Furthermore, we study its connections to diffusion on Riemannian manifolds and Wasserstein gradient flows. Bayesian inference serves as a main application area for ALDI. In case of a forward problem with additive Gaussian measurement errors, ALDI allows for a gradient-free approximation in the spirit of the ensemble Kalman filter. A computational comparison between gradient-free and gradient-based ALDI is provided for a PDE constrained Bayesian inverse problem.