arXiv subjects

Nikolas Nüsken

Publications and source records attributed to Nikolas Nüsken.

21 records · Page 2Linked to original sources

arXiv2020

Affine invariant interacting Langevin dynamics for Bayesian inference

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.

math.NA↗

arXiv2019

Note on Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler by Garbuno-Inigo, Hoffmann, Li and Stuart

An interacting system of Langevin dynamics driven particles has been proposed for sampling from a given posterior density by Garbuno-Inigo, Hoffmann, Li and Stuart in Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler (arXiv:1903:08866v2). The proposed formulation is primarily studied from a formal mean-field limit perspective, while the theoretical behaviour under a finite particle size is left as an open problem. In this note we demonstrate that the particle-based covariance interaction term requires a non-trivial correction. We also show that the corrected dynamics samples exactly from the desired posterior provided that the empirical covariance matrix of the particle system remains non-singular and the posterior log-density satisfies the standard Bakry-Emery criterion.

math.DS↗

arXiv2019

State and Parameter Estimation from Observed Signal Increments

The success of the ensemble Kalman filter has triggered a strong interest in expanding its scope beyond classical state estimation problems. In this paper, we focus on continuous-time data assimilation where the model and measurement errors are correlated and both states and parameters need to be identified. Such scenarios arise from noisy and partial observations of Lagrangian particles which move under a stochastic velocity field involving unknown parameters. We take an appropriate class of McKean-Vlasov equations as the starting point to derive ensemble Kalman-Bucy filter algorithms for combined state and parameter estimation. We demonstrate their performance through a series of increasingly complex multi-scale model systems.

math.NA↗