@article{GarbunoInigoNueskenReich2020,
  author    = {Garbuno-Inigo, Alfredo and N{\"u}sken, Nikolas and Reich, Sebastian},
  title     = {Affine invariant interacting Langevin dynamics for Bayesian inference},
  series = {SIAM journal on applied dynamical systems},
  volume    = {19},
  journal   = {SIAM journal on applied dynamical systems},
  number    = {3},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {1536-0040},
  doi       = {10.1137/19M1304891},
  pages     = {1633 -- 1658},
  year      = {2020},
  abstract  = {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 nondegeneracy 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.},
  language  = {en}
}
@article{NueskenPavhotis2019,
  author    = {N{\"u}sken, Nikolas and Pavhotis, Grigorios A.},
  title     = {Constructing Sampling Schemes via Coupling},
  series = {SIAM ASA journal on uncertainty quantification / Society for Industrial and Applied Mathematics ; American Statistical Association},
  volume    = {7},
  journal   = {SIAM ASA journal on uncertainty quantification / Society for Industrial and Applied Mathematics ; American Statistical Association},
  number    = {1},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {2166-2525},
  doi       = {10.1137/18M119896X},
  pages     = {324 -- 382},
  year      = {2019},
  abstract  = {In this paper we develop a general framework for constructing and analyzing coupled Markov chain Monte Carlo samplers, allowing for both (possibly degenerate) diffusion and piecewise deterministic Markov processes. For many performance criteria of interest, including the asymptotic variance, the task of finding efficient couplings can be phrased in terms of problems related to optimal transport theory. We investigate general structural properties, proving a singularity theorem that has both geometric and probabilistic interpretations. Moreover, we show that those problems can often be solved approximately and support our findings with numerical experiments. For the particular objective of estimating the variance of a Bayesian posterior, our analysis suggests using novel techniques in the spirit of antithetic variates. Addressing the convergence to equilibrium of coupled processes we furthermore derive a modified Poincare inequality.},
  language  = {en}
}
@article{NueskenReichRozdeba2019,
  author    = {N{\"u}sken, Nikolas and Reich, Sebastian and Rozdeba, Paul J.},
  title     = {State and parameter estimation from observed signal increments},
  series = {Entropy : an international and interdisciplinary journal of entropy and information studies},
  volume    = {21},
  journal   = {Entropy : an international and interdisciplinary journal of entropy and information studies},
  number    = {5},
  publisher = {MDPI},
  address   = {Basel},
  issn      = {1099-4300},
  doi       = {10.3390/e21050505},
  pages     = {23},
  year      = {2019},
  abstract  = {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.},
  language  = {en}
}