@article{MaoutsaReichOpper2020,
  author    = {Maoutsa, Dimitra and Reich, Sebastian and Opper, Manfred},
  title     = {Interacting particle solutions of Fokker-Planck equations through gradient-log-density estimation},
  series = {Entropy},
  volume    = {22},
  journal   = {Entropy},
  number    = {8},
  publisher = {MDPI},
  address   = {Basel},
  issn      = {1099-4300},
  doi       = {10.3390/e22080802},
  pages     = {35},
  year      = {2020},
  abstract  = {Fokker-Planck equations are extensively employed in various scientific fields as they characterise the behaviour of stochastic systems at the level of probability density functions. Although broadly used, they allow for analytical treatment only in limited settings, and often it is inevitable to resort to numerical solutions. Here, we develop a computational approach for simulating the time evolution of Fokker-Planck solutions in terms of a mean field limit of an interacting particle system. The interactions between particles are determined by the gradient of the logarithm of the particle density, approximated here by a novel statistical estimator. The performance of our method shows promising results, with more accurate and less fluctuating statistics compared to direct stochastic simulations of comparable particle number. Taken together, our framework allows for effortless and reliable particle-based simulations of Fokker-Planck equations in low and moderate dimensions. The proposed gradient-log-density estimator is also of independent interest, for example, in the context of optimal control.},
  language  = {en}
}
@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}
}