Adaptive thermostats for noisy gradient systems

Benedict Leimkuhler, Xiaocheng Shang

Research output: Contribution to journalArticlepeer-review

22 Citations (Scopus)
150 Downloads (Pure)


We study numerical methods for sampling probability measures in high dimension where the underlying model is only approximately identified with a gradient system. Extended stochastic dynamical methods are discussed which have application to multiscale models, nonequilibrium molecular dynamics, and Bayesian sampling techniques arising in emerging machine learning applications. In addition to providing a more comprehensive discussion of the foundations of these methods, we propose a new numerical method for the adaptive Langevin/stochastic gradient Nosé--Hoover thermostat that achieves a dramatic improvement in numerical efficiency over the most popular stochastic gradient methods reported in the literature. We also demonstrate that the newly established method inherits a superconvergence property (fourth order convergence to the invariant measure for configurational quantities) recently demonstrated in the setting of Langevin dynamics. Our findings are verified by numerical experiments.
Original languageEnglish
Pages (from-to)A712-A736
Number of pages25
JournalSIAM Journal on Scientific Computing
Issue number2
Publication statusPublished - 1 Mar 2016


  • stochastic differential equations
  • adaptive thermostat
  • Bayesian sampling
  • machine learning
  • invariant measure
  • ergodicity


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