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Correction to: Hybrid Monte Carlo methods for sampling probability measures on submanifolds

  • Tony Lelièvre
  • Mathias Rousset
  • Gabriel StoltzEmail author
Correction
  • 55 Downloads

1 Correction to: Numerische Mathematik (2019) 143:379–421  https://doi.org/10.1007/s00211-019-01056-4

Shiva Darshan and Miranda Holmes–Cerfon (Courant Institute, NYU) pointed out a mistake in the projection functions to enforce the momentum constraint when rewriting the algorithm in Numerical Algorithm A of Section 3.1. Two different projection functions are actually needed, see indeed the formula for the Lagrange multiplier \(\lambda ^{n+1}\) after Equation (7) for the RATTLE step, and Remark 5 for the Ornstein–Uhlenbeck step.

We provide below a corrected version of the pseudo-code for the complete algorithm; see Numerical algorithms 1, 2 and 3. Changes are highlighted in blue. The sampling algorithm consists in iterating procedure ConstrainedGHMC of the algorithm (Numerical algorithm 1), which uses the procedures LAGRANGE_MOMENTUM_OU (Numerical algorithm 2) and LAGRANGE_MOMENTUM_RATTLE (Numerical algorithm 3) to compute the Lagrange multiplier for momentum constraints in the fluctuation/dissipation and RATTLE steps, respectively. The procedure NEWTON to compute the Lagrange multiplier for position constraints is unchanged.

Numerical algorithms 2 and 3 differ by a multiplication by \(\left( \mathrm{Id} + \Delta t \gamma M^{-1}/4\right) ^{-1}\), which arises from the specific choice of the discretization of the fluctuation/dissipation part in Algorithm 3.

Notes

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Tony Lelièvre
    • 1
  • Mathias Rousset
    • 2
  • Gabriel Stoltz
    • 1
    Email author
  1. 1.CERMICS (ENPC), InriaUniversité Paris-EstMarne-la-ValléeFrance
  2. 2.SIMSART Team-ProjectInria RennesFrance

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