Symmetric Langevin methods

Of particular interest for sampling the canonical distribution are symmetric Langevin methods. We believe these are likely to be the most useful class of methods for practitioners, as by symmetrizing the expansion the odd order terms in (7.16) vanish identically using the Jacobi identity in the BCH expansions. This implies that a symmetric scheme gives a second order error in computed averages. Many symmetric methods can be constructed that require only one evaluation of the force per iteration (effectively making them as inexpensive as a first order method). 

Building on these schemes, the OU solves can instead be placed in the center of the step, or we may choose to have more than one OU solve distributed through the iteration. As it happens, some of these methods are of particular interest. We refer to the method obtained by inserting the OU solve in the middle of the Verlet method either by the strings ABOBA (Position-Verlet version) and BAOAB] (Velocity-Verlet version) or by the more pronounceable names Symmetric Langevin Position-Verlet or Symmetric Langevin Velocity-Verlet, respectively. 

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