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In this paper, structure-preserving time-integrators for rigid body-type mechanical systems are derived from a discrete Hamilton–Pontryagin variational principle. From this principle, one can derive a novel class of variational partitioned Runge– Kutta methods on Lie groups. Included among these integrators are generalizations of symplectic Euler and(More)
This paper presents a Lie–Trotter splitting for inertial Langevin equations (geometric Langevin algorithm) and analyzes its long-time statistical properties. The splitting is defined as a composition of a variational integrator with an Ornstein–Uhlenbeck flow. Assuming that the exact solution and the splitting are geometrically ergodic, the paper proves the(More)
Metropolized integrators for ergodic stochastic differential equations (SDEs) are proposed that (1) are ergodic with respect to the (known) equilibrium distribution of the SDEs and (2) approximate pathwise the solutions of the SDEs on finite-time intervals. Both these properties are demonstrated in the paper, and precise strong error estimates are obtained.(More)
Keywords: Molecular dynamics Metropolis–Hastings Verlet RATTLE RESPA a b s t r a c t This paper extends the results in [8] to stochastic differential equations (SDEs) arising in molecular dynamics. It implements a patch to explicit integrators that consists of a Metropolis–Hastings step. The 'patched integrator' preserves the SDE's equilibrium distribution(More)
This paper presents a Lie-Trotter splitting for inertial Langevin equations (Geometric Langevin Algorithm) and analyzes its long-time statistical properties.The splitting is defined as a composition of a vari-ational integrator with an Ornstein-Uhlenbeck flow. Assuming the exact solution and the splitting are geometrically ergodic, the paper proves the(More)
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This note is to point out an error in the theory part of the publication [1]. We will follow the notations and definitions of [1] unless stated otherwise. Contrary to what is claimed in Section 2.2 of [1], the modified Metropolis-Hastings acceptance criterion (eqn. (6) in [1]) does not satisfy a modified detailed balance condition for the choice of a linear(More)
By treating tippe top inversion as a dissipation-induced instability, we explain tippe top inversion through a system we call the modified Maxwell–Bloch equations. We revisit previous work done on this problem and follow Or's mathematical model [SIAM J. A linear analysis of the equations of motion reveals that only the equilibrium points correspond to the(More)