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## Section: New Results

### Stable and accurate schemes for Langevin dynamics with general kinetic energies

Participants : Zofia Trstanova, Gabriel Stoltz.

We studied integration schemes for Langevin dynamics with a kinetic energy different from the standard, quadratic one in order to accelerate the sampling of the Boltzmann–Gibbs distribution. We considered two cases: kinetic energies which are local perturbations of the standard kinetic energy around the origin, where they vanish (this corresponds to the so-called adaptively restrained Langevin dynamics); and more general non-globally Lipschitz energies. We developed numerical schemes which are stable and of weak order two, by considering splitting strategies where the discretizations of the fluctuation/dissipation are corrected by a Metropolis procedure. We used the newly developed schemes for two applications: optimizing the shape of the kinetic energy for the adaptively restrained Langevin dynamics, and reducing the metastability of some toy models with non-globally Lipschitz kinetic energies.

Figure 19. Comparison of the mean exit times for 2D double-well potential with the standard and the modified kinetic energy function (2000 realizations) as a function of the inverse temperature $\beta \in \left\{3,4,5,6,7,8,9,10\right\}$. Thanks to the modified kinetic energy, the transition between two metastable states occurs on average three times faster.