High-order sampling of the invariant distribution of ergodic stochastic dynamics: preconditioning and postprocessing

Gilles Vilmart (University of Geneva)

Wednesday: May 27th, 2026, 3 pm - TBC

Abstract: Preconditioning techniques for ergodic stochastic dynamics are attractive to enhance the convergence to the invariant measure, especially in high dimension (or infinite dimension for stochastic PDEs), to tackle the issues of stiffness and multi-modal target distributions. Achieving high order of accuracy is however challenging in the context. In this talk, we present an integrator for overdamped Langevin dynamics with position dependent diffusion tensor, with a postprocessor that is second order accurate for sampling the invariant measure, while requiring only one force evaluation per timestep. Analysis of the sampling bias is performed using the algebraic framework of exotic aromatic Butcher-series. We show that such high-order of accuracy can also be achieved for stochastic PDEs. We obtain to the best of our knowledge the first integrators with rigorous analysis that achieves high-orders of accuracy for sampling the invariant distribution of a class of ergodic parabolic semilinear SPDEs with gradient structure.

References:
-C.-E. Bréhier, A. Busnot Laurent, A. Debussche, G. Vilmart, Preconditioning for the high-order sampling of the invariant distribution of parabolic semilinear SPDEs,
Submitted (2025)
-E. Bronasco, B. Leimkuhler, D. Phillips, and G. Vilmart, Efficient Langevin sampling with position-dependent diffusion,
Submitted (2025)

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