Deterministic and stochastic particle methods for the Fokker-Planck equation in S--formulation with application to point set registration
Abstract
We present two particle methods for point set registration in bounded domains based on the Fokker-Planck equation. The first method relies on a moving least squares discretization of the S--formulation, in which moving grid points (particles) are advected by the drift with the target distribution, diffusion is resolved on a dynamically evolving particle cloud. This setting naturally leads to strong compression and expansion of the particle cloud. Obstacles are handled by enforcing reflective boundary conditions through a novel ghost point method. The second method is a Monte Carlo solver for the associated Langevin stochastic differential equation. It relies on a local approximation of the logarithmic gradient of the evolving particle density to extract macroscopic osmotic paths from individual stochastic trajectories. Owing to its inherent parallelism, this method exhibits excellent scalability and is well suited for high-dimensional registration problems. We illustrate the main features and performance of both approaches through extensive numerical experiments.
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BibTeX
@misc{willems2026deterministic,
title = {{Deterministic and stochastic particle methods for the Fokker-Planck equation in S--formulation with application to point set registration}},
author = {Klaas Willems and Angelo Iollo and Giovanni Russo and Tommaso Taddei},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01700},
eprint = {2610.01700},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01700},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Deterministic and stochastic particle methods for the Fokker-Planck equation in S--formulation with application to point set registration
AU - Klaas Willems
AU - Angelo Iollo
AU - Giovanni Russo
AU - Tommaso Taddei
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01700
UR - https://arxiv.org/abs/2610.01700
AB - We present two particle methods for point set registration in bounded domains based on the Fokker-Planck equation. The first method relies on a moving least squares discretization of the S--formulation, in which moving grid points (particles) are advected by the drift with the target distribution, diffusion is resolved on a dynamically evolving particle cloud. This setting naturally leads to strong compression and expansion of the particle cloud. Obstacles are handled by enforcing reflective boundary conditions through a novel ghost point method. The second method is a Monte Carlo solver for the associated Langevin stochastic differential equation. It relies on a local approximation of the logarithmic gradient of the evolving particle density to extract macroscopic osmotic paths from individual stochastic trajectories. Owing to its inherent parallelism, this method exhibits excellent scalability and is well suited for high-dimensional registration problems. We illustrate the main features and performance of both approaches through extensive numerical experiments.
ER -