UlamMethod.jl

This package is an implementation of Ulam's method ^{1 2} (see also Galerkin projection ³) for the discretization of a stochastic operator using pure Julia. Given a set of one-step trajectories

defined in a subset of $\mathbb{R}^N$, the essential goal of Ulam's method is to partition the domain into a series of non-intersecting regions and construct a transition probability matrix $P$ on these regions.

• Built on the Meshes.jl computational geometry framework.
• Supports trajectory data in arbitrary dimensions.
• Supports automatic boundary construction.
• Multiple 2D algorithms for partitioning to triangles, rectangles, hexagons and adaptively sized Voronoi cells.
• Multiple stochasticization algorithms.
• One-line plotting conveniences in 2D for easy visualization.

Documentation

This package is in the Julia General Registry. In the Julia REPL, run the following code and follow the prompts:

import Pkg
Pkg.add("UlamMethod")

Access the functionality of the package in your code by including the following line:

using UlamMethod

The highest level function is

ulam_method(traj, nbins; nirvana)

where

• traj: A Trajectories object, holding the short-range trajectory data.
• nbins: The number of bins requested. When using this high-level convenience function, these will be uniform segements (1D), rectangles (2D) and hypercubes (≥3D).
• nirvana: The boundary of the bins will be set such that the fraction of data in nirvana⁴ is equal to this value. In other words, a fraction nirvana of the data will be outside of the binned region.

In general, your trajectory data can be loaded simply as traj = Trajectories(x0, xT), where x0 and xT are Dim x N matrices such that Dim is the dimension of the data and N is the number of points. For testing, the function Trajectories(Dim, N) is provided that generates random trajectories automatically. We will generate 1000 trajectories in 2D and apply Ulam's method with 200 bins.

using UlamMethod
import Random; Random.seed!(1234) # reproducible randomness

traj = Trajectories(2, 1000)
ulam = ulam_method(traj, 200)

After computing ulam, use the following functions:

• P_closed(ulam) gives the full transition probability matrix.
• bins(ulam) gives the bins and points(bins(ulam)) gives their vertices.
• membership(points, ulam) returns the bin membership of a Dim x n_points matrix points.

In order to use the plotting functionality, a Makie backend is required. For this example, this amounts to including the line import CairoMakie. This provides access to the function viz(object; kwargs...) which can visualize any UlamMethod object.

import CairoMakie
viz(traj)

viz(ulam)

Note that bins colored black are bins that contained data but were not a member of the largest strongly connected component.

Please use the following citation if you use this package in your research.

@article{bonner2023improving,
  title={Improving the stability of temporal statistics in transition path theory with sparse data},
  author={Bonner, Gage and Beron-Vera, FJ and Olascoaga, MJ},
  journal={Chaos: An Interdisciplinary Journal of Nonlinear Science},
  volume={33},
  number={6},
  year={2023},
  publisher={AIP Publishing}
}

Initial development of this package was supported by the National Science Foundation.