UlamMethod.jl

Introduction

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 two-dimensional, one-step trajectories

defined in a domain, 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. In UlamMethod.jl, this is accomplished in two main steps

1. The user provides a polygon containing the data, and covering of the the domain is generated by polygons according to one of several different [binning algorithms](@ref binning).
2. The number of trajectories beginning in polygon $i$ and ending in polygon $j$ is used to create the entry $P_{i, j}$ of $P$ such that the edges of the domain are handled by a [stochasticization algorithm](@ref stoc).

The polygons which form the covering and the transition probability matrix are the main outputs.

Installation

In the Julia REPL, run the following code and follow the prompts:

using Pkg
Pkg.add("UlamMethod")

Make the functions in this package available to use in your code by including the following line:

using UlamMethod

Quickstart

The core functionality is provided by

ulam_method(traj::UlamTrajectories, domain::UlamDomain)

where traj contains information about trajectories and domain contains information about the domain and covering. Here are 10000 random trajectories in the domain $[0, 10]^2$

using UlamMethod

n_data = 10000
x0, y0, xT, yT = 10*rand(n_data), 10*rand(n_data), 10*rand(n_data), 10*rand(n_data)
traj = UlamTrajectories(x0 = x0, y0 = y0, xT = xT, yT = yT)

We will take our domain to be the rectangular subset $[3, 5] \times [4, 8]$ and generate a covering with 40 rectangles.

xmin, xmax, ymin, ymax = 3, 5, 4, 8
domain = UlamDomain(xmin, xmax, ymin, ymax, poly_type = "rec", poly_number = 40)

Run Ulam's method.

ulam = ulam_method(traj, domain)    # the main calculation

ulam.P_closed                       # the transition matrix
pt = PolyTable(ulam.polys)          # PolyTable makes a simple list of nodes and edges
pt.nodes                            # |x|y| table of polygon nodes
pt.edges[:,3]                       # the index of the polygon that the i'th node belongs to

References