Updated scripts

Integrated simplified B matrix

Removed config.py

All scripts updated

6D example script added

Refactored grid container

Refactored code to make initialization faster

Refactored marginalize function in scripts

Fixed bug in 6D example

Updated scripts

Updated README

Assets added

Update README.md

Update README.md

.gitignore

@@ -0,0 +1,5 @@
+*.npy
+sparselib.egg-info/
+build/
+*.png
+debug/

README.md

@@ -1,61 +1,72 @@
-Funcs:
-	- Variables:
-		- dynamics: list(function)
-		- uncertainty: function
-
-Domain:
-	- Variables
-		- max_level: (int)
-		- domain: (Dx2 array)
-		- dim: int
-
-SparseGrid:
-	- Methods:
-		- init:
-			- Inputs: domain, max level
-			- Outputs: N/A
-		- fit:
-			- Inputs: func
-			- Outputs: N/A
-		- eval:
-			- Inputs: coords (NxD array)
-			- Outputs: vals (N array)
-
-GridContainer:
-	- Methods:
-		- fit:
-			- Inputs: Funcs class
-			- Output: N/A
-
-SolverParams:
-	- max_level
-	- dimension
-	- domain
-	- dynamics function(s)
-	- uncertainty function
-	- funcs: [uncertainty, dynamics]
-
-SpectralDiscretization:
-	- Methods:
-		- init:
-			- Inputs: SolverParams
-			- Outputs: N/A
-		- solve:
-			- Inputs: time (float)
-					  coords (NxD array)
-			- Output: vals (NxD array)
-
-
-solverParams --> SpectralDiscretization
-SpectralDiscretization():
-	1) Constructs GridContainer
-	2) Calls gridContainer.fit()
-SpectralDiscretization.solve(t, xs):
-	1) Runs propagation script
-
-
-
-FIXES:
-	- Fix issue with basis_der always multiplying 1j * k[d] for each
-		dim even though der is not over each dim
-	- Similar fix for basis function
+# Uncertainty Propagation Using Half-Densities Via Sparse Spectral Discretization on Compact Manifolds
+Parker Ewen, Lucas Lymburner, Henry Jacobs, Ram Vasudevan<br>
+| [Webpage](https://roahmlab.github.io/sparselib_page/) | [Full Paper](https://repo-sam.inria.fr/fungraph/3d-gaussian-splatting/3d_gaussian_splatting_high.pdf)<br>
+<p align="center">
+  <img src="assets/2D_example.gif" />
+</p>
+
+This repository contains the *sparselib* implementation associated with the paper "Uncertainty Propagation Using Half-Densities Via Sparse Spectral Discretization on Compact Manifolds".
+
+## Setup
+This package can be installed using the pip package installer:
+
+```shell
+git clone https://github.com/ParkerEwen5441/sparselib.git
+cd sparselib
+pip install .
+```
+
+## Overview
+
+The codebase has 4 main components:
+- A grid object used for computing the hyperbolic Fourier coefficients
+- A grid container for storing all grid objects used to approximate the uncertainty and vector fields
+- A spectral scheme for computing solutions to the linear advection equation
+
+Any of these components may be modified to suit the user's particular implementation.
+Algorithms used in the component implementations are constructive so as to alleviate the curse-of-dimensionality.
+
+
+## Running
+
+To run the an example script for the 1D example seen in the paper, simply use
+
+```shell
+cd scripts
+python 1D_example.py
+```
+
+Additionally, other scripts are available for the 2D, 3D, and 6D cases.
+To run the 2D and 3D cases, there is an additional dependency to plot the uncertainty over the torus:
+
+```shell
+pip install s3dlib==1.0.0
+```
+
+## Abstract
+Uncertainty propagation is a critical component in various applications such as stochastic optimal control, optimal transport, probabilistic inference, and filtering.
+This paper frames uncertainty propagation of a dynamical system with compact support thru the lens of advecting a probability density through a vector field via the transport and continuum equations.
+These equations exhibit a number of conservation laws.
+For example, scalar multiplication is conserved by the transport equation, while positivity of probabilities is conserved by the continuum equation.
+Certain discretization techniques, such as particle based methods, conserve these properties, but converge slower than spectral discretization methods on smooth data.
+Standard spectral discretization methods, on the other hand, do not conserve the invariants of the transport equation and the continuum equation.
+This article constructs a novel sparse spectral discretization technique that conserves these important invariants, namely positivity of probability measure, while simultaneously preserving spectral convergence rates. 
+The performance of this proposed method is illustrated on several numerical experiments and the quantitative convergence rate of the proposed approach is derived.
+
+If you use this library, please cite the following work.
+
+<section class="section" id="BibTeX">
+  <div class="container is-max-desktop content">
+    <h2 class="title">BibTeX</h2>
+    <pre><code>@Article{kerbl3Dgaussians,
+      author       = {Ewen, Parker and Lymburner, Lucas and Jacobs, Henry and Vasudevan, Ram},
+      title        = {Uncertainty Propagation Using Half-Densities Via Sparse Spectral Discretization on Compact Manifolds},
+      journal      = {In Submission},
+      number       = {1},
+      volume       = {1},
+      month        = {September},
+      year         = {2024},
+      url          = {https://roahmlab.github.io/sparselib_page/}
+}</code></pre>
+  </div>
+</section>