Monotone schemes for curvature-driven PDEs

by Jeff Calder (UMN) and Wonjun Lee (UMN)

• Paper: arXiv
• Jeff Calder, School of Mathematics, University of Minnesota: website
• Wonjun Lee, Institute for Mathematics and Its Applications, Uniersity of Minnesota: website

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Introduction

This repository contains C++ and python codes for running the monotone algorithm to solve curvature-driven PDEs. Here are list of PDEs that can be solved using this algorithm. Let $\Omega = [0,1]^d$ be a domain in $\mathbb{R}^d$ and $\partial \Omega$ be a boundary of $\Omega$.

Eikonal equation

$$ |\nabla u(x)| = f(x),\quad x \in \Omega $$

$$ u(x) = 0,\quad x \in \partial \Omega $$

Mean curvature PDE

$$|\nabla u(x)|\kappa(x) = f(x),\quad x \in \Omega $$

$$ u(x) = 0,\quad x \in \partial \Omega $$

where $\kappa(x) = - \text{div}\left( \frac{\nabla u}{|\nabla u|} \right)$ is the mean curvature of the level set surface of $u$ passing through $x$.

Affine flows PDE

$$|\nabla u(x)|\kappa(x)_+^{\alpha} = f(x),\quad x \in \Omega $$

$$u(x) = 0,\quad x \in \partial \Omega $$

where $\alpha \in (0,1]$ is a constant depending on the dimension $d$ and $(t)_+ := \max(0,t)$.

Tukey Depth

$$ |\nabla u(x)| = \int_{(y-x)\cdot \nabla u(x) = 0} \rho(y) dS(y),\quad x \in \Omega $$

$$ u(x) = 0,\quad x \in \partial \Omega $$

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Tutorial

Prerequisites

• pip
• python >= 3.6

Follow this link to see the instruction for the installation of pip: https://pip.pypa.io/en/stable/installation/.

Installing the package

Install the package by running the following command:

    pip install monotonescheme

Running the codes

You can find the example python script files and notebook files in tests folder. The notebook files in the folder solve the following problems:

1. Affine flows in 2D Cartesian grid.

• affine_PDE_2D.ipynb
• affine_PDE_2D.py
• An example solution of an affine flow equation where $f=1$ on two balls domain.
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2. Tukey depth eikonal equation in 2D Cartesian grid.

• tukey_PDE_2D.ipynb
• tukey_PDE_2D.py
• An example solution of a Tukey depth eikonal equation where the data density $\rho$ is a uniform distribution on two balls domain.
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3. Motion by curvature PDE in 3D Cartesian grid.

• curvature_PDE_3D.ipynb
• curvature_PDE_3D.py
• An example solution of a curvature PDE where $f=1$ on a cube in 3D domain.
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4. Eikonal equation and Tukey depth eikonal equation in unstructured grids

• Eikonal equations
  • Eikonal_PDE_graph.ipynb
  • Eikonal_PDE_graph.py
• Tukey depth eikonal equations
  • tukey_PDE_graph.ipynb
  • tukey_PDE_graph.py
• An example solution of eikonal equation where $f=1$ on a circular domain.
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