monotonescheme 0.0.1

Python wrapper for C++ codes for the monotone scheme for curvature motion PDEs

Monotone schemes for solving curvature motion PDEs

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

• Paper: arXiv
• Jeff Calder: website
• Wonjun Lee: website

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Outline

This repository contains c++ and python codes for running the monotone algorithm to solve curvature motion PDEs. Here are list of PDEs that can be solved using this algorithm.

Eikonal equation

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

Mean curvature PDE

$$ |\nabla u(x)|\kappa(x) = f(x), \quad x \in \Omega $$ $$ x = 0, \quad x \in \partial \Omega $$ 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 $x$.

Tukey Depth

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

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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

First install the package by running the following command:

    pip install MonotoneScheme

(TO BE CONTINUED)