monotonescheme 0.0.4

Python wrapper for the monotone discretization problem in 2D and 3D Cartesian grids (join work with Jeff Calder (UMN))

monotone schemes for solving curvature motion PDEs

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

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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

Prerequisite

• 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)