monotonescheme 0.0.16

Python wrapper for C++ codes for the monotone scheme for curvature-driven PDEs

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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Outline

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 $\Omega \subset \mathbb{R}^d$ be a bounded domain and $\partial \Omega$ be a boundary of $\Omega$.

Eikonal equation

$$ \begin{align*} |\nabla u(x)| &= f(x), && x \in \Omega \ x &= 0, && x \in \partial \Omega \end{align*} $$

Mean curvature PDE

$$ \begin{align*} |\nabla u(x)|\kappa(x) &= f(x), && x \in \Omega \ x &= 0, && x \in \partial \Omega \end{align*}$$ 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

$$ \begin{align*} |\nabla u(x)|\kappa(x)+^{\alpha} &= f(x), && x \in \Omega \ x &= 0, && x \in \partial \Omega \end{align*}$$ where $\alpha \in (0,1]$ is a constant depending on the dimension $d$ and $(t)+ := \max(0,t)$.

Tukey Depth

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

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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)