Math and Computer Science Honors Theses

Persistent Homology in Ballistic Deposition simulation models

Date of Creation

5-16-2020

Document Type

Departmental Honors Thesis - Restricted Access

Department

Mathematics

First Advisor

David B. Damiano

Abstract

In this project, we are interested in the granular material surface roughness and porous structures. In the first experiment, we used random walk simulation to investigate the relationship between the expected cover time and the complexity of the Ballistic Deposition (BD) voids. In the second experiment, which is the major part of this thesis, we developed a Dynamic Deposition Model (DDM) based on BD and introduced three physical events to generate fractal and self-affine features. The three events are diffusion, absorption, and desorption, which were inspired by Björk and Deng’s paper, “Discrete simulation models of surface growth”[6]. Our background about the Upper Box Dimension and Persistent Homology Dimension is from Benjamin Schweinhart’s article “Persistent Homology and the Upper Box Dimension.” Our DDM simulation shows the box dimension of the BD voids are likely to be greater than 1.5. Combining Schweinhart’s theorem, we conclude that the Upper Box dDimension is, in fact, the 1-Persistent Homology Dimension of the Rips complex for the voids. We also generated persistence diagram and bar codes to analyze the shape of the void in each stage of DDM simulations. The results suggest that the DDM construction and simulation create an infinite set of points with fractal-like behavior. We conclude that the first Persistent Homology Dimension is equivalent to the Upper Box Dimension and is strictly greater than 1 by the DDM simulation.

Comments

Reader: Eric Ruggieri

This document is currently not available here.

Share

COinS