Ubiquitous Laplacian eBook

This book is designed for graduate students in applied and computational mathematics and is also accessible to students in engineering and computer science. It serves as a textbook for an introductory graduate-level course on numerical methods for solving partial differential equations (PDEs), with a focus on the Laplacian operator — a fundamental and ubiquitous tool in scientific computing and data science.

A distinctive feature of the book is its emphasis on the connections between numerical PDEs and modern data science. It presents a broad scope of applications across computational mathematics, including image processing, optimal transport, point clouds, shape matching, and data processing.

The book is organized into two parts. The first part covers classical numerical methods for the Laplacian or Poisson equation on structured grids, including conventional topics such as finite difference and finite element methods. The second part focuses on the Laplace–Beltrami operator on surfaces approximated by triangular meshes, and discrete Laplacians for point cloud representations of manifolds.

Throughout, the book includes homework-level problems and research-oriented projects suitable for undergraduate, junior graduate, and research-training assignments.

Contents:

• Preface
• About the Authors
• Introduction
• Cartesian Grids and Meshes:
  • Finite Difference Method
  • Finite Element Method on Rectangular Meshes
  • Applications and Examples on Cartesian Grids
• Unstructured Meshes and Point Clouds:
  • A Brief Introduction to Di¿erential Geometry
  • Laplacian on Triangular Mesh Discretization of Surfaces
  • Laplacian on Point Cloud Representation of Manifolds
  • Applications of Laplacian on Manifold-Structured Data
• Appendices:
  • Linear Algebra
  • Iterative Methods for Solving Linear Systems
  • Splitting Methods in Convex Minimizatio
• Bibliography
• Index

Readership: Introductory level graduate courses for numerical PDEs, finite element methods, finite difference methods, etc; Professors who are familiar with traditional numerical PDEs but also wish to cover modern data science applications would be very interested in such a book; Advanced undergraduate level courses for introducing numerical PDEs as well as data science.