Ubiquitous Laplacian: An Introduction To Numerical Pdes With Applications In Data Science

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.