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

This book is designed for first-year graduate students in applied and computational mathematics, while also being accessible to students in engineering and computer science. It serves as a textbook for an introductory graduate course on numerical methods for solving partial differential equations (PDEs), with a focus on the Laplacian operator - a fundamental tool in scientific computing and data science.A key feature of the book is its emphasis on the connections between numerical PDEs and data science. It explores a broad range of applications, including image processing, optimal transport, point clouds, graph Laplacians, shape matching, and data classification.The book is structured into two parts. The first part covers classical numerical methods for the Laplacian and Poisson equations on structured grids, including finite difference and finite element methods. The second part extends to the Laplace-Beltrami operator on surfaces, discrete Laplacians for point cloud representations of manifolds, and graph Laplacians.Throughout, the book includes homework problems and research-oriented projects suitable for undergraduate and junior graduate students.