Advanced Linear Algebra with Applications - Part I (Numerical linear algebra for PDEs, machine learning, and data assimilation)

Abstract

These lecture notes form the first part of a master's-level course on advanced numerical linear algebra. Their aim is not only to present the classical algorithms, but to show why the subject has become considerably more central than it was a generation ago. Numerical linear algebra grew up alongside the numerical solution of partial differential equations, and for a long time that is where its large sparse systems came from. Ranking the nodes of a network, assimilating observations into a weather forecast, and fitting a model to a large noisy data set now lead to problems of the same kind: too large to factorise, structured, and accessible only through matrix-vector products. Strikingly few ideas are needed for all of them. Each chapter therefore develops a standard topic and then puts it to work outside its original setting. We treat norms, factorisations, conditioning and floating-point arithmetic; sparse matrices arising from finite differences, from graphs and from machine learning; stationary iterations and the smoothing property; the conjugate gradient and Lanczos methods, with spectral clustering and regularisation by early stopping; Arnoldi and GMRES, with PageRank and large least squares; and finally preconditioning, Schwarz domain decomposition and multigrid. We assume a first course in linear algebra. Every section closes with a summary of what should be retained and every chapter with exercises, several drawn from past examinations. Accompanying Python code reproduces the numerical illustrations.