Hilbert Space Splittings and Iterative Methods by Michael Griebel

Hilbert Space Splittings and Iterative Methods by Michael Griebel

This book is about the theory of so-called Schwarz methods for solving variational problems in a Hilbert space V arising from linear equations and their associated quadratic minimization problems. Schwarz methods are based on the construction of a sequence of approximate solutions by solving auxiliary variational problems on a set of (smaller, finite-dimensional) Hilbert spaces $V_i$ in a certain order, combining them, and using the combined approximations in an iterative procedure. The spaces $V_i$ form a so-called space splitting for V, they need not necessarily be subspaces of V, and their number can be finite or infinite.

The convergence behavior of Schwarz methods is influenced by certain properties of the space splittings they are based on. These properties are identified, and a detailed treatment of traditional deterministic and more recent greedy and stochastic orderings in the subproblem solution process is given, together with an investigation of accelerated methods. To illustrate the abstract theory, the numerical linear algebra analogs of the iterative methods covered in the book are discussed. Its standard application to the convergence theory of multilevel and domain decomposition methods for solving PDE problems is explained, and links to optimization theory and online learning algorithms are given.

Providing an introduction and overview of iterative methods which are based on problem decompositions and suitable for parallel and distributed computing, the book could serve as the basis for a one- or two-semester course for M.S. and Ph.D. students specializing in numerical analysis and scientific computing. It will also appeal to a wide range of researchers interested in scientific computing in the broadest sense.

Peter Oswald is a performer, poet, and dramatist. From 1998 until 2009, he was the Writer-in-Residence at Shakespeare's Globe. His verse plays have been presented in the National Theatre, the West End, Broadway, and all around the world, and have been translated into a number of languages. The South Bank Prize was given to his staging of Schiller's Mary Stuart. In 2002, he co-founded The Abyss Theatre Company, with which he collaborates as a writer and actor.

In 2015, he performed extended story-poems based on Icelandic sagas and folktales at the Folger Theatre in Washington, DC, the Hay Festival, and the Wells Festival. He and Sean Borodale collaborated on a collection of poems called Dyad, as well as pamphlets with his wife Alice, with whom he frequently performs. In Devon, he lives with her and their three children. In 2016, he was awarded a Society of Authors travel scholarship.