Mathematics, Department of Mathematics and Computer Science, Spiegelgasse 5, Basel, 5th floor, seminar room 05.002
Perlen-Kolloquium: Prof. Dr. Martin J. Gander (Université de Genève)
Iterative methods for linear systems were invented for the same reasons as they are used today, namely to reduce computational cost. Gauss states in a letter to his friend Gerling in 1823: "you will in the future hardly eliminate directly, at least not when you have more than two unknowns". Richardson's paper from 1910 was then very influential, and is a model of a modern numerical analysis paper: modeling, discretization, approximate solution of the discrete problem, and a real application. More general vector extrapolation methods were then introduced and studied in the pioneering work of Bresinzki, and they can be shown to be equivalent to Krylov method. It was however the work of Stiefel, Hestenes and Lanczos in the early 1950 that sparked the success story of Krylov methods with the invention of the conjugate gradient method, and there are now many Krylov methods to choose from. For general linear systems, they come in two main classes, the ones that minimize the residual in a Krylov space, and the ones that make the residual orthogonal to it. This will bring us finally to the modern iterative methods for solving partial differential equations, which also come in two main classes: domain decomposition methods and multigrid methods. Domain decomposition methods go back to the alternating Schwarz method invented by Herman Amandus Schwarz in 1869 to close a gap in the proof of Riemann's famous Mapping Theorem. Multigrid goes back to the seminal work by Fedorenko in 1961, with main contributions by Brandt and Hackbusch in the Seventies.
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