MATH 5485 / CS 5485
Numerical Analysis and Software I

 

Numerical Analysis and Software I

Iterative Methods for Large Sparse Linear Systems and Eigenvalue Problems - From Basics to Current Research

Iterative methods for linear systems and eigenvalue problems play a fundamental role in virtually all large-scale simulation and optimization problems. Hence, they are of great importance for computational science and engineering applications. In this course, we will introduce and discuss the most popular iterative methods and preconditioners, analyze their theoretical properties (including convergence issues), and evaluate solvers and preconditioners for various applications, including numerical experimentation. Students can use their own research problems for such experiments. Furthermore, we will discuss how variations of these methods can be used for solving large-scale eigenvalue problems, and how these methods relate to nonlinear systems and optimization (more on this will be taught in part II in the spring course on nonlinear inverse problems). Finally, we will consider a selection from important current research topics, such as convergence improvement by various projections, efficiently solving long sequences of linear systems, preconditioners for saddle-point problems (incompressible flow, optimization), multilevel methods and preconditioners, solving linear systems with approximate matrices, and recent convergence results and error bounds.

Although some material on iterative methods is taught in our general 4445/6 and 5465/6 numerical analysis courses, this course treats the topic much more extensively and in greater theoretical depth.

Instructor: Eric de Sturler (click to check what I do the rest of my time)

Prerequisites: A senior level numerical analysis course, such as math 4445/4446, or a numerical linear algebra course, including basic (but not advanced) programming skills.

Overview

 

               

Design of an optimal beam

 

Lecture Notes

Textbook:

Iterative Krylov Methods for Large Linear Systems,

Cambridge University Press Series: Cambridge Monographs on Applied and Computational Mathematics (No. 13),

Henk A. van der Vorst, Universiteit Utrecht, The Netherlands.

The lecture notes will cover significant additional material.

Other useful books on iterative methods are:

Some books focusing on preconditioning are:

Homework:

    Your homework is now officially late!

Quizzes will only be given on request J

Course Policies

Class Projects (link will be added shortly)