[Fall 2020]

Prerequisites

Students should have linear algebraic capabilities that surpass the mere ability to perform linear algebraic computions and that include geometric intuition in normed spaces and inner product spaces. They should be acquainted with the basic principles of numerical mathematics and have programming skills that allow them to work in MatLab or to learn it quickly and independently.

This usually requires that apart from a first-year BSc course in Linear Algebra, the student has followed an advanced course in lineair algebra and/or a course in numerical linear algebra. They have also followed an introductory course in Numerical Mathematics, preferably one that includes the concepts of:

1) Finite Precision Arithmetic
2) Conditioning of a problem, stability of an algorithm

in the context of LU- and QR-factorization. They know elementary unitary maps like plane rotations and reflectors in hyperplanes and their matrices, understand both the Classical and the Modified Gram-Schmidt algorithm, and know the Spectral Theorems for selfadjoint, normal and unitary linear  transformations and the Schur-,Jordan-, and Singular Value factorizations.

These prerequisites and assumed prior knowledge can for example be obtained from:

[1] L.N. Trefethen and D. Bau (1997). Numerical Linear Algebra, SIAM Society for Industrial and Applied Matematics. Lectures 1-31.

[2] A. Quarteroni, R. Sacco and F. Saleri (2006). Numerical Mathematics. Springer Verlag, 2nd edition. Chapters 1-5.

OBTAINING AND TESTING YOUR ASSUMED PRIOR KNOWLEDGE:

The first lecture (3x45 minutes) will be spent on reviewing this material.

At the beginning of the second lecture, a quick multiple-choice test will be given.

The grade for this test will not count towards the final grade of the course and is only meant to inform the student about his/her mastering of these prerequisites and to serve as an indicator in the decision to follow the course or not.

Aim of the course:

This course is a first introduction into the main aspects of iterative methods to approximate the solutions of finite- but high-dimensional linear equations, eigenvalue-, and singular value problems. Many of these methods are based on the clever reduction of the problem to an approximating problem of much smaller dimensions. The smaller problem yields an approximate solution of the original problem but simultaneously provides information how to set up the next reduced problem whose corresponding approximation is better than the previous one. This leads to a sequence of smaller problems that need to be solved in order to get increasingly better approximations of the solution of the original problem.

The aim is to teach students how to approximate solutions of large scale linear algebra problems by cleverly designed small scale linear algebra problems, how to analyse the approximation properties mathematically, and how to iplement the corresponding methods in MatLab. Students are taught how to perform experiments in MatLab and how to discuss their outcomes.

This course can be part of an MSc program in Applied Mathematics and will be of great added value in the other MasterMath courses Parallel Algorithms, Systems and Control, and Numerical Bifurcation Analysis of Large-scale systems. It also supplements Applied Finite Elements and Numerical Methods for PDEs.

Keywords: Iterative methods, subspace method, Krylov methods. In particular CG, GMRES, BiCG, Lanczos, Arnoldi and Jacobi-Davidson. Abstract embedding. Matlab.

Lecturers

Jan Brandts, Korteweg-de Vries Institute for Mathematics, UvA