PREREQUISITES
Students should have linear algebraic capabilities that surpass the mere ability
to perform linear algebraic computations 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 orthogonal 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 two lectures will be spent on reviewing this material. Note that reviewing
is not the same as explaining in detail. If you have not seen the material before it
may be hard to absorb everything in just these two weeks.
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.
The focus will be on mathematical ideas and theorems. Instead of covering a
large number of algorithms, we study a number of central algorithms in greater
detail, from their defining mathematical principles to their implementation.
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 Numerical Methods for PDEs (stationary of time-dependent).
LECTURER
Jan Brandts, Korteweg-de Vries Institute for Mathematics, UvA
- Docent: Jan Brandts