Linear system of equations form one of the most fundamental problem classes in computation in science and engineering. Many linear systems of equations stemming from applications are expressed in terms of large sparse matrices, which are solved by iterative algorithms. The purpose of this course is to learn about the iterative algorithms for large-scale linear systems and how to improve them by using matrix structures and the techniques called preconditioning.
Information for research students about course offerings
Content and learning outcomes
- Iterative methods (Krylov methods, Gauss-Seidel methods)
- Convergence theory (eigenvalues, pseudospectra, right-hand side dependence)
- General preconditioners
- Problem specific preconditioners
Intended learning outcomes
A students which has passed this course should know
which iterative methods are available for linear systems, and integration of preconditioning.
how to apply and adapt convergence theory for the iterative methods.
apply general preconditioners based on, diagonal, LU-factorization.
apply problem specific preconditioners, such as domain decomposition, Schur-complement and adapted for partial differential equations such as Helmholtz problem.
characterize the quality of a preconditioner experimentally and theoretically.
Lectures, seminar, problem solving, problem composition.
Literature and preparations
This course is designed for PhD students in applied and computational mathematics, but it is suitable also for other PhD students with a background in computation with mathematical interests. The students are expected to have taken a basic and a continuation course in numerical analysis or acquired equivalent knowledge in a different way, and preferrably also a course in matrix computations or numerical linear algebra, e.g., SF3580 and/or SF2524.
It will be announced on the course web page 3 weeks before course starts
Examination and completion
If the course is discontinued, students may request to be examined during the following two academic years.
- INL1 - Assignment, 7.5 credits, grading scale: P, F
Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.
The examiner may apply another examination format when re-examining individual students.
Other requirements for final grade
Problems solved, posed, seminar presented and homeworks solved.
Opportunity to complete the requirements via supplementary examination
Opportunity to raise an approved grade via renewed examination
- All members of a group are responsible for the group's work.
- In any assessment, every student shall honestly disclose any help received and sources used.
- In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.