Course syllabus

# Iterativ lösning av storskaliga system i beräkningsteknik

Iterative Solution of Large Scale Systems in Scientific Computing

## FMNN30, 7,5 credits, A (Second Cycle)

## General Information

## Aim

## Learning outcomes

## Contents

## Examination details

## Admission

## Reading list

## Contact and other information

Iterative Solution of Large Scale Systems in Scientific Computing

Valid for: 2019/20

Decided by: PLED F/Pi

Date of Decision: 2019-03-26

Elective for: F4, F4-bs, Pi4-bs

Language of instruction: The course will be given in English

A core problem in Scientific Computing is the solution of nonlinear and linear systems. These arise in the solution of boundary value problems, stiff ODEs and in optimization. Particular difficulties appear when the systems are large, meaning millions of unknowns. This is often the case when discretizing partial differential equations which model important phenomenas in science and technology. Due to the size of the systems they may only be solved using iterative methods.

The aim of this course is to teach modern methods for the solution of such systems.

The course is a direct follow up of the course *FMNN10
Numerical Methods for Differential Equations,* and expands the
student's toolbox for calculating approximative solutions of
partial differential equations.

Knowledge and understanding

For a passing grade the student must

- understand basic iterative methods for linear and nonlinear equations, and understand their mathematical differences
- understand the framwork of Jacobian-free Newton-Krylov methods
- understand multigrid methods and their application to some model problems.

Competences and skills

For a passing grade the student must

- be able to implement an inexact Jacobian-free Newton-Krylov method
- be able to implement a multigrid method for model problems
- be able to implement basic iterative solvers as computer programs.

Judgement and approach

For a passing grade the student must

be able to decide, given information about a nonlinear or linear system, which solver to use and which not to.

- Where do large scale linear and nonlinear systems arise in Scientific Computing?
- Speed of convergence
- Termination criteria
- Fixed Point mehtods and convergence theory
- Newton's method, its convergence theory and its problems
- Inexact Newton's method and its convergence theory
- Methods of Newton type and associated convergence theory
- Linear systems
- Krylov subspace methods and GMRES
- Preconditioning GMRES
- Jacobian-free Newton-Krylov methods
- Multigrid methods in one and two dimensions
- Multigrid methods for nonstandard equations and for nonlinear systems

Grading scale: TH - (U,3,4,5) - (Fail, Three, Four, Five)

Assessment: Computational project with written report. Take home exam and/or oral exam, to be decided by the examiner at the start of the course.

The examiner, in consultation with Disability Support Services, may deviate from the regular form of examination in order to provide a permanently disabled student with a form of examination equivalent to that of a student without a disability.

Required prior knowledge: FMNN10 Numerical Methods for Differential Equations.

The number of participants is limited to: No

The course overlaps following course/s: NUMN30, FMNN15

- Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, 1995, ISBN: 9780898713527. Covers the first part of the course.
- Wesseling, P.: An Introduction to Multigrid Methods. R T Edwards, 2004, ISBN: 9781930217089. Covers the second part of the course.

Teacher: Philipp Birken, philipp.birken@na.lu.se

Course coordinator: Anders Holst, studierektor@math.lth.se

Course administrator: Student Office, expedition@math.lth.se

Course homepage: http://www.maths.lu.se/utbildning/numerisk-analys/courses-in-numerical-analysis/