Announcements – Description – Prerequisites – Tutorials – Registration – Exam – Contents – Assignments – Literature

• 2021-07-21: Sample solution to Assignment 06 online.
  Lecture notes updated: Corrected a sign error in the last line of Eq.(7.1.39).
• 2021-07-19: Lecture notes updated: Changed a formulation on page 82 (reference version) which wrongly stated that, for the conjugate gradient scheme, the correction is given by the residual.
• 2021-07-13: Assignment 06 is online.
  You can submit this assignment until 2021-07-20, 14:00 via e-mail to Matthias Augustin.
  Lecture notes updated (all numbers are with respect to the reference version of the lecture notes):
  • Corrected index in Eq. (5.2.19), first line, in the expression directly before the equality sign with the ! above.
  • Corrected parentheses on left-hand side of Eq. (6.3.6).
  • Corrected a sign in (6.3.11) and cancelled out the \tau on the right-most side.
  • Added parentheses around the k+1 on the left-hand side of Eq. (6.3.20).
  • Extended Definition 6.5 and Remark 6.6 slightly to clarify the kinds of limits that need to be considered.
  • Changed Eq. (6.4.2) such that k is an upper index to the finite difference operator, not an argument.
  • Added missing index to norm of matrix in Eq. (6.4.13).
  • Added missing index j to t on right-hand side of Eq. (6.4.19).
  • Changed n to k in Eqs. (6.4.20) and (6.4.22) as well as in the line before Eq. (6.4.24).
  • Added a missing factor to Eq. (6.4.35) and added a line before that equation to explain where the factor comes from.
  • Added missing factor h in the norms in Eq. (6.4.45), (6.4.46), and (6.4.47).

Target group: Students in the Master Programme Visual Computing
Lecture aim: Provide some concepts which are useful for the numerical treatment of partial differential equations. This includes

• iterative solvers for linear systems of equations,
• (short) introduction to partial differential equations (PDEs),
• finite-difference (FD) schemes,
• schemes for PDEs of potential and diffusion type,
• properties of PDEs which transfer to FD schemes,
• hyperbolic problems and upwinding.

This course is suitable for students of visual computing, mathematics, and computer science.
Students attending this course should be familiar with basic concepts of (multi-dimensional) calculus and linear algebra as covered in introductory maths course (such as Mathematik für Informatiker I-III). Mathematical prerequisites which exceed the basic mathematics courses are provided within the lecture notes. All material will be in English. Knowledge from image processing may be helpful, but is not required.

Due to the current sitation caused by SARS-Cov2, this lecture will be closer to an inverted classroom / blended learning setting. This means that for each live online sessions, students will be required to prepare using the available material. A schedule detailing the amount of content that you need to prepare will be made available at the start of the lecturing period. Live sessions themselves are intended to follow a Q&A structures and might present further exercise material.

There will be a total of 6 homework assignments which will be graded. Assignments will be published on the Teams file repository and this website. Students are expected to submit their solutions to these assignments via email to Matthias Augustin within one week after publications. Working together in groups of up to 3 students is permitted and encouraged. Some assignments contain programming exercises.

In order to qualify for the final exam, it is necessary to achieve 50% of the points of all assignment sheets in total. All exams will be oral.

Registration is closed since the submitting date of the first assignment passed (2021-05-01).

According to the regulations concerning storage and processing of personal data (Art. 6 (1) Datenschutzgrundverordnung (DSGVO)) we store and process your personal data for the purpose of lecture and tutorial organisation only. I.e. we may use them to contact you, to inform you about your grade, and to transmit your grades to the examination office.

In order to qualify for the final exam, it is necessary to achieve 50% of the points of all assignment sheets in total.

There will be two oral exams, one at the beginning and one at the end of the semester break. In case of qualification, you are allowed to take part in both exams. The better grade counts, but each exam will count as an attempt individually. Please remember that you have to register online for the exam in the HISPOS system of the Saarland University for each attempt separately.

The first exam takes place on
Monday, August 02, 2020.

The second exam takes place on
Thursday, September 30, 2020.

Further information about the exam can be found here

Organizational slides from the first online session

Lecture notes
reference version
newest version
The first chapter of these lecture notes is intended to be a summary of content from linear algebra, in particular concerning things needed in the context of systems of linear equations. It is intended to allow students to quickly refresh their knowledge in this particular field. The actual content of this lecture starts with Chapter 2.
Links to additional exercises will be added to the schedule after each live session.

Date	Please prepare	Pages
04/13	Introduction and Organization	--
04/15	Section 2.1	47-53
04/20	Section 2.2	53-61
04/22	Section 2.3 to end of Section 2.3.1	61-67
04/27	Section 2.3.2 and Section 2.3.3	67-72
04/29	Section 2.3.4 and Section 2.4.1	72-77
05/04	Section 2.4.2 to Equation (2.4.45)	77-83
05/06	Definition 2.48 to Equation (2.4.81)	77-83
05/11	Section 2.5	90-94
05/13	Ascension Day	--
05/18	Chapter 3 up to Remark 3.5 & Assignment 01	97-101
05/20	Pages 102 to 108	102-108
05/25	Section 3.3 to Section 4.2	109-113
05/27	Assignment 02	--
06/01	Section 4.3 and Section 4.4	114-118
06/03	Corpus Christi	--
06/08	Chapter 5 up to Equation (5.2.22)	119-125
06/10	Remark 5.5 to Equation (5.2.41) & Assignment 03	125-133
06/15	Section 5.3 to Remark 5.13	133-138
06/17	Section 5.3.3 to Remark 5.18	138-145
06/22	Section 5.5 to Remark 6.2	145-152
06/24	Assignment 04	--
06/29	Section 6.3 to Remark 6.4	152-157
07/01	After Remark 6.4 to Remark 6.13	157-164
07/06	Section 6.4.2 and Section 6.4.3	164-170
07/08	Assignment 05	--
07/13	Section 6.4.4 and Section 6.4.5	170-174
07/15	Chapter 7 up to Equation (7.1.37)	191-196
07/20	Definition 7.2 to Theorem 7.11	196-204
07/22	Assignment 06	--

No.	Title	Date		Submit until
Assignment 01	Iterative Solvers -- Splitting Methods	05/04	[download]	05/11
Assignment 02	Iterative Solvers -- Gradient and Projection Methods	05/18	[download]	05/25
Assignment 04	Laplace Equation in 1D	06/15	[download]	06/22
Assignment 05	Parabolic Schemes	06/29	[download]	07/06

No.	Title	Date		Submit until
Assignment 01	Iterative Solvers -- Splitting Methods	05/04	[download]	05/11
Assignment 02	Iterative Solvers -- Gradient and Projection Methods	05/18	[download]	05/25
Assignment 03	PDEs, Classification, Finite Differences	06/01	[download]	06/08
Assignment 04	2D Stencils, Higher Order Consistency, Numerics for Laplace Equation in 1D	06/15	[download]	06/22
Assignment 05	Parabolic PDEs	06/29	[download]	07/06
Assignment 06	von Neumann Analysis	07/13	[download]	07/20

No.	Title	Date
Assignment 01	Iterative Solvers -- Splitting Methods	05/13	[download]
Assignment 01	Sample Program: Iterative Solvers -- Splitting Methods	05/13	[download]
Assignment 02	Iterative Solvers -- Gradient and Projection Methods	05/26	[download]
Assignment 02	Sample Program: Iterative Solvers -- Gradient and Projection Methods	05/26	[download]
Assignment 03	PDEs, Classification, Finite Differences	06/08	[download]
Assignment 04	2D Stencils, Higher Order Consistency, Numerics for Laplace Equation in 1D	06/23	[download]
Assignment 04	Sample Program: Laplace Equation in 1D	06/23	[download]
Assignment 05	Parabolic PDEs	07/06	[download]
Assignment 05	Sample Programm: Parabolic Schemes	07/06	[download]
Assignment 06	von Neumann Analysis	07/21	[download]

There is no specific text book for this class, but here is a selection of some books covering many of the topics in this course, giving background material and providing further reading:

• Numerical Solution of Partial Differential Equations in Science and Engineering
  L. Lapidus, G. F. Pinder, Wiley, 1999.
• Numerical Partial Differential Equations
  Volume I: Finite Difference Methods
  Volume II: Conservation Laws and Elliptic Equations
  J. W. Thomas, Springer, 1995 (Vol. I) / 1999 (Vol. II).
• Finite Difference Methods for Ordinary and Partial Differential Equations
  R. J. LeVeque, SIAM, 2007.
• Numerical Methods for Conservation Laws
  R. J. LeVeque, Birkhäuser, 1992.
• Numerical Solution of Partial Differential Equations
  K. W. Morton, D. Mayers, Cambridge University Press, 2005.
• The Finite Difference Method in Partial Differential Equations
  A. R. Mitchell, D. F. Griffiths, Wiley, 1985.
• Numerical Solution of Differential Equations
  W. E. Milne, Wiley, 1960.
• Introduction to Numerical Analysis
  J. Stoer, R. Bulirsch, Springer, 1993.
  English translation of the originally german version.

Most of these and further books can be found in the mathematics and computer science library.

Further references will be provided during the lecture as needed.