Peter Arbenz: Catalogue data in Autumn Semester 2014

Name Prof. em. Peter Arbenz
E-mailarbenz@inf.ethz.ch
URLhttp://www.inf.ethz.ch/personal/arbenz/
DepartmentComputer Science
RelationshipRetired Adjunct Professor

NumberTitleECTSHoursLecturers
263-5001-00LIntroduction to Finite Elements and Sparse Linear System Solving Information 4 credits2V + 1UP. Arbenz, T. Kaman
AbstractThe finite element (FE) method is the method of choice for (approximately) solving partial differential equations on complicated domains. In the first third of the lecture, we give an introduction to the method. The rest of the lecture will be devoted to methods for solving the large sparse linear systems of equation that a typical for the FE method. We will consider direct and iterative methods.
ObjectiveStudents will know the most important direct and iterative solvers for sparse linear systems. They will be able to determine which solver to choose in particular situations.
ContentI. THE FINITE ELEMENT METHOD

(1) Introduction, model problems.

(2) 1D problems. Piecewise polynomials in 1D.

(3) 2D problems. Triangulations. Piecewise polynomials in 2D.

(4) Variational formulations. Galerkin finite element method.

(5) Implementation aspects.


II. DIRECT SOLUTION METHODS

(6) LU and Cholesky decomposition.

(7) Sparse matrices.

(8) Fill-reducing orderings.


III. ITERATIVE SOLUTION METHODS

(9) Stationary iterative methods, preconditioning.

(10) Preconditioned conjugate gradient method (PCG).

(11) Incomplete factorization preconditioning.

(12) Multigrid preconditioning.

(13) Nonsymmetric problems (GMRES, BiCGstab).

(14) Indefinite problems (SYMMLQ, MINRES).
Literature[1] M. G. Larson, F. Bengzon: The Finite Element Method: Theory, Implementation, and Applications. Springer, Heidelberg, 2013.

[2] H. Elman, D. Sylvester, A. Wathen: Finite elements and fast iterative solvers. OUP, Oxford, 2005.

[3] Y. Saad: Iterative methods for sparse linear systems (2nd ed.). SIAM, Philadelphia, 2003.

[4] T. Davis: Direct Methods for Sparse Linear Systems. SIAM, Philadelphia, 2006.

[5] H.R. Schwarz: Die Methode der finiten Elemente (3rd ed.). Teubner, Stuttgart, 1991.
Prerequisites / NoticePrerequisites: Linear Algebra, Analysis, Computational Science.
The exercises are made with Matlab.
401-0663-00LNumerical Methods for CSE Information 7 credits4V + 2UP. Arbenz
AbstractThe course gives an introduction into fundamental techniques and algorithms of numerical mathematics which play a central role in numerical simulations in science and technology. The courses focuses
on fundamental ideas and algorithmic aspects of numerical methods.
The exercises involve actual implementation of numerical methods.
Objective* Knowledge of the fundamental algorithms in numerical mathematics
* Knowledge of the essential terms in numerical mathematics and the
techniques used for the analysis of numerical algorithms
* Ability to choose the appropriate numerical method for concrete problems
* Ability to interpret numerical results
* Ability to implement numerical algorithms afficiently
Content1. Direct Methods for linear systems of equations
2. Interpolation
3. Iterative Methods for non-linear systems of equations
4. Krylov methods for linear systems of equations
5. Eigensolvers
6. Least Squares Techniques
7. Filtering Algorithms
8. Approximation of Functions
9. Numerical Quadrature
10. Clustering Techniques
11. Single Step Methods for ODEs
12. Stiff Integrators
13. Structure Preserving Integrators
Lecture notesLecture slides will be made available to participants.
LiteratureW. Dahmen, A. Reusken "Numerik für Ingenieure und Naturwissenschaftler", Springer 2006.
M. Hanke-Bourgeois "Grundlagen der Numerischen Mathematik und des wissenschaftlichen Rechnens", BG Teubner, 2002
C. Moler, Numerical computing with MATLAB, SIAM, 2004
P. Deuflhard and A. Hohmann, "Numerische Mathematik I", DeGruyter, 2002
Prerequisites / NoticeThe course will be accompanied by programming exercises relying on the high level programming language MATLAB. A brief introduction to Matlab will be given during the first week.
406-0663-AALNumerical Methods for CSE
Enrolment only for MSc students who need this course as additional requirement.
7 credits15RP. Arbenz
AbstractIntroduction into fundamental techniques and algorithms of numerical mathematics which play a central role in numerical simulations in science and technology.
Objective* Knowledge of the fundamental algorithms in numerical mathematics
* Knowledge of the essential terms in numerical mathematics and the techniques used for the analysis of numerical algorithms
* Ability to choose the appropriate numerical method for concrete problems
* Ability to interpret numerical results
* Ability to implement numerical algorithms afficiently
ContentThe course will cover the following chapters:

1. Direct Methods for linear systems of equations
2. Interpolation
3. Iterative Methods for non-linear systems of equations
4. Krylov methods for linear systems of equations
5. Eigensolvers
6. Least Squares Techniques
7. Filtering Algorithms
8. Approximation of Functions
9. Numerical Quadrature
10. Clustering Techniques
11. Single Step Methods for ODEs
12. Stiff Integrators
Lecture notesComprehensive lecture materials are available upon request from the lecturer.
LiteratureW. Dahmen, A. Reusken "Numerik für Ingenieure und Naturwissenschaftler", Springer 2006
M. Hanke-Bourgeois "Grundlagen der Numerischen Mathematik und des wissenschaftlichen Rechnens", BG Teubner, 2002
C. Moler, "Numerical computing with MATLAB", SIAM, 2004
P. Deuflhard and A. Hohmann, "Numerische Mathematik I", DeGruyter, 2002
Prerequisites / NoticeSolid knowledge about fundamental concepts and technques from linear algebra & calculus as taught in the first year of science and engineering curricula.