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401-4671-00L  Advanced Numerical Methods for CSE

SemesterHerbstsemester 2018
DozierendeR. Hiptmair, C. Jerez Hanckes
Periodizitätjährlich wiederkehrende Veranstaltung
LehrspracheEnglisch


KurzbeschreibungThis course discusses modern numerical methods involving complex algorithms and intricate data structures that render an efficient implementation non-trivial. The focus will be on boundary element methods, hierarchical matrix techniques, convolution quadrature, and algebraic multigrid methods.
Lernziel- Appreciation of the interplay of functional analysis, advanced calculus, numerical linear algebra, and sophisticated data structures in modern computer simulation technology.
- Knowledge about the main ideas and mathematical foundations underlying boundary element methods, hierarchical matrix techniques, convolution quadrature, and reduced basis methods.
- Familiarity with the algorithmic challenges arising with these methods and the main ways on how to tackle them.
- Knowledge about the algorithms' complexity and suitable data structures.
- Ability to understand details of given implementations.
- Skills concerning the implementation of algorithms and data structures in C++.
Inhalt1 Boundary Element Methods (BEM)
1.1 Elliptic Model Boundary Value Problem: Electrostatics . . . . . . . .
1.2 Boundary Representation Formulas . . . . . . . . . . . . . . . . . .
1.3 Boundary Integral Equations (BIEs) . . . . . . . . . . . . . . . . . .
1.4 Boundary Element Methods in Two Dimensions . . . . . . . . . . . . . . . . . . .
1.5 Boundary Element Methods on Closed Surfaces . . . . . . . . . . . . . . . . . . .
1.6 BEM: Various Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Local Low-Rank Compression of Non-Local Operators
2.1 Examples: Non-Local Operators . . . . . . . . . . . . . . . . . . . . .
2.2 Approximation of Kernel Collocation Matrices . . . . . . . . . . . . . . .
2.3 Clustering Techniques . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Hierarchical Matrices . . . . . . . . . . . . . . . . . . .
3 Convolution Quadrature
3.1 Basic Concepts and Tools
3.2 Convolution Equations: Examples . . . . . . . . . . . . . .
3.3 Implicit-Euler Convolution Quadrature . . . . . . . . . . . .
3.5 Runge-Kutta Convolution Quadrature . . . . . . . . . . . .
3.6 Fast Oblivious Convolution Quadrature . . . . . . . .
4 Algebraic Multigrid Methods
SkriptLecture material will be created during the course and will be made available online and in chapters.
LiteraturS. Sauter and Ch. Schwab, Boundary Element Methods, Springer 2010
O. Steinbach, Numerical approximation methods for elliptic boundary value problems, Springer 2008
M. Bebendorf, Hierarchical matrices: A means to efficiently solve elliptic boundary value problems, Springer 2008
W. Hackbusch, Hierarchical Matrices, Springer 2015
S. Boerm, Efficient Numerical Methods for Non-Local Operators: H2-Matrix Compression, Algorithms and Analysis, EMS 2010
S. Boerm, Numerical Methods for Non-Local Operators, Lecture Notes Univ. Kiel 2017
M. Hassell and F.-J. Sayas, Convolution Quadrature for Wave Simulations
J.-C. Xu and L. Zikatanov, Algebraic Multirgrid Methods, Acta Numerica, 2017
Ch. Wagner, Introduction to Algebraic Multigrid, Lecture notes IWR Heidelberg, 1999, https://perso.uclouvain.be/alphonse.magnus/num2/amg.pdf
Voraussetzungen / Besonderes- Familiarity with basic numerical methods
(as taught in the course "Numerical Methods for CSE").
- Knowledge about the finite element method for elliptic partial differential equations (as covered in the course "Numerical Methods for Partial Differential Equations").