Linear and Nonlinear Multiscale Solution Strategies
Professor: Rolf Krause
Assistant: Dorian Krause
Course type: Lecture
Value in ECTS: 6
Bibliographic references available on the University Library website
Academic year 2012/2013 - Spring semester – not offered
Prerequisites: PDEs – Mathematical Modeling and Numerical Simulations.
Multilevel methods are a widely used and highly efficient tool in computational science for solving large scale linear and nonlinear system. In this course, we present the state of the arte for linear as well as nonlinear multilevel and multigrid methods.
The solution of large linear and nonlinear systems of equations is one of the most important tasks in numerical simulation. Since standard solution methods as, e.g., Gaußian elimination do not scale optimally with respect to the number of unknowns, alternative solution strategies have been developed during the last decades. In particular multilevel or multiscale solution strategies have been developed, which are often employed due to their high efficiency. Prominent examples are multilevel methods for linear elliptic problems, which allow for the solution of symmetric positive definite systems with computational effort proportional to the number of unknowns, which turns them into "optimal" solution methods. However, in particular for nonlinear and constrained problems, the derivation of reliable as well as efficient multilevel solution strategies is far from trivial, since convergence is much more difficult to achieve than in the linear case. In this course, we start from well known subspace correction methods for linear problems and proceed to more recent developments as are nonlinear multigrid and monotone multigrid. We will see in what way multilevel hierarchies can be adapted to certain smooth and non-smooth nonlinearities and in what way robust and efficient multilevel methods for nonlinear problems can be derived. Finally, we will consider (recursive) trust-region methods and their application to minimization problems in computational mechanics. For all methods, we will also discuss their parallelisation. Besides the theoretical presentation and analysis of the algorithm, we will also comment on their implementation. Bothe, theoretical as well as practical exercises are part of this course.
- Parallel Linear Multigrid Methods
- Geometric and Algebraic Multigrid Methods
- Subspace Correction Methods
- Constrained Minimisation
- Trust Region Methods
- Parallel Nonlinear Multilevel Methods
William L. Briggs, Van Emden Henson, and Steve F. McCormick, A Multigrid Tutorial, Second Edition, SIAM, 2000 (book home page), ISBN 0-89871-4621.
Wolfgang Hackbusch, Multigrid Methods and Applications, Springer, 1985.
Pieter Wesseling, An Introduction to Multigrid Methods, Corrected Reprint. Philadelphia: R.T. Edwards, Inc., 2004. ISBN 1-930217-08-0.
Gene H. Golub and Charles F. Van Loan, Matrix computations
Toselli, Widlund, Domain Decomposition Methods Nocedal Wright, Numerical Optimisation.
Conn Gould Toint, Trust-Region Methods.
R. Fletcher, Practical Methods of Optimisation.
Numerical Optimization, Series: Springer Series in Operations Research and Financial Engineering , Nocedal, Jorge, Wright, Stephen 2nd ed., 2006, XXII, 664 p. 85 illus., http://www.springer.com/mathematics/book/978-0-387-30303-1