Fast Solvers:
Algebraic Multigrid and Algebraic Multilevel Methods
| Course leader: | Mr. Dr. J.Kraus |
| Lecture (L): | Monday, 15.30-17.00 p.m., Room T1010 (First lecture: Monday, March 5, 2007) |
| Tutorial (T): | Monday, 14.30-15.15 p.m., Room T1010 (First tutorial: Monday, March 12, 2007) |
| Target audience: | Students of the branch "Technische Mathematik" in the 6th term (or above) and other interested students |
| Course type: | elective |
| Course hours: | 2 L, 1 T |
- This lecture course provides an introduction to iterative methods for the solution of large sparse systems of linear equations arising from discretization of elliptic boundary-value problems. The main emphasis is on algebraic multigrid and algebraic multilevel preconditioning methods.
- Lecture:
- Iterative methods for systems of linear algebraic equations with symmetric positive definite matrices
- Linear stationary methods
- Polynomial acceleration
- Conjugate Gradients
- Preconditioning via incomplete factorization
- (Relative) condition number
- Simple two-grid methods
- Block factorization (BF)
- Hierarchical basis block-diagonal (HBBD) preconditioning
- Hierarchical basis block-factorization (HBBF) preconditioning
- Analysis of two-grid methods based on the Cauchy-Bunyakowski-Schwarz (CBS) inequality
- Direct extension to multigrid methods
- Recursive BF, HBBD, and HBBF preconditioners
- Approximation of the pivot block
- Schur complement approximation
- Algebraic Multilevel Iteration (AMLI)
- Linear AMLI
- Nonlinear AMLI
- Convergence theory
- Algebraic multigrid (AMG)
- Interplay of coarse-grid correction and smoothing
- M-matrices and the Ruge-Stüben algorithm
- Algebraic Interpolation and the CBS constant
- Two-grid convergence estimates
- Recent developments (energy-minimizing interpolation, smoothed aggregation, adaptive AMG)
- Tutorial:
- Schedule and exercises:
| No. | Date | Topic | ||
| 1 | Mon, March 12, 2007 | Basic Iterative Methods | ps | |
| 2 | Mon, March 19, 2007 | Optimal Chebyshev Methods | ps | |
| 3 | Mon, March 26, 2007 | Cojugate Gradient Method | ps | |
| 4 | Mon, April 16, 2007 | Condition Number, ILU, PCG | ps | |
| 5 | Mon, April 30, 2007 | M-matrices, Existence of ILU Factorization | ps | |
| 6 | Mon, May 7, 2007 | Two-level Preconditioners, CBS Inequality | ps | |
| 7 | Mon, May 14, 2007 | Two-level FEM, Hierarchical Basis | ps | |
| 8 | Mon, May 21, 2007 | Local CBS Constant | ps | |
| 9 | Mon, June 25, 2007 | AMLI, Implementation | ps |
