Algebraic Multi Grid Algebraic Multi Grid AMG Brandt

Algebraic Multi. Grid

Algebraic Multi. Grid – AMG (Brandt 1982) § General structure § Choose a subset of variables: the C-points such that every variable is “strongly connected” to this subset § Define the interpolation (aggregation) weights of each fine variables to the C-points § Construct the coarse level equations § Repeat until a small enough problem § Interpolate (disaggregate) to the finer level § Classical versus weighted aggregation

Data structure For each node i in the graph keep 1. A list of all the graph’s neighbors: for each neighbor keep a pair of index and weight 2. … 3. … 4. Its current placement 5. The unique square in the grid the node belongs to For each square in the grid keep 1. A list of all the nodes which are mostly within Ø Defines the current physical neighborhood 2. The total amount of material within the square

Data structure For each node i in the graph keep 1. A list of all the graph’s neighbors: for each neighbor keep a pair of index and weight 2. A list of finer level vertices belonging to i 3. A list of coarse level aggregates i contributes to 4. Its current placement 5. The unique square in the grid the node belongs to For each square in the grid keep 1. A list of all the nodes which are mostly within Ø Defines the current physical neighborhood 2. The total amount of material within the square

Influence of (pointwise) Gauss-Seidel relaxation on the error Poisson equation, uniform grid Error of initial guess Error after 10 relaxations Error after 5 relaxation Error after 15 relaxations

The basic observations of ML § Just a few relaxation sweeps are needed to converge the highly oscillatory components of the error => the error is smooth § Can be well expressed by less variables § Use a coarser level (by choosing every other line) for the residual equation § Smooth component on a finer level becomes more oscillatory on a coarser level => solve recursively § The solution is interpolated and added

TWO GRID CYCLE Fine grid equation: 1. Relaxation Approximate solution: Smooth error: Residual equation: residual: 2. Coarse grid equation: Approximate solution: ~v 2 h 3. Coarse grid correction: h h = u~old + u~ new 4. Relaxation ~v 2 h

TWO GRID CYCLE MULTI-GRID Fine grid equation: 1. Relaxation 1 Approximate solution: Smooth error: Residual equation: residual: 2. Coarse grid equation: Approximate solution: 3. Coarse grid correction: 4. Relaxation 2 4 ~v 2 h 3 by recursion 5 h h = u~old + u~ new 6 Correction Scheme ~v 2 h

V-cycle: V(n 1, n 2) h 2 h . . h 0/4 . h 0/2 h 0 * interpolation (order m) of corrections relaxation sweeps * residual transfer enough sweeps or direct solver

G 1 G 2 G 3 Gl Hierarchy of graphs G 3 Gl Apply grids in all scales: 2 x 2, 4 x 4, … , n 1/2 xn 1/2 Solve the large systems of equations by multigrid! Coarsening Interpolate and relax

Graph drawing example