Algebraic MultiGrid

Algebraic MultiGrid – 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  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  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 5 relaxation Error after 10 relaxations 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 Approximate solution: Fine grid equation: 2. Coarse grid equation: h2 v ~~~ h old h new uu  Residual equation: Smooth error: 1. Relaxation residual: h2 v ~ Approximate solution: 3. Coarse grid correction: 4. Relaxation

TWO GRID CYCLE Approximate solution: Fine grid equation: 2. Coarse grid equation: h old h new uu h2 v ~~~  Residual equation: Smooth error: 1. Relaxation residual: h2 v ~ Approximate solution: 3. Coarse grid correction: 4. Relaxation by recursion MULTI-GRID CYCLE Correction Scheme

interpolation (order m) of corrections relaxation sweeps residual transfer enough sweeps or direct solver *... * h0h0 h 0 /2 h 0 /4 2h h V-cycle: V     

Hierarchy of graphs Apply grids in all scales: 2x2, 4x4, …, n 1/2 xn 1/2 Coarsening Interpolate and relax Solve the large systems of equations by multigrid! G1G1 G2G2 G3G3 GlGl G1G1 G2G2 G3G3 GlGl

Graph drawing example