Traditional Algebraic Multigrid Method

AMG method was first introduced by Brandt in [19]. It is developed for solving problems with irregular or unknown geometry properties. In contract to GMG, AMG uses only information from the system matrix.

In this section, we focus our discussion on the traditional AMG. Considering an

alge-Level 1 Relax

Relax

Relax

Solve

Relax Relax

Relax

Interpolate Interpolate Restrict

Restrict

2

3

L

Fig. 2.7: The Multigrid V-Cycle

braic equation, Ax = b, AMG determines the inter-grid mapping operators, coarse grid from the matrix A and the graph of it. Each row of the matrix A can be represented as a node and its connection edge in a graph. The coefficients of the matrix A represent the connections of the graph. For example, if |aij| = 0, there is no edge between node i and j in the graph of A. If |a_ij| ≥ θ|a_ii|, we say that node j strongly influences i. If

|a_ij| ≤ θ|a_ii|, node j weakly influences i. Here, θ is a coefficient from 0 to 1 and is often chosen to be 0.25.

With these definitions, we can construct the matrix graph of A and determine the coarse grid by the color scheme algorithm [18]. This method begins by assigning a measure to each node i of its potential quality to be a coarse node. The weight of node i is determined by counting the number of nodes strongly influenced by node i. Then, we choose the node i with maximum weight to be the starting coarse grid since it has good potential to approximate other nodes. The nodes strongly influenced by node i are defined as fine nodes since they can be approximated well by node i. It’s logical that the nodes strongly influence the new fine nodes should be defined as coarse nodes since they can approximate the new fine nodes well. Thus ,we increase the weights of the nodes strongly influence the new fine nodes by 1 and repeat the coarse node selection until all nodes of

Algorithm of Color Scheme

Input: The Graph of System Matrix A of Nodes 1, 2, ..., n and the Related Weights w1, w2, ..., wnof These Nodes Output: The Sets of Coarse and Fine Nodes

1 Begin

2 NodeCounter=0

3 While NodeCounter!=n 4 MaxWeight=0, StartNode=1 5 For each node i

6 If node i is not defined as a coarse or fine node 7 If w_i >MaxWeight

8 Then MaxWeight=w_i, StartNode=i

9 EndFor

10 StartNode is defined as a coarse node, NodeCounter++

11 For each node j that is strongly influenced by StartNode 12 If node j is not defined as a coarse or fine node

13 node j is defined as a fine node, NodeCounter++

14 For each node k that strongly influences node j 15 If node k is not defined as a coarse or fine node

16 w_k+ +

17 EndFor

18 EndFor

19 End.

Table 2.1: Algorithm of Color Scheme

the matrix graph of A are defined as coarse or fine nodes. The algorithm of color scheme is shown in Table 2.1 and an example of it is given

Fig. 2.8 shows an example of color scheme. The description of each step is shown below:

• Example of Color Scheme:

– Step a: A matrix graph of A is given with node number 1 to 14.

– Step b: The weight of each node i is determined.

– Step c: Node 3 is defined as the starting coarse node with maximum weight of 4 and node 1, 4, 13, 14 are defined as fine nodes. The weights of node 2, 5, 9, 11 are increased.

Strong connection Weak connection

Fig. 2.8: Example of Color Scheme

– Step d: Node 11 is defined as the new coarse node with maximum weight of 5 and node 12 is defined as new fine node. The weight of node 9 is increased by 1.

– Step e: Node 9 is defined as the new coarse node with maximum weight of 5 and node 6 is defined as new fine node.

– Step f: Node 5 is defined as the new coarse node with maximum weight of 3 and node 7 is defined as new fine node. The weight of node 8 is increased by 1.

– Step g: Node 8 is defined as the new coarse node with maximum weight of 3 and node 10 is defined as new fine node.

– Step h: Node 2 is defined as the new coarse node with maximum weight of 2.

All nodes in the matrix graph of A are defined and the color scheme finishes.

By using the color scheme algorithm, we can get the coarse grid and every fine node i can be approximated well by the coarse nodes strongly influence i. However, the selected coarse grid only considers local connections of each node i and may choose bad coarse nodes that will decrease the convergence rate. To overcome this defect, we propose a global mapping operator construction to build the global-considering coarse grids.

To further discuss the grid mapping operator, we continue the discussion of inter-grid transfer operator. Since the key to the efficiency of the multiinter-grid method depends on the complementarity of the relaxation and coarse-grid correction steps. We begin the discussion of inter-grid transfer operator with the property of algebraic smoothness, (Ae)i ≈ 0 which means that residual become small after several iterative iterations for each row i. The equation can be rewritten as

a_iie_i ≈ −^X

j6=i

a_ije_j (2.28)

We define that the DOFs of fine grid is C ∪ F , where C is the set of coarse-level nodes and F is the set of remaining fine-level nodes. Rewriting Equation (2.28), we can get

a_iie_i ≈ − ^X

j∈Ci

a_ije_j − ^X

k∈Fi

a_ike_k (2.29)

where C_i = C ∩ N_i, F_i = F ∩ N_i, and N_i means the neighboring nodes of node i.

For further discussion, we divide the F_i into F_i^sand F_i^w where F_i^sis the set of nodes which strongly influence i in F_i, and F_i^w is the set of nodes which weakly influence i in F_i. Equation (2.29) can be rewritten as

a_iie_i ≈ − ^X

j∈Ci

a_ije_j − ^X

k∈F_i^s

a_ike_k− ^X

m∈F_i^w

a_ime_m (2.30)

From Equation (2.30), we can try to define an interpolation structure since the ei for each node is approximated by the neighboring coarse nodes Ci and fine nodes Fi. If we can approximate the value of Fi as a sum of the values of Ci, ei can be approximated by C_i only and an interpolation can be defined.

Since the values of F_i^s nodes are large compared to a_ii, we approximate e_k by C_i in

Substituting Equation (2.31) into Equation (2.30) and adding the values of F_i^w points into aii, we can get the following equation



From Equation (2.32), an interpolation formula for i ∈ F , e_i = ^P

j∈Ci is simply added to a_ii. However, the selection of F_i^s and F_i^w is fully determined by their coefficients in Equation (2.30), and this would cause the bad choice of F_i^sand F_i^w. Some nodes of F_i^w with large errors should be labeled in the set of F_i^s. This behavior will decrease the convergence rate of standard AMG. One of the main object of our AbAMG is to overcome this defect.

After introducing the concepts of color scheme and weights calculation, the flowchart of traditional AMG is shown in Fig. 2.9. At first, a cycle construction is performed to construct the multigrid V-cycle. In the fine grid, a color scheme is performed to determine the coarse grid and the weights of intergrid transfer operator can be calculated by Equation (2.33). The coarse grid operator A_c can be derived from the Galerkin operator A_c = RA_fP . We apply these steps repeatedly until the coarsest grid operator is coarse enough.

After the step of traditional AMG cycle construction, we can derive the multigrid V-cycle and the answer of x can be solved by the multilevel solver mentioned in section 2.3.

Input

Color scheme

Weights calculation Coarse grid construction

Can level (i+1) be solved?

i=1

i=i+1 No

Yes

Output

Level 1