Del Estatuto Nacional de Estupefacientes (ENE) al Código Penal, más de

As in the geometric case, the coarse grid Ωl+1 _{= C}l _{should meet two demands:}

1. The smooth error components e should be well represented on Ωl+1.

2. The coarse grid should be a constant (independent of |Ωl_{|) factor smaller than the}

fine grid.

Now, algebraic multigrid is often used if geometric multigrid fails, i.e. a geometrically obtained coarsening like mesh width doubling does not reflect the behavior of smooth error components very well. This can for example be due to anisotropies in the operator

(a) Initialized weights (b) First coarse grid point (c) Neighbors set to fine

(d) Update weights for neigh- bors of neighbors

(e) Second coarse grid point (f) Neighbors set to fine

(g) Update weights for neigh- bors of neighbors

(h) Third coarse grid point (i) Final coarse grid

Figure 2.6.: Coarsening process for a 5-point stencil. The numbers denote the weights of still undecided points, the red points belong to the coarse grid and the white points belong to the fine grid. Updated weights are given in blue. The blue edges indicate which connections are considered within the graph at the current step. Note that the strength matrix is symmetric here.

(see e.g. Example 2.1) or different PDE coefficients in different parts of the domain. Furthermore, the geometry of the discretization domain may not allow a straightforward coarsening.

Instead of relying on geometrical information, the key to the construction of an algebraic coarse grid is to use the information that we know about the structure of the smooth error. In the previous section, we have already seen that for relevant PDE problems the smooth error varies slowly along the strong connections. In other words, the value ei at

a point i ∈ Ω can easily be approximated from the values of strongly connected points j ∈ Si. The coarsening demands mentioned above can now be expressed in terms of S.

C1 For any fine grid point i ∈ F and every strongly connected point j ∈ Si let either

j be a coarse grid point, or j itself depend on a coarse grid point k ∈ Si∩ C.

C2 Let C be a maximal set so that two points in C are not strongly connected to each other.

The first condition ensures that for any fine grid point i ∈ F all strong connections are reflected in the coarsening process, not only those that directly lead to a coarse grid point j ∈ C. The reason for this is the that we will construct the interpolation operators (see Section 2.8) to interpolate along the strong connections, i.e. the value at a fine grid point i ∈ F will be interpolated from the values ej, where j is contained in the set of strongly

connected coarse neighbors Si ∩ C. On the other hand, the value ek at any strongly

connected coarse grid point k ∈ Si should be represented in the interpolation scheme.

Now, as long as ek, k ∈ Si∩ F , also depends on and interpolates from ej, j ∈ Si∩ C, ek

cannot differ “too much” from these ej and hence from ei.

If C fulfills Condition C2, it is called a maximal independent set (MIS). An indepen- dent set I ⊂ V in a graph (V, E) is a set of vertices such that two vertices i, j ∈ I are not directly connected to each other, i.e. there is no edge (i, j) in E. A maximal independent set is an independent set I such that no additional vertex i can be added to I without loosing the independent set property. This should not be confused with a maximum independent set, which is an independent set with a maximum number of vertices. The task of constructing a coarse grid hence turns into the task of finding a maximal independent set, for which many greedy algorithms are available, e.g. [Lub86]. In this particular case, however, the set of coarse grid points has to satisfy condition C1 to ensure the robustness of the interpolation. Therefore, the construction of coarse grids is split into two phases. In the first phase, an independent set is created. The second phase then enforces condition C1, thereby sacrificing the independent nature of the set C.

In Algorithm 2.5 we give the first phase of the classical coarsening algorithm. Most important here is that we assign a weight λi to each point i, which measures the “use-

fulness” of i as coarse grid point. Before starting the iteration, we initialize λi to the

number of points j that depend on i, i.e. |ST

i | (Figure 2.6(a)). A point with maximal

weight is chosen and added to the set of coarse grid points C (Figure 2.6(b)). Then, all neighbors j that depend on i (j ∈ S_iT) can interpolate from i and hence we assign them to the set of fine grid points (Figure 2.6(c)). On the other hand, interpolation should

algorithm 2.5 AmgPhaseI(Ω, S, ST, C, F ) ([RS87], algorithm A2) begin U ← Ω; C ← ∅; F ← ∅; for i ∈ Ω do λi ← |SiT|; od;

while maxi∈Uλi 6= 0 do

i ← arg maxj∈Uλj; pick coarse grid point

C ← C ∪ {i}; U ← U \ {i}; for j ∈ ST

i ∩ U do

F ← F ∪ {j}; U ← U \ {j}; add neighbors to fine grid

for k ∈ Sj ∩ U do

λk ← λk+ 1; increase weights of neighbors of neighbors

od; od;

for j ∈ Si∩ U do

λj ← λj − 1; decrease weights of point that influence i

od; od;

F ← F ∪ U ; end

be from a large portion of connection of j, not only one strong coupling (see Section 2.8 for an intensive discussion). This motivates us to increase, for all newly assigned fine grid points j, the weight of all k ∈ Sj such that they are more likely to become coarse

in future iterations (Figure 2.6(d)).

Furthermore, as i is now coarse, the strong connections of i, Si are not needed for in-

terpolation anymore. Hence, we can reduce the weights λj for all j ∈ Si if j is still

undecided. Then, we restart the iteration and again pick a point with maximal weight λi (Figure 2.6(e)) until all points are assigned either fine or coarse (Figure 2.6(i)). It is

easy to see that the iteration needs at most O(N · nzr2_{) steps, where N is the number}

of unknowns and nzr the maximal number of nonzeros per row, which is assumed to be small and independent of N for relevant PDE problems.

The second phase algorithm 2.6, which is derived from Algorithm A3 in [RS87], takes as input the C/F -splitting determined by the first pass. We loop over all fine grid points i ∈ F and their strongly coupled fine neighbors j ∈ Si ∩ F . Then we check whether

these points share a common C-point, i.e. a point k such that both k ∈ Si and k ∈ Sj.

If such a k does not exist, either i or j is added to the set of coarse grid points, where i is given precedence if two or more additional coarse points would be needed to ensure stability for interpolating ei. This algorithm also takes O(n · nzr2) operations (outer

(a) Strong Couplings (b) After first coarsening phase (red points are coarse)

(c) After second coarsening phase (red and green points are coarse)

Figure 2.7.: Finite difference discretization of a PDE with a jumping diffusion coefficient (Example 2.3). The blue line gives the phase boundary; the arrows denote a strong coupling only in the indicated direction.

algorithm 2.6 AmgPhaseII(Ω, S, C, F ) begin for i ∈ F do ˜ C ← ∅; for j ∈ Si∩ F do if Sj ∩ Si∩ C = ∅ then if ˜C 6= ∅

then change i, because otherwise 2 points need to be changed C ← C ∪ {i};

F ← F \ {i}; GOTO NEXTi;

else j could be an interpolation point for i

˜ C ← {j}; fi; fi; od; C ← C ∪ ˜C; F ← F \ ˜C; NEXTi : od; end

Figure 2.8.: Standard coarsening for a three-dimensional Laplacian. The blue points are fine grid points, the red and green points belong to the coarse grid. On the coarse level, the point in the center has connections to 18 other points. After the application of aggressive coarsening using the sets ˆS_i(2,2), all green points are also fine grid points and the original 7-point stencil is recovered on the coarse level.

Example 2.3 We consider the PDE

−∇ · η(x, y)∇u(x, y) = f(x, y)

on a square domain with Dirichlet boundary conditions, and

η(x, y) = (

1 if x + y > 0.9 1000 else.

In Figure 2.7(a) we show the strong couplings (α = 0.25) for a finite difference discretiza- tion. The blue line indicates the phase boundary for η. We see that the strength matrix is not symmetric (the arrows denote the strong couplings that are only present in one direction). Figure 2.7(b) shows the coarse grid (red points) after phase I (Algorithm 2.5) of the coarsening scheme. We see that the fine grid points at the upper right side to the phase boundary are strongly dependent on the points at the lower left side of the boundary, these points however are also fine grid points that do not depend on the same set of coarse grid points, thus violating condition C1. Algorithm 2.6 repairs this problem by inserting the green points in Figure 2.7(c).