AN EVALUATION OF PARALLEL MULTIGRID AS A SOLVER AND A ...

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92 C. W. OOSTERLEE AND T. WASHIOΓ2Γ30λ 1 λ 2 λ3x x x1λ n-lΓ 1FIG. 2. ¯Γi for a case where most eigenvalues are near one.Remark 2.1.2. Inequality (9) shows that ‖r (k) ‖ 2 is not larger than the norm ofthe (k − l)th residual of GMRES(m) with initial residual ¯r (l) that is included in thesubspace corresponding to the eigenvalues close to one {λ i | i > l}.Remark 2.1.3. As already mentioned, (I − AK −1 ) gives the residual transformationfor the Richardson iterative process based on the matrix splitting A =K +(A−K). Hence, from (10), the asymptotic convergence of GMRES(m) is at leastfaster than Richardson with an initial residual; that does not include componentscorresponding to the large eigenvalues {λ i | i ≤ l} of AK −1 .GMRES polynomial. We construct the GMRES polynomial p k to understand theway the GMRES search directions are constructed. Here, the GMRES polynomialmeans p k ∈ P k , which gives the residual vector r (k) of the kth GMRES iterativeprocess as r (k) = p k (Ã)r(0) . As mentioned in the proof of Theorem 2.1, p k gives theminimal L 2 -norm of the residual under the condition p k (0) = 1. The minimal residualpolynomial p k looks like(16)p k (Ã)r(0) = b k Ã k r (0) + · · · + b 1 Ã 1 r (0) + r (0) .In order to find the coefficients b k the following function f should be minimized:(17)Therefore,(18)f(b 1 , . . . , b k ) := (p k (Ã)r(0) , p k (Ã)r(0) )k∑:= b i b j (Ãi r (0) , Ãj r (0) ) + 2∂f∂b i= 2i,j=1k∑b i (Ãi r (0) , r (0) ).i=1k∑b j (Ãi r (0) , Ãj r (0) ) + 2(Ãi r (0) , r (0) ) = 0.j=1The coefficients b k are obtained by solving matrix equation (18).The solutions of p k (λ ∗ ) = 0 will suggest how GMRES optimizes p k (Ã)r(0) ; thesesolutions are calculated in section 3. Then it can be seen whether the polynomial (13)assumed in Theorem 2.1 is appropriate for the problems investigated.2.2. Three multigrid methods. For a given positive number M, a standardsequence of grids G 1 , G 2 , . . . , G M for multigrid is defined as follows:(19)(20)G M = G,G L = {(i, j)|(2i − 1, 2j − 1) ∈ G L+1 }, 1 ≤ L ≤ M − 1.
MULTIGRID AS A SOLVER AND A PRECONDITIONER 93(2j+1)J+1w3Xw4= coarse (and fine) grid point(2j)X X XX= fine grid point(2j-1)Jw1w2XI I+1(2i-1) (2i) (2i+1)FIG. 3. One coarse grid cell with the prolongation weights and four fine grid cells for standardmultigrid (for coarse grid indices capital letters are used and the fine grid indices are in brackets).The three multigrid preconditioners that are implemented and whose performance iscompared are now discussed.2.2.1. Prolongation operators. In the first two algorithms different matrixdependentprolongation operators are implemented. Figure 3 shows one coarse andfour fine grid cells with indices for the explanation of the different prolongationweights. The first operator is well known in the literature ([3], [6]) and has beenused before by the authors in [23]. Problems on grids with arbitrary grid sizes can besolved with a similar efficiency. So it is not necessary to choose powers of two (+1)for the grid sizes. The prolongation mapping P L+1 : u L ∈ Q GL ↦→ u L+1 ∈ Q GL+1with weights w1, w2, and w3 appears as follows:• for (2i, 2j − 1) ∈ Q GL+1d 1 = a 1 2i,2j−1 + a4 2i,2j−1 + a7 2i,2j−1 ,d 2 = a 3 2i,2j−1 + a6 2i,2j−1 + a9 2i,2j−1 ,d = −(a 2 2i,2j−1 + a5 2i,2j−1 + a8 2i,2j−1 ),w1 2i,2j−1 = d1d ; w2 2i,2j−1 = d2d ;• for (2i − 1, 2j) ∈ Q GL+1d 1 = a 1 2i−1,2j + a2 2i−1,2j + a3 2i−1,2j ,d 2 = a 7 2i−1,2j + a8 2i−1,2j + a9 2i−1,2j ,d = −(a 4 2i−1,2j + a5 2i−1,2j + a6 2i−1,2j ),w1 2i−1,2j = d1d ; w3 2i−1,2j = d2d .The construction of d differs from the operators in [23] and results in an accurateprolongation, for example, near Dirichlet boundaries, when the grid size is not idealfor multigrid.On the remaining points the prolongation is found as follows:(21)(22)on points (2i-1,2j-1) u L+12i−1,2j−1 = uL i,j;on points (2i,2j) u L+12i,2j is determined so that AL+1 P L+1 u L = 0 at (2i,2j).Some matrix components in the algorithm disappear at boundaries. Therefore, theprolongation is also valid at a boundary. This first algorithm is denoted by MG1.
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