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160 G. Bencteux et al. insensitive to entries D ij with indices (i, j) such that |H ij | is below some cut-off value. In all the calculations presented below, the global step is performed with groups consisting of two blocks (r = 2), and the algorithm is, therefore, exactly that displayed in Figure 4. 5.2 Sequential Computations The numerical results presented in this section have been obtained with a single 2.8 GHz Xeon processor. Density matrices have been computed for a series of matrices H and S of types P 1 , P 2 ,andP 3 , using (1) the MDD algorithm, (2) a diagonalization procedure (the dsbgv.f routine from the LAPACK library), and (3) the DMM method [LNV93]. The latter method belongs to the class of linear scaling algorithms. An important feature of the DMM method is that linear scaling is achieved through cut-offs on the matrix entries. We have chosen here a cut-off strategy based on a priori defined patterns, that may be suboptimal. Our implementation of DMM converges to a fairly good approximation of the exact density matrix and scales linearly, but the prefactor might possibly be improved by more refined cut-off strategies. A detailed presentation of the comparison between the three methods is provided in [BCHL07]. Our new approach for computing the Newton direction in the global step (see Section 3) further improves the efficiency of MDD: with the new implementation of MDD, and with respect to the former implementation reported on in [BCHL07], CPU time is divided by 2 for P 1 type molecules, by 5 for P 2 , and by 10 for P 3 , and the memory required is now lower for MDD than for DMM. These results are shown for P 2 in Figures 6 and 7. They clearly demonstrate that the MDD algorithm scales linearly with respect to the parameter n m (in both CPU time and memory occupancy). Let us also notice that for P 2 , the crossover point between diagonalization and MDD (as far as CPU time is concerned) is now shifted to less than 2,000 basis functions. 5.3 Parallel Computations We conclude with some tests of our parallel implementation of the MDD algorithm described in Section 4. These tests have been performed on a 8 node Linux cluster in dedicated mode, consisting of 8 biprocessors DELL Precision 450 (Intel(R) Xeon(TM) CPU 2.40GHz), with Gigabit Ethernet connections. They concern the polyethylene family P 3 , for which the size of each ghost block is about 150 Ko. We only test here the highest level of parallelism of the MDD algorithm, consistently with the relatively low number of processors that have been used in this first study. We plan to test multilevel parallelism in a near future.
Domain Decomposition Approach for Computational Chemistry 161 1e+08 1e+07 LAPACK DMM MDD 1e+06 CPU Time in seconds 100000 10000 1000 100 10 100 1000 10000 100000 1e+06 N b Fig. 6. Requested CPU time for computing the density matrix of a molecule of type P 2 as a function of the number of basis functions. 1e+09 LAPACK DMM MDD 1e+08 Memory requirement in Kbytes 1e+07 1e+06 100000 10000 1000 100 1000 10000 100000 1e+06 N b Fig. 7. Requested memory for computing the density matrix of a molecule of type P 2 as a function of the number of basis functions. In particular, the local step in each block, as well as the global step in each group, will be parallelized. Tables 2 and 3 report on the speedup (ratio between the wall clock time with one processor and the wall clock time for several processors) and efficiency (ratio between the speedup and the number of processors) of our parallel MDD algorithm. The scalability, namely the variation of the wall clock time when the number of processors and the size of the matrix proportionally grow, is reported in Table 4, for a molecule of type P 3 .
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Partial Differential Equations
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Partial Differential Equations Mode
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Dedicated to Olivier Pironneau
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VIII Preface computers has been at
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Contents List of Contributors .....
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List of Contributors Yves Achdou UF
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List of Contributors XV Claude Le B
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Discontinuous Galerkin Methods Vive
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Discontinuous Galerkin Methods 5 2
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Discontinuous Galerkin Methods 7 g
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Discontinuous Galerkin Methods 9 (
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Discontinuous Galerkin Methods 15 W
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Let a h and b h denote the bilinear
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Discontinuous Galerkin Methods 19 t
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Discontinuous Galerkin Methods 21 l
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Table 1. Primal DG for transport Di
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Discontinuous Galerkin Methods 25 [
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Mixed Finite Element Methods on Pol
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Mixed FE Methods on Polyhedral Mesh
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4 Hybridization and Condensation Mi
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is symmetric and positive definite,
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with some coefficient α ∈ R wher
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44 E.J. Dean and R. Glowinski so fa
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54 E.J. Dean and R. Glowinski Fig.
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60 E.J. Dean and R. Glowinski Fig.
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62 E.J. Dean and R. Glowinski Assum
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Higher Order Time Stepping for Seco
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u n+1 h Optimal Higher Order Time D
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Optimal Higher Order Time Discretiz
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96 I. Sazonov et al. To provide a p
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Computing the Eigenvalues of the La
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Eigenvalues of the Laplace-Beltrami
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A Fixed Domain Approach in Shape Op
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Shape Optimization Problems with Ne
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