### Table 1 Block coordinate descent algorithm

"... In PAGE 6: ... The resulting CG scheme is ensured to converge to the unique minimizer of J as a function of k, under constraint (4). See Table1 for the detailed algorithm.... ..."

### Table IV. init2D-A3: Convex iteration space. block block-like cyclic-like cyclic

### Table 1 Convex

"... In PAGE 4: ... The next step of the analysis, however, provided a more promising result. As shown in Table1 , the number of people visible from each convex space was consistently correlated not only with the visual range of the space but also with its integration into the setting as a whole. That more people are visible from spaces which have a stronger visual range is hardly surprising.... In PAGE 5: ... However, these correlations were neither very strong or consistent. Table1 . Correlation between the Number of People Visible from Each Convex Space with Convex Configuration Variables.... ..."

### Table 5: Convex quadratics: log barrier method

"... In PAGE 30: ... The convergence criterion for the inner iteration was krx (xk;j; k)k 10?7krx (xk;0; k)k: We used 0 := 105 and := 2 10?1. Table5 gives the results using the log barrier algorithm. Table 6 gives the results using the reciprocal barrier algorithm.... ..."

### Table 2: Programs in our experiment set and their characteristics ( + = block, block-cyclic(4) or cyclic; * = not distributed).

1997

"... In PAGE 14: ... communication in SendSet Pending 1 f(1,0),(2,1),(3,2),(4,3),(5,4),(6,5),(7,6)g f(0,1),(1,2),(2,3),(3,4),(4,5),(5,6),(6,7)g 2 f(0,1),(1,2),(2,3)g f(1,0),(2,1),(3,2)g 3 f(3,4),(4,5),(5,6),(6,7)g f(4,3),(5,4),(6,5),(7,6)g PENDING OUT for iteration 1 iteration 2 1 ff(0;1);(1;2);(2;3);(3;4);(4;5);(5;6);(6;7)g;f;g;f;gg ff;g;f;g;f;gg 2 ff(3;4);(4;5);(5;6);(6;7)g;f;g;f;gg ff;g;f;g;f;gg 3 ff;g;f;g;f;gg ff;g;f;g;f;gg nication and n2 loops after the communication; consequently it is more powerful than the one offered by Gupta and Schonberg in [6]. 6 Preliminary Results The applications used in our study and their characteristics are listed in Table2 . We experimented with BLOCK, B-CYC (block-cyclic) and CYCLIC distributions on 8 and 32 processors to measure the static improvements in communication volume, number of data messages and number of synchronization messages.... ..."

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### Table 2: Programs in our experiment set and their characteristics ( + = block, block-cyclic(4) or cyclic; * = not distributed).

1997

"... In PAGE 14: ... communication in SendSet Pending 1 f(1,0),(2,1),(3,2),(4,3),(5,4),(6,5),(7,6)g f(0,1),(1,2),(2,3),(3,4),(4,5),(5,6),(6,7)g 2 f(0,1),(1,2),(2,3)g f(1,0),(2,1),(3,2)g 3 f(3,4),(4,5),(5,6),(6,7)g f(4,3),(5,4),(6,5),(7,6)g PENDING OUT for iteration 1 iteration 2 1 ff(0;1);(1;2);(2;3);(3;4);(4;5);(5;6);(6;7)g;f;g;f;gg ff;g;f;g;f;gg 2 ff(3;4);(4;5);(5;6);(6;7)g;f;g;f;gg ff;g;f;g;f;gg 3 ff;g;f;g;f;gg ff;g;f;g;f;gg nication and n2 loops after the communication; consequently it is more powerful than the one offered by Gupta and Schonberg in [6]. 6 Preliminary Results The applications used in our study and their characteristics are listed in Table2 . We experimented with BLOCK, B-CYC (block-cyclic) and CYCLIC distributions on 8 and 32 processors to measure the static improvements in communication volume, number of data messages and number of synchronization messages.... ..."

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### Table 1: Comparison of convergence results for the energy and other crucial quantities for di erent nite element methods. See the text for explanation of the notation.

1998

"... In PAGE 3: ... However, for the present problem of a non-convex energy density, the results are rather sobering: In general, it can only be shown that a minimizing deformation uh 2 Ah satis es E(uh) Ch1=2; (10) where C denotes a generic constant that may depend on the topology of the quasiuniform triangulation Th and the domain but not on the mesh-size h, see [8, 21, 22], and [7] for a de nition of quasiuniformity. For a complete list of results for important quantities, see Table1 . Moreover, it turns out that the quality of the approximation depends strongly on the degree of alignment of the numerical mesh with the physical laminates.... In PAGE 4: ... To this end, we present a new algorithm based on discontinuous nite elements. It will be shown that this algorithm allows much improved convergence rate estimates for the energy, namely O(h2), and other quantities of interest as they are given in Table1 . In particular, the resolution of laminate microstructure on general meshes is much better than by the classical (non-)conforming discussed above.... In PAGE 7: ... In the case (d), it additionally depends on the choice of the value of . Again, we stress the fact that these convergence results are much better than those derived for the conforming (using (bi-, tri-)linear ansatz functions, see [21]) or classical non-conforming (using piecewise rotated (bi-,tri)linear ansatz functions, see [20, 21]) nite element methods, see also Table1 . This re ects the increased accuracy of the ansatz for non-aligned meshes: The misaligned triangulation does not lead to a dramatic pollution of the computed solution anymore.... ..."

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### Table 1 shows that the choice of whether to use wavelet{vaguelette decomposition or vaguelette{wavelet decomposition does not make a strong di erence in terms of average mean square error, nor does the choice between the di erent wavelet functions. The largest discrepancy between the two wavelet-based methods is the improvement a orded by using vaguelette{wavelet decomposition for the HeaviSine function.

1998

"... In PAGE 10: ... The mother wavelets D4 and D8 were used in the wavelet{vaguelette decomposition and D5 and D9 in the vaguelette{wavelet decom- position. The various average mean square errors yielded by the application of the exact risk formulae are given in Table1 . To make a comparison with singular value decomposition, Table 1 near here we calculated the minimal average mean square error for a truncated singular value de- composition estimator with optimally chosen cut-o point M.... In PAGE 10: ... To make a comparison with singular value decomposition, Table 1 near here we calculated the minimal average mean square error for a truncated singular value de- composition estimator with optimally chosen cut-o point M. The results are also given in Table1 . Table 2 gives the optimal values of the thresholds in terms of , each found by a grid search at grid interval 0:05 , where is the standard deviation of the noise.... In PAGE 11: ... Thus, we can examine the convergence for di erent methods by studying their performance as a function of signal-to-noise ratio. Table1 clearly indicates that the rates of convergence for the wavelet-based methods are much faster than that of the singular value decomposition approach, especially for the `blocks apos; and `Doppler apos; functions. The theoretical ground for this phenomenon is given in Section 5.... In PAGE 17: ...19] Tikhonov, A.N. amp; Arsenin, V.Y. (1977). Solutions of Ill-posed Problems. New York: Wiley. Table1 : Exact ideal average mean square error for the estimation of various test functions using the singular value decomposition (SVD), wavelet{vaguelette decomposition(WVD), and vaguelette{wavelet decomposition(VWD) approaches, for various levels of the signal- to-noise ratio SNR. SNR SVD WVD (D4) VWD (D5) WVD (D8) VWD (D9) Bumps 10 0.... ..."

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