### Table 3: Preconditioned convergence factors for the multi-level method (accelerated by CGS) applied to the `staircase apos; problem.

1997

"... In PAGE 20: ...and, as discussed in [27], re ects the error better than that of the residual itself since the preconditioned system is better-conditioned than the original one.) The results in Table3 are as expected from Theorem 1 in the sense that the preconditioned convergence factor increases slowly with L so long as L 3. The deterioration of the convergence rate when L = 4 is used is due to the large p3 obtained in this case, which implies a large upper bound in (18).... ..."

Cited by 2

### Table 2: Multi-level density analysis on results from existing fixed-dissection filling methods. Notation: T/W/r: Layout / window size / r-dissection; LP: linear programming method; Greedy: Greedy method; MC: Monte-Carlo method; IGreedy: iterated Greedy method; IMC: iterated Monte-Carlo method; OrgDen: density of original layout; FD: fixed-dissection density analysis; Multi-Level: multi-level density analysis; MaxD: maximum window density; MinD: minimum window density; DenV: density

in ABSTRACT

"... In PAGE 3: ...rom an industry standard-cell layout4 (Table 2.3). Benchmark L1 is the M2 layer from an 8,131-cell design and benchmark L2 is the M3 layer from a 20,577-cell layout. Table2 shows that underestimation of the window density varia- tion as well as violation of the maximum window density in fixed- 3 For the test cases used in this paper, the runtimes of the multi- level analysis with accuracy =1:5#25 appear reasonable. Our (un- optimized) implementation has the following runtimes for Min-Var LP solutions and the spatial density model: L1/32 (45 sec), L1/16 (183 sec), L2/28 (99 sec), L2/14 (390 sec), For the effective density model, the runtimes are: L1/32 (49 sec), L1/16 (194 sec), L2/28 (109 sec), L2/14 (416 sec).... ..."

### Table 2: Multi-level density analysis on results from existing fixed-dissection filling methods. Notation: T/W/r: Layout / window size / r-dissection; LP: linear programming method; Greedy: Greedy method; MC: Monte-Carlo method; IGreedy: iterated Greedy method; IMC: iterated Monte-Carlo method; OrgDen: density of original layout; FD: fixed-dissection density analysis; Multi-Level: multi-level density analysis; MaxD: maximum window density; MinD: minimum window density; DenV: density

in ABSTRACT

"... In PAGE 3: ...rom an industry standard-cell layout4 (Table 2.3). Benchmark L1 is the M2 layer from an 8,131-cell design and benchmark L2 is the M3 layer from a 20,577-cell layout. Table2 shows that underestimation of the window density varia- tion as well as violation of the maximum window density in fixed- 3 For the test cases used in this paper, the runtimes of the multi- level analysis with accuracy =1:5#25 appear reasonable. Our (un- optimized) implementation has the following runtimes for Min-Var LP solutions and the spatial density model: L1/32 (45 sec), L1/16 (183 sec), L2/28 (99 sec), L2/14 (390 sec), For the effective density model, the runtimes are: L1/32 (49 sec), L1/16 (194 sec), L2/28 (109 sec), L2/14 (416 sec).... ..."

### Table 2: Multi-level density analysis on results from existing xed-dissection lling methods. Notation: T/W/r: Layout / window size / r-dissection; LP: linear programming method; Greedy: Greedy method; MC: Monte-Carlo method; IGreedy: iterated Greedy method; IMC: iterated Monte-Carlo method; OrgDen: density of original layout; FD: xed-dissection density analysis; Multi-Level: multi-level density analysis; MaxD: maximum window density; MinD: minimum window density; DenV: density variation.

"... In PAGE 4: ... (a): a xed-dissection, where each window consists of 5 5 cells (the same size as tiles); (b): a xed-dissection for post- lling den- sity analysis, where each window consists of 10 10 smaller tiles and each cell consists of 2 2 tiles. Table2 shows that underestimation of the window density variation as well as violation of the maximum window den- sity in xed-dissection lling can be severe: e.g.... ..."

### Table 2. Multi-level approach to workplace studies.

2002

"... In PAGE 7: ... This addresses the widely recognized problem of ethnographic approaches that, while they can provide an understanding of current work practices, they are not intended to explore the consequences of socio-technical change. Table2 shows a multi-level structure for workplace studies, with level 1 consisting of a survey of the existing organizational structures and schedules, levels 2 and 3 providing an analysis of situated practices and interactions of those for whom the technology is intended, and level 4 offering a synthesis of the findings in terms of designs for new socio-technical systems. The four levels of the approach give an overview of workplace activity leading to more detailed investigation of particular problem areas, with each level illuminating the situated practices, and also providing a set of issues to be addressed for the next level.... ..."

Cited by 9

### Table 2 shows the performance improvement of the multi-level optimization over the usual optimization method.

"... In PAGE 10: ...5 W 0 0 100 200 300 400 CPU Time (msec) Figure 8: Convergence Curve for Bending Cow. Table2 : Performance Improvement by Multi-Level Optimization. 7, and lt; are the total CPU time (msec) without and with Multi Level Optimization respectively.... ..."

### Table I Total execution time and accuracy for LDM and Multi-level method on Calcium molecule and BPTI molecule

### Table 3: Preconditioned convergence factors for the multi-level method (accelerated by CGS) applied to the `staircase apos; problem.

"... In PAGE 27: ... This is an advantage of left preconditioning over right and symmetric ones. The results in Table3 are as expected from Theorem 1 in the sense that the preconditioned convergence factor increases slowly with L so long as L 3. The deterioration of the convergence rate when L = 4 is used is due to the large p3 obtained in this case, which implies a large upper bound in (21).... ..."

### Table 1: Scalability study for two trebecular bone datasets using the Multi-Level Image Cache system. The rates are

2003

Cited by 2