### 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 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 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: Multi-level MRS for Test Problem 1 with two cycling algorithms.

"... In PAGE 7: ...ycle algorithms. We applied MRS on the rst, second and third nest grids. The com- bined multi-level MRS schemes are referred to as single-level MRS ( rst nest grid only), double-level MRS ( rst and second nest grids only) and triple-level MRS ( rst, second and third rst grids). The iteration counts are given in Table1 . The convergence histories of a particular case are depicted in Fig.... In PAGE 7: ... 1 shows that the multi-level MRS schemes are very e cient as an acceleration technique and the multi-level MRS performed better than the single-level MRS. These claims are further supported by results in Table1 . Even in the case of quot; = 10?1, we did not see a reduction in iteration count, we did observe acceleration in our experiments, which was just not substantial enough to reduce one additional iteration because the standard multigrid method converged very fast.... ..."

### Table 1: Multi-level MRS for Test Problem 1 with two cycling algorithms.

"... In PAGE 7: ...ycle algorithms. We applied MRS on the rst, second and third nest grids. The com- bined multi-level MRS schemes are referred to as single-level MRS ( rst nest grid only), double-level MRS ( rst and second nest grids only) and triple-level MRS ( rst, second and third rst grids). The iteration counts are given in Table1 . The convergence histories of a particular case are depicted in Fig.... In PAGE 7: ... 1 shows that the multi-level MRS schemes are very e cient as an acceleration technique and the multi-level MRS performed better than the single-level MRS. These claims are further supported by results in Table1 . Even in the case of quot; = 10?1, we did not see a reduction in iteration count, we did observe acceleration in our experiments, which was just not substantial enough to reduce one additional iteration because the standard multigrid method converged very fast.... ..."

### Table 1: Multi-level MRS for Test Problem 1 with two cycling algorithms.

"... In PAGE 7: ...ycle algorithms. We applied MRS on the rst, second and third nest grids. The com- bined multi-level MRS schemes are referred to as single-level MRS ( rst nest grid only), double-level MRS ( rst and second nest grids only) and triple-level MRS ( rst, second and third rst grids). The iteration counts are given in Table1 . The convergence histories of a particular case are depicted in Fig.... In PAGE 7: ... 1 shows that the multi-level MRS schemes are very e cient as an acceleration technique and the multi-level MRS performed better than the single-level MRS. These claims are further supported by results in Table1 . Even in the case of quot; = 10?1, we did not see a reduction in iteration count, we did observe acceleration in our experiments, which was just not substantial enough to reduce one additional iteration because the standard multigrid method converged very fast.... ..."

### Table 1. Algorithm parameters during multi-level non-rigid registration.

2002

"... In PAGE 5: ... In parallel, the image data resolution was refined from 4 mm voxel size to 2 mm, 1 mm, until at the final stage the orig- inal image data was used. Table1 gives an overview of the parameters used at each level for fixed-constraint and alternating-constraints non-rigid registration. A key question to answer is whether a given technique provides a way of com- pensating motion artifacts while preventing volume loss of contrast-enhancing structures.... ..."

Cited by 3

### 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).... ..."