### Table 2: Results for the CELAR problems using Double-Update Boltzmann Machine

"... In PAGE 15: ... According to this approach, the connection weights are not computed and stored before network operation, but the value of each weight is determined only when it is used in a computation. Table2 summarizes the obtained results, which concern the quality of the solution, as well as mean and standard deviation of execution time. For the rst four problems the displayed solution gure concerns the number of di erent frequencies used, whereas the largest frequency used is shown for scen05.... ..."

### Table 4: Weight discretization in other neural network models.

"... In PAGE 5: ...2 Quantization E ects in Other Neural Network Models Also for other neural network models the e ects of a coarse quantization of the weightvalues on recall and learning have been investigated. The small number of weight discretization algorithms proposed can be partly explained from the fact that the required accuracy for successful learning in these models is lower than for gradient descent learning in multilayer networks ( Table4 ). An interesting example of a hardware implementation is Bellcore apos;s implementation of a Boltzmann machine and Mean-Field learning, whichallows on-chip learning with only 5-bit weights [Alspector-92].... ..."

### Table 1: Annealing Schedule for the CELAR problems Double-update Boltzmann Machine CELAR optimal best found average CPU time standard deviation

"... In PAGE 12: ... The starting temperature T0 and the number of double-update trials per temperature were di erent for each problem instance. Table1 shows the various con gurations used, which were determined through systematic experimentation. The frequency of trials on the zeroth row was one per temperature step.... ..."

### Table 1: Comparative results for the Boltzmann and the Cauchy Machine = 0:25 = 0:50 = 0:75 n Avg. Time Avg. Cost Avg. Time Avg. Cost Avg. Time Avg. Cost

"... In PAGE 8: ... The results concern both the e ectiveness (quality of the obtained solutions) and the e ciency (execution time). The results shown in Table1 concern the Cauchy and the Boltzmann Machine considered individually. It can be observed that the Boltzmann Machine is clearly superior in both the quality of the obtained solutions and the execution time in sequential implementation.... In PAGE 10: ... Note that speedup is computed in reference to the sequential execution time of the algorithm on the same machine. If we combine the results shown in Tables 2 and 4, referring to execution of the sequential and the parallel implementation of the proposed method on the same ma- chine (shared memory), and compare them with the results shown in Table1 , it is interesting to see that the hybrid optimization method clearly outperforms the Boltzmann Machine in terms of execution time when implemented in parallel, for any instance of the problem. Even better execution times are expected if more processors are used.... ..."

### Table 4: Results of the Boltzmann machine implemented for the mapping problem Parallel Implementation: The Boltzmann Machine is from its nature a massively parallel algorithm that should be easy to implement on existing parallel systems. Unfortunately even small problems result in a large and very dense unit graph. Representing each unit by one process cuts down the e ciency due to the large number of context switches on each processor. Therefor the processors run only single processes simulating all units that are mapped onto them. 23

1993

"... In PAGE 24: ...between the processors. Table4 shows the results of the Boltzmann machine applied to the mapping problem. The solution quality is slightly better than that one of the genetic algorithm, although the results are far away for those that are achieved by our simulated annealing implementation.... ..."

Cited by 3

### Table 5: Speedup for distributed memory implementation. especially for large problem instances. If we compare Tables 1, 2 and 4, we will see that for n = 1600 the execution time of parallel Group Update using 8 processors is even less than that of the Boltzmann Machine, while for the smaller sizes execution times are comparable.

1995

"... In PAGE 16: ... Then it informs the worker processors about the changes and continues until the stopping criterion is met. The results in Table5 show good speedup obtained on the Paragon machine, especially in large instances of our problem. From Tables 4{5 we can see that in both machines we obtained very good speedup,... ..."

Cited by 2

### TABLE A.2. Stability restrictions on = = x for advection. Stability restrictions for di usion equation ct = cxx are obtained in a similar way. The relevant eigenvalues for the standard di usion discretizations of order 2 and 4 are

### Table 1. Benchmarks for restricted and unrestricted IAP.

2008

"... In PAGE 12: ... Thus, beyond 8 agents, it is hard to know whether reduced speedups are due to our parallelization and implementation or to limitations of the machine. Although most of the benchmarks we use are quite well-known, Table1 pro- vides a brief description. Speedups appear in Tables 2 (which contains only programs parallelized using restricted [N]SIAP, as in Figure 3) and 3 (which ad- ditionally contains unrestricted IAP programs, as in Figure 4).... ..."

Cited by 2

### Table 1. Benchmarks for restricted and unrestricted IAP.

2008

"... In PAGE 12: ... Thus, beyond 8 agents, it is hard to know whether reduced speedups are due to our parallelization and implementation or to limitations of the machine. Although most of the benchmarks we use are quite well-known, Table1 pro- vides a brief description. Speedups appear in Tables 2 (which contains only programs parallelized using restricted [N]SIAP, as in Figure 3) and 3 (which ad- ditionally contains unrestricted IAP programs, as in Figure 4).... ..."

Cited by 2

### Table 1. Identi cations for Multicomponent Lattice Boltzmann Gas Lattice Boltzmann Helium II

"... In PAGE 9: ... However, gradients in the passive scalar eld a ect the ow be- cause the equation of state for the uid depends on this passive scalar. Table1 lists the lattice Boltzmann quantities in correspondence to like quantities in Helium II. For small Mach number and in the incompressible regime, the continuity equation,... ..."