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Ant Stigmergy on the Grid - Computer Systems @ JSI - IJS

Ant Stigmergy on the Grid - Computer Systems @ JSI - IJS

4 Experiments and

4 Experiments and results 4.1 Experimental environment Evaluation of cooling patterns and their assessment with respect to the cost function (Eq. 9) was done by means of a numerical simulator [15]. Its principal task is to dynamically track the temperature field in the slab as a function of process parameters. It involves a 3D model of the slab and finite element numerical approximation. In this study it was applied under the assumption of steady-state caster operation, and the search for optimal cooling patterns performed in the off-line manner. A single simulator run takes about 25 seconds on a 1.8 GHz AMD Opteron computer. Table 2. Parameter discretizations Pos. Zone no. Parameter p step i [m 3 /h] s i 1 p 1 0.1 191 2 p 2 0.1 348 c 3 p 3 0.1 267 e 4 p 4 0.1 65 n 5 p 5 0.1 74 t 6 p 6 0.1 58 e 7 p 7 0.1 53 r 8 p 8 0.1 49 9 p 9 0.1 51 1 p 10 0.1 191 2 p 11 0.1 348 c 3 p 12 0.1 267 o 4 p 13 0.1 24 r 5 p 14 0.1 21 n 6 p 15 0.1 6 e 7 p 16 0.1 53 r 8 p 17 0.1 49 9 p 18 0.1 51 Note that the parameter discretization is important part of the optimization process. According to experimental analyze of different discretizations [9], the uniform step size was chosen (see Table 2). The number of possible parameter settings can be obtained as follows. For a parameter p i from the interval [p min i ,p max i ] with step size p step i ,thereares i = ⌊(p max i − p min i )/p step i ⌋ +1 values ∏ possible, and the total number of settings is s = D i=1 s i,whereD is the number of parameters. This results in s =4.7 · 10 33 for step size p step i =0.1m 3 /h. The computer platform used to perform the experiments was an eight-node Grid connected via Giga-bit switch, each node consisted of two AMD Opteron TM 1.8 GHz processors and 2 GB of RAM. We used a homogeneous Grid, so we could estimate the speed-up. But this is not a requirement for a real-life applications. Clients and server used a standard TCP/IP protocol to communicate between each other. In the experiments, the stopping criterion for the DMASA was given by the number of solution evaluations. It was set to 768 (eight ants × eight levels × 12 climb-downs per level) evaluations per run and this value was chosen considering the computational complexity of the optimization procedure. Other algorithm parameters were set as follows. The DMASA operated with eight levels, at each level ants in sub-colony climbs down the graph until the “level ending condition” is not met (96 evaluations per level in all sub-colonies together). Total number of ants in all sub-colonies is eight, while the number of ants in each sub-colony depends on the number of nodes in the Grid as shown in Table 3. Table 3. Distribution of ants Num. of nodes in the Grid 1 2 4 8 Number of sub-colonies 1 2 4 8 ong>Antong>s per sub-colony 8 4 2 1 Table 4. The optimized coolant flows Pos. Zone no. T [ ◦ C] Parameter p ′ i [m 3 /h] 1 1048.5 p 1 17.2 2 1039.3 p 2 33.0 c 3 978.3 p 3 20.9 e 4 977.6 p 4 7.9 n 5 960.3 p 5 9.2 t 6 950.0 p 6 5.8 e 7 940.0 p 7 3.1 r 8 930.1 p 8 3.2 9 919.8 p 9 2.1 1 880.5 p 10 8.4 2 863.3 p 11 22.8 c 3 803.5 p 12 15.6 o 4 836.0 p 13 3.5 r 5 804.7 p 14 4.4 n 6 783.0 p 15 2.8 e 7 773.8 p 16 1.3 r 8 760.9 p 17 1.0 9 720.6 p 18 1.2 The cost of this solution is 9070.4. For each distribution (N = 1, 2, 4 or 8), we ran the algorithm 30 times and compared the solution quality and computational time.

4.2 Results In this section we present and discuss the results of the experimental evaluation of the DMASA algorithm employed with different degrees of distribution (number of nodes used in the Grid). Table 4 shows the best parameter setting p ′ found so far [9]. This is the solution obtained in every run regardless of the degree of distribution. For this reason we further concentrate only on the time needed to compute the solution and not on the solution quality. Table 5. Computation time T [hours:minutes] needed by the DMASA and average speed-up Num. of nodes in the Grid 1 2 4 8 T min 8:43 5:06 3:57 3:02 T max 10:55 7:56 6:36 5:15 T avg 9:40 6:30 4:54 4:05 Std 0:34 0:33 0:33 0:38 speed-up 1.00 1.49 1.97 2.37 In Table 5 one can see the computational times needed by the DMASA on different number of nodes in the Grid. It is shown that we managed to decrease time quite noticeably from almost ten hours on one node to under four hours on the Grid with eight nodes. The standard deviation remained almost the same through all evaluations, between 33 and 41 minutes. c(T) 100000 80000 60000 40000 20000 c(T') = 9070.4 N=1 N=2 N=4 N=8 0 0 2 4 6 8 10 time [h] Figure 4. Convergence of the DMASA Figures 4 and 5 show the convergence and speed-up speed-up 2.5 2 1.5 1.49 1.97 1 1 2 3 4 5 6 7 8 N (nodes in the Grid) Figure 5. Speed-up of the DMASA of the DMASA that was achieved with different number of nodes. We can see that the convergence is improved with higher number of nodes and the speed-up is not linear but still quite evident. 5 Conclusions Optimization of coolant flow settings in continuous casting of steel is a key to higher product quality. It is nowadays to a high degree performed through virtual experimentation involving numerical process simulators and advanced optimization techniques. In this study of optimizing 18 cooling water flows for a Ruukki casting machine under steady-state conditions, a new distributed search algorithm based on stigmergy perceived in ant colony was applied and its performance evaluated. The results indicate the importance of the applied search space discretization [9] and suggest the construction of a hybrid algorithm to find near-optimal solutions in smaller number of solution evaluations. We have shown that with the Grid computing the computation time can be drastically decreased (from half a day to a few hours) without any decrease in the solution quality. It is our interest to further investigate the performance of the proposed DMASA algorithm under different scenarios. The work can be extended with Grid middleware [19], such as the Globus Toolkit [11], Condor [21], or UNICORE (Uniform Interface to Computer Resources) [17]. References [1] N. Chakraborti, R. Kumar, D. Jain. A study of the continuous casting mold using a Pareto- 2.37

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