1 00:00:00,469 --> 00:00:03,859 in our lecture on numerical methods 2 00:00:03,859 --> 00:00:06,810 today I'd like to do a small coding 3 00:00:06,810 --> 00:00:11,130 session and we discussed the Monte Carlo 4 00:00:11,130 --> 00:00:14,969 methods and somewhere in the middle 5 00:00:14,969 --> 00:00:18,600 there was this little experiment 6 00:00:18,600 --> 00:00:21,630 described how you can use a Monte Carlo 7 00:00:21,630 --> 00:00:26,910 approximation to approximate P okay so 8 00:00:26,910 --> 00:00:31,560 you find this in the script the thing is 9 00:00:31,560 --> 00:00:33,950 that four times the moniker integral of 10 00:00:33,950 --> 00:00:39,930 the indicator function on a two 11 00:00:39,930 --> 00:00:46,470 dimensional random vector ax so 12 00:00:46,470 --> 00:00:51,870 eggs and why 13 00:00:51,870 --> 00:01:00,539 uniform on zero one gives you an 14 00:01:00,539 --> 00:01:06,250 integral and this integral is actually 15 00:01:06,250 --> 00:01:11,310 equal to P so we can approximate P by 16 00:01:11,310 --> 00:01:14,380 calculating this on the counter in 17 00:01:14,380 --> 00:01:17,259 sequence and the idea behind this was 18 00:01:17,259 --> 00:01:18,549 also in the script 19 00:01:18,549 --> 00:01:21,280 just to have you this if you sample 20 00:01:21,280 --> 00:01:26,590 random points in the unit hypercube then 21 00:01:26,590 --> 00:01:30,270 the probability to be inside the circle 22 00:01:30,270 --> 00:01:35,069 is the ratio of the area of the circle 23 00:01:35,069 --> 00:01:39,190 with respect to the area of the unit 24 00:01:39,190 --> 00:01:42,970 hypercube okay and so then we do the 25 00:01:42,970 --> 00:01:47,830 transformation to just the upper 26 00:01:47,830 --> 00:01:50,920 secretary of the quarter from 0 to 1 and 27 00:01:50,920 --> 00:01:55,989 then we get four times where this 28 00:01:55,989 --> 00:02:03,310 integral okay this gives us P okay so 29 00:02:03,310 --> 00:02:07,269 and then later in the script we have the 30 00:02:07,269 --> 00:02:12,470 remark that Monte Carlo 31 00:02:12,470 --> 00:02:17,060 is a very good technique if you like to 32 00:02:17,060 --> 00:02:21,460 perform parallelization so why is that 33 00:02:21,460 --> 00:02:28,460 okay so there was the experiment here 34 00:02:28,460 --> 00:02:32,840 that we should use this algorithm of 35 00:02:32,840 --> 00:02:38,780 approximating P and parallelize it say 36 00:02:38,780 --> 00:02:41,290 for example to make use of multiple 37 00:02:41,290 --> 00:02:47,630 processor cores so I would like to do a 38 00:02:47,630 --> 00:02:50,330 small excursion on parallelization so 39 00:02:50,330 --> 00:02:53,090 teach you a little bit about this and 40 00:02:53,090 --> 00:02:55,040 actually you will see that with some 41 00:02:55,040 --> 00:02:57,920 frameworks some Java frameworks it's 42 00:02:57,920 --> 00:03:00,050 actually very easy it's just writing one 43 00:03:00,050 --> 00:03:03,280 additional board and you are done 44 00:03:03,280 --> 00:03:06,920 okay so the point here is that the Monte 45 00:03:06,920 --> 00:03:15,620 Carlo method uses a sequence of iid so 46 00:03:15,620 --> 00:03:18,610 it uses a sequence of independent 47 00:03:18,610 --> 00:03:21,860 identically distributed random variables 48 00:03:21,860 --> 00:03:25,580 and the important aspect here is the 49 00:03:25,580 --> 00:03:27,950 independent so if the random variables 50 00:03:27,950 --> 00:03:31,370 are independent actually this directly 51 00:03:31,370 --> 00:03:34,370 implies that the calculations we have to 52 00:03:34,370 --> 00:03:38,420 perform on these samples are independent 53 00:03:38,420 --> 00:03:44,549 so each individual calculation 54 00:03:44,549 --> 00:03:48,760 can be performed independently so we can 55 00:03:48,760 --> 00:03:52,719 very easily split the vac into say 56 00:03:52,719 --> 00:03:57,549 smaller parts and then distribute these 57 00:03:57,549 --> 00:04:00,489 smaller parts to their own processor 58 00:04:00,489 --> 00:04:06,879 core or like on the GPU - or to a GPU 59 00:04:06,879 --> 00:04:16,090 set okay so the idea is that we use a 60 00:04:16,090 --> 00:04:23,050 random sequence and we just split it in 61 00:04:23,050 --> 00:04:30,550 two blocks so say we have blocks of size 62 00:04:30,550 --> 00:04:41,680 M so then we have J times M plus I 63 00:04:41,680 --> 00:04:46,840 from i from 0 to m minus 1 so each 64 00:04:46,840 --> 00:04:52,070 subsequence has m elements and then we 65 00:04:52,070 --> 00:04:56,720 have K such sucks subsequence such that 66 00:04:56,720 --> 00:05:02,830 K times M is the total number so we have 67 00:05:02,830 --> 00:05:14,979 K sub sequences J from 0 to K minus 1 68 00:05:14,979 --> 00:05:19,280 okay and then we can perform a Monte 69 00:05:19,280 --> 00:05:23,120 Carlo integral on each of these sub 70 00:05:23,120 --> 00:05:25,819 sequences calculate the sum and then the 71 00:05:25,819 --> 00:05:28,300 total sum is just the sum of the 72 00:05:28,300 --> 00:05:35,289 individuals okay so to implement this 73 00:05:35,289 --> 00:05:38,900 parallelization actually sometimes there 74 00:05:38,900 --> 00:05:42,740 are frameworks that help us in splitting 75 00:05:42,740 --> 00:05:47,150 up things so actually we don't even have 76 00:05:47,150 --> 00:05:53,419 to do this splitting so how do we get 77 00:05:53,419 --> 00:05:57,259 these sub sequences so we have several 78 00:05:57,259 --> 00:06:04,129 options of generating such sub sequences 79 00:06:04,129 --> 00:06:07,639 so first of all if you get back to this 80 00:06:07,639 --> 00:06:14,010 slide you see here that 81 00:06:14,010 --> 00:06:26,550 this sequence here starts at XJ times M 82 00:06:26,550 --> 00:06:31,350 plus zero and 83 00:06:31,350 --> 00:06:43,230 hence at XJ times M plus n minus 1 so if 84 00:06:43,230 --> 00:06:45,330 we would like to do Monte Carlo we would 85 00:06:45,330 --> 00:06:51,150 like to start all the sub tasks at the 86 00:06:51,150 --> 00:06:54,780 same time so actually we like to have a 87 00:06:54,780 --> 00:06:59,400 sequence where we can cut out a sub 88 00:06:59,400 --> 00:07:04,230 sequence and perform Monte Carlo on this 89 00:07:04,230 --> 00:07:05,870 sub sequence so if you have for example 90 00:07:05,870 --> 00:07:11,460 1,000 Monte Carlo samples so n is 1,000 91 00:07:11,460 --> 00:07:13,590 and you would like to have 10 sub 92 00:07:13,590 --> 00:07:16,350 sequences each with 100 Monte Carlo 93 00:07:16,350 --> 00:07:19,050 samples you would like to have a 94 00:07:19,050 --> 00:07:23,910 sequence that can start at X 100 X 200 X 95 00:07:23,910 --> 00:07:29,030 300 and so on so in order to do this a 96 00:07:29,030 --> 00:07:31,890 sequence where you can calculate the 97 00:07:31,890 --> 00:07:35,190 number at a certain index directly from 98 00:07:35,190 --> 00:07:38,390 that index has a certain advantage 99 00:07:38,390 --> 00:07:44,150 recall that for linear congruential 100 00:07:44,150 --> 00:07:56,260 generator 101 00:07:56,260 --> 00:08:08,180 we generated the next point from the 102 00:08:08,180 --> 00:08:12,950 previous point okay and so if we would 103 00:08:12,950 --> 00:08:15,740 like to start at X 100 maybe we have to 104 00:08:15,740 --> 00:08:21,620 run from X 0 to X 99 to actually get the 105 00:08:21,620 --> 00:08:24,770 starting point which is a bit useless 106 00:08:24,770 --> 00:08:26,330 because it means that we have to 107 00:08:26,330 --> 00:08:30,350 generate all other random numbers in in 108 00:08:30,350 --> 00:08:32,630 a sequence so we cannot paralyze the 109 00:08:32,630 --> 00:08:35,029 general generation of the random numbers 110 00:08:35,029 --> 00:08:37,310 so sometimes this is not an issue 111 00:08:37,310 --> 00:08:39,020 because the generation of the random 112 00:08:39,020 --> 00:08:41,300 numbers is fast so you first generate 113 00:08:41,300 --> 00:08:43,460 all the random numbers and then you 114 00:08:43,460 --> 00:08:46,820 calculate your function f of X when you 115 00:08:46,820 --> 00:08:49,430 do the Montecarlo integral so sometimes 116 00:08:49,430 --> 00:08:53,180 you only paralyze the evaluation of the 117 00:08:53,180 --> 00:08:55,430 function but you do not paralyze the 118 00:08:55,430 --> 00:09:01,130 generation of the random numbers if you 119 00:09:01,130 --> 00:09:04,360 are sold by going from the bottom 120 00:09:04,360 --> 00:09:07,070 through the top if you are in the 121 00:09:07,070 --> 00:09:11,290 special situation that your secrets 122 00:09:11,290 --> 00:09:15,410 calculates an element X I explicitly 123 00:09:15,410 --> 00:09:20,839 from the index so like for example in 124 00:09:20,839 --> 00:09:22,960 the funda coreboot and Horton sequence 125 00:09:22,960 --> 00:09:26,030 so then you can very easily paralyze 126 00:09:26,030 --> 00:09:28,400 this because you just say start at a 127 00:09:28,400 --> 00:09:34,660 certain index if this is not possible 128 00:09:34,660 --> 00:09:37,760 sometimes there are certain random 129 00:09:37,760 --> 00:09:41,650 number generators that have some kind of 130 00:09:41,650 --> 00:09:50,990 skip ahead possibility so you can find a 131 00:09:50,990 --> 00:09:55,130 formula that generates element 100 from 132 00:09:55,130 --> 00:09:58,400 say element 0 so you can find a formula 133 00:09:58,400 --> 00:10:01,459 that generates an archer step so you 134 00:10:01,459 --> 00:10:04,970 first have to do 135 00:10:04,970 --> 00:10:08,960 this step and if this is not possible 136 00:10:08,960 --> 00:10:11,180 say for example you have a pseudo-random 137 00:10:11,180 --> 00:10:13,940 number generator for example like math 138 00:10:13,940 --> 00:10:16,340 and twister and you do not find a 139 00:10:16,340 --> 00:10:18,650 formula that allows you to do this skip 140 00:10:18,650 --> 00:10:22,220 ahead then there is a dirty trick you 141 00:10:22,220 --> 00:10:24,160 can do yeah but it's maybe not 142 00:10:24,160 --> 00:10:27,740 recommended so if the sequence is 143 00:10:27,740 --> 00:10:29,300 generated from a seed 144 00:10:29,300 --> 00:10:32,270 yes so there is some initial value then 145 00:10:32,270 --> 00:10:35,510 you can consider to generate different 146 00:10:35,510 --> 00:10:41,300 sequences with different random seeds so 147 00:10:41,300 --> 00:10:43,220 this works there 148 00:10:43,220 --> 00:10:45,590 you know it's but it's maybe not an 149 00:10:45,590 --> 00:10:49,400 accurate solution because if you just 150 00:10:49,400 --> 00:10:53,950 consider that your sequence is like that 151 00:10:53,950 --> 00:10:58,550 okay these are now your numbers you 152 00:10:58,550 --> 00:11:01,610 generated and you are Montecarlo seed of 153 00:11:01,610 --> 00:11:04,310 the sequence is here this initial value 154 00:11:04,310 --> 00:11:06,710 this is the seed the starting point and 155 00:11:06,710 --> 00:11:09,050 it's generating the next point from the 156 00:11:09,050 --> 00:11:11,210 previous point and then you choose a 157 00:11:11,210 --> 00:11:16,640 random lis other seat and here you 158 00:11:16,640 --> 00:11:19,160 generate now a different Monte Carlo 159 00:11:19,160 --> 00:11:25,490 sequence then it may happen by chance 160 00:11:25,490 --> 00:11:28,930 that actually your Monte Carlo seed is 161 00:11:28,930 --> 00:11:32,170 identically to some number that appeared 162 00:11:32,170 --> 00:11:35,600 early in the other sequence which then 163 00:11:35,600 --> 00:11:38,180 means that actually the other sequence 164 00:11:38,180 --> 00:11:40,910 already contains 165 00:11:40,910 --> 00:11:45,610 these numbers and you see that the 166 00:11:45,610 --> 00:11:49,490 second sequence only has three 167 00:11:49,490 --> 00:11:52,519 additional points so the gain in 168 00:11:52,519 --> 00:11:57,110 accuracy is a little bit lower okay 169 00:11:57,110 --> 00:12:03,079 but still this technique of using random 170 00:12:03,079 --> 00:12:08,060 seats different seats to generate a 171 00:12:08,060 --> 00:12:10,560 sequence 172 00:12:10,560 --> 00:12:14,579 yeah so this is the seat 173 00:12:14,579 --> 00:12:22,080 and it may happen that the seed appears 174 00:12:22,080 --> 00:12:29,010 in the other sequence so it might happen 175 00:12:29,010 --> 00:12:35,030 there two sequences 176 00:12:35,030 --> 00:12:51,090 overlap overlap eloped okay so these are 177 00:12:51,090 --> 00:12:53,760 some options on how we can parallel eyes 178 00:12:53,760 --> 00:12:57,240 and now before I discuss something here 179 00:12:57,240 --> 00:13:03,930 on the coding let's do a code session on 180 00:13:03,930 --> 00:13:07,470 this so today I like to do coding with 181 00:13:07,470 --> 00:13:13,440 you and I will use two to api's two 182 00:13:13,440 --> 00:13:16,740 frameworks this Java stream API and the 183 00:13:16,740 --> 00:13:18,470 execute offender 184 00:13:18,470 --> 00:13:22,840 okay so let's go here and create a new 185 00:13:22,840 --> 00:13:24,500 [Music] 186 00:13:24,500 --> 00:13:27,090 experiment a new class so let's call 187 00:13:27,090 --> 00:13:33,890 this Monte Carlo integration experiment 188 00:13:33,890 --> 00:13:38,880 or Monte Carlo integration then 189 00:13:38,880 --> 00:13:42,270 experiment I would like to have a main 190 00:13:42,270 --> 00:13:50,840 method and I just implement some ways of 191 00:13:50,840 --> 00:13:58,310 paralyzing this approximation of P here 192 00:13:58,310 --> 00:14:03,090 okay so first as a benchmark maybe I 193 00:14:03,090 --> 00:14:07,940 would like to know the analytic value so 194 00:14:07,940 --> 00:14:14,690 let's define this as a constant here 195 00:14:14,690 --> 00:14:20,130 so this is our analytic value okay and 196 00:14:20,130 --> 00:14:27,750 later I will try to compare this absurd 197 00:14:27,750 --> 00:14:32,340 has to go here compare our numerical 198 00:14:32,340 --> 00:14:34,259 solution with this so 199 00:14:34,259 --> 00:14:37,799 the first one is to use a heart and 200 00:14:37,799 --> 00:14:44,479 sequence and let's define the number of 201 00:14:44,479 --> 00:14:47,699 Montecarlo samples let's be a little bit 202 00:14:47,699 --> 00:14:51,569 bold I would like to be twenty million 203 00:14:51,569 --> 00:14:54,809 one two three one two three this is 10 204 00:14:54,809 --> 00:14:57,419 to the power of seven well let's use 10 205 00:14:57,419 --> 00:15:03,089 to the power of eight 200 million so 206 00:15:03,089 --> 00:15:07,609 this is two times ten to the power of 207 00:15:07,609 --> 00:15:16,470 eight and with 20 to 200 million samples 208 00:15:16,470 --> 00:15:21,079 I would like to approximate P so let's 209 00:15:21,079 --> 00:15:25,679 test with the Horten sequence and I 210 00:15:25,679 --> 00:15:28,759 would like to test this with the Java 211 00:15:28,759 --> 00:15:33,509 stream API and it may be first I do an 212 00:15:33,509 --> 00:15:37,289 example where we are not using the 213 00:15:37,289 --> 00:15:39,709 parallelization so let's call this 214 00:15:39,709 --> 00:15:43,230 sequential okay I create dedicated 215 00:15:43,230 --> 00:15:48,449 masses to do this okay so we have a 216 00:15:48,449 --> 00:15:51,749 little bit more nicer overview in our 217 00:15:51,749 --> 00:15:53,179 kochiya 218 00:15:53,179 --> 00:16:01,079 okay so approximation of P using Monte 219 00:16:01,079 --> 00:16:04,259 Carlo with the Horton sequence so the 220 00:16:04,259 --> 00:16:08,220 Horton sequence we had it in our course 221 00:16:08,220 --> 00:16:14,220 before so in random numbers we had the 222 00:16:14,220 --> 00:16:16,379 funder corporate sequence and the Horton 223 00:16:16,379 --> 00:16:19,289 sequence so Horton sequence is just a 224 00:16:19,289 --> 00:16:21,359 funder corporate sequence where each 225 00:16:21,359 --> 00:16:24,539 dimension has a different base and the 226 00:16:24,539 --> 00:16:28,970 funder corporate sequence was this 227 00:16:28,970 --> 00:16:36,499 algorithm here that 228 00:16:36,499 --> 00:16:41,689 performed the calculation of the old 229 00:16:41,689 --> 00:16:46,219 number of under corporate number by form 230 00:16:46,219 --> 00:16:52,249 directly from the index okay so we have 231 00:16:52,249 --> 00:16:54,639 a sequence of indices and we can 232 00:16:54,639 --> 00:16:58,069 directly calculate the number from the 233 00:16:58,069 --> 00:17:02,089 index so I create a stream so a stream 234 00:17:02,089 --> 00:17:06,049 is just a sequence in in this framework 235 00:17:06,049 --> 00:17:12,459 I create a stream of numbers from 0 to 236 00:17:12,459 --> 00:17:17,059 number of samples so if you look here 237 00:17:17,059 --> 00:17:20,000 this first one is included the last one 238 00:17:20,000 --> 00:17:22,370 is not included so this is just 0 to n 239 00:17:22,370 --> 00:17:29,360 minus 1 ok then this stream I would like 240 00:17:29,360 --> 00:17:33,139 to apply a function that map's each 241 00:17:33,139 --> 00:17:36,740 index on a floating point double number 242 00:17:36,740 --> 00:17:39,679 so I use here map to double and the 243 00:17:39,679 --> 00:17:42,230 argument should be a function that map's 244 00:17:42,230 --> 00:17:48,769 an integer an integer to a floating 245 00:17:48,769 --> 00:17:53,389 point double number ok so how do we 246 00:17:53,389 --> 00:17:55,399 calculate this floating for double 247 00:17:55,399 --> 00:17:58,879 number so first we calculate our hand 248 00:17:58,879 --> 00:18:02,600 and vector so we calculate x and y so X 249 00:18:02,600 --> 00:18:09,070 is 2 times the port number so 250 00:18:09,070 --> 00:18:21,759 Horten sequence George let's part this 251 00:18:21,759 --> 00:18:26,500 that's yours 252 00:18:26,500 --> 00:18:33,400 a heart in sequence from here no obvious 253 00:18:33,400 --> 00:18:35,290 wasn't any missing that's what's the 254 00:18:35,290 --> 00:18:38,230 reason we cannot find it okay and this 255 00:18:38,230 --> 00:18:40,540 this function get heart and number four 256 00:18:40,540 --> 00:18:46,260 given base and I pass the index I and 257 00:18:46,260 --> 00:18:50,800 then the base so we was base - for the 258 00:18:50,800 --> 00:18:55,300 first time ention and we was base 3 for 259 00:18:55,300 --> 00:18:59,230 the second dimension so horton sequence 260 00:18:59,230 --> 00:19:00,670 is between 0 & 1 261 00:19:00,670 --> 00:19:03,310 i subtract 1/2 then it's between minus 262 00:19:03,310 --> 00:19:04,780 1/2 and plus 1/2 263 00:19:04,780 --> 00:19:07,090 i multiplied with 2 then it is between 264 00:19:07,090 --> 00:19:11,290 minus 1 and plus 1 so x is 2 okay so 265 00:19:11,290 --> 00:19:16,630 next number is the horton sequence for 266 00:19:16,630 --> 00:19:20,160 base 3 so we generate a two dimensional 267 00:19:20,160 --> 00:19:23,620 horton sequences with base two and three 268 00:19:23,620 --> 00:19:28,000 for the two dimensions next thing is 269 00:19:28,000 --> 00:19:31,600 that we have to check the indicator 270 00:19:31,600 --> 00:19:38,950 function so we integrate from 0 from 271 00:19:38,950 --> 00:19:42,370 minus 1 to 1 in both dimensions and the 272 00:19:42,370 --> 00:19:44,170 next thing is that we have to calculate 273 00:19:44,170 --> 00:19:48,370 F which is the indicator function are we 274 00:19:48,370 --> 00:19:54,940 in the unit circle or not so if x 275 00:19:54,940 --> 00:20:00,370 squared plus y squared is smaller than 1 276 00:20:00,370 --> 00:20:04,780 we are in the unit circle so we return 1 277 00:20:04,780 --> 00:20:11,979 otherwise we return 0 278 00:20:11,979 --> 00:20:16,009 okay so this maps the sequence of 279 00:20:16,009 --> 00:20:19,820 integers to a sequence of indicator 280 00:20:19,820 --> 00:20:22,849 function values which we then just sum 281 00:20:22,849 --> 00:20:28,039 up okay so this is now on our Monte 282 00:20:28,039 --> 00:20:31,940 Carlo some so we have to divide by the 283 00:20:31,940 --> 00:20:35,629 number of symbols so divided by the 284 00:20:35,629 --> 00:20:39,589 number of samples and actually the 285 00:20:39,589 --> 00:20:44,929 integral is just as we just transformed 286 00:20:44,929 --> 00:20:51,259 here two times so that the new DX is two 287 00:20:51,259 --> 00:20:54,019 times the old DX so we have to multiply 288 00:20:54,019 --> 00:21:00,849 by four okay so that was our little 289 00:21:00,849 --> 00:21:06,909 integration problem and let's print out 290 00:21:06,909 --> 00:21:11,739 the result so we are using here a 291 00:21:11,739 --> 00:21:23,349 sequential and the Java stream RP and 292 00:21:23,349 --> 00:21:29,239 what is the error the error is P orchard 293 00:21:29,239 --> 00:21:36,619 minus P analytic is Tierra okay so let's 294 00:21:36,619 --> 00:21:43,009 try if this box so I run this little 295 00:21:43,009 --> 00:21:51,549 program okay so now this takes a while 296 00:21:51,549 --> 00:21:54,559 you see here processors are working a 297 00:21:54,559 --> 00:21:58,190 little bit now but not too much you are 298 00:21:58,190 --> 00:22:03,460 not not everyone is here busy 299 00:22:03,460 --> 00:22:06,049 hmm maybe I should have got used here 300 00:22:06,049 --> 00:22:10,219 the number of samples okay so we get a 301 00:22:10,219 --> 00:22:14,059 10 to the minus 6 and it took a few 302 00:22:14,059 --> 00:22:15,880 seconds 303 00:22:15,880 --> 00:22:19,210 at reasons we like to do paralyzation i 304 00:22:19,210 --> 00:22:24,809 would like to know how long it takes so 305 00:22:24,809 --> 00:22:28,539 if we like to have a half measure for 306 00:22:28,539 --> 00:22:32,529 time we can ask here the system for the 307 00:22:32,529 --> 00:22:35,769 current time in milliseconds before we 308 00:22:35,769 --> 00:22:39,399 do the calculation and after we do the 309 00:22:39,399 --> 00:22:46,299 calculation okay and then we calculate 310 00:22:46,299 --> 00:22:53,950 the time in seconds as the time and - 311 00:22:53,950 --> 00:22:58,200 the time start converted to a double 312 00:22:58,200 --> 00:23:06,700 divided by 1000 okay and we can prevent 313 00:23:06,700 --> 00:23:09,789 in addition here maybe the time that it 314 00:23:09,789 --> 00:23:17,190 took to calculate this 315 00:23:17,190 --> 00:23:24,549 Monte Carlo integral okay so maybe add 316 00:23:24,549 --> 00:23:28,300 it to use here a little bit 317 00:23:28,300 --> 00:23:35,679 the numbers 2f the result a little bit 318 00:23:35,679 --> 00:23:39,670 faster okay so you see this took here 319 00:23:39,670 --> 00:23:45,130 five seconds so if I go back to 20 it 320 00:23:45,130 --> 00:23:48,850 will take maybe 20 seconds to calculate 321 00:23:48,850 --> 00:23:52,929 this okay so wait while it's running 322 00:23:52,929 --> 00:23:59,350 let's do this now in parallel okay so I 323 00:23:59,350 --> 00:24:06,480 just copy this code here I rename this 324 00:24:06,480 --> 00:24:11,770 with stream say parallel and here on the 325 00:24:11,770 --> 00:24:14,890 top I also call this other test which 326 00:24:14,890 --> 00:24:19,900 performs the calculation in panel okay 327 00:24:19,900 --> 00:24:22,360 so now it's actually in the same code I 328 00:24:22,360 --> 00:24:25,090 haven't done the parallelization yet so 329 00:24:25,090 --> 00:24:27,880 and I would like to show you how easy it 330 00:24:27,880 --> 00:24:30,880 is to colonize this because you can tell 331 00:24:30,880 --> 00:24:34,360 the stream API that he should perform 332 00:24:34,360 --> 00:24:37,870 calculations in parallel and actually 333 00:24:37,870 --> 00:24:40,300 you can also decide when he should do 334 00:24:40,300 --> 00:24:44,320 this so here he is generating a stream 335 00:24:44,320 --> 00:24:48,100 of integer numbers and I would like to 336 00:24:48,100 --> 00:24:50,740 tell him after you have generated this 337 00:24:50,740 --> 00:24:53,590 everything that comes after this can be 338 00:24:53,590 --> 00:24:56,620 done in parallel and you can do this by 339 00:24:56,620 --> 00:25:00,940 just adding here the Verte parallel okay 340 00:25:00,940 --> 00:25:05,200 actually that's it yeah so this code is 341 00:25:05,200 --> 00:25:08,860 now performing this calculation here in 342 00:25:08,860 --> 00:25:19,840 parallel okay so let's print this using 343 00:25:19,840 --> 00:25:25,260 stream so this is parallel 344 00:25:25,260 --> 00:25:34,420 oops so let's check how long it takes if 345 00:25:34,420 --> 00:25:37,480 we perform it in parallel so I run the 346 00:25:37,480 --> 00:25:47,250 program 347 00:25:47,250 --> 00:25:52,030 okay and while it's running you see the 348 00:25:52,030 --> 00:25:54,790 reason why he can easily paralyze this 349 00:25:54,790 --> 00:25:58,150 is that the stuff which is inside here 350 00:25:58,150 --> 00:26:02,410 is completely independent yeah so for 351 00:26:02,410 --> 00:26:05,320 every eye he's doing something but the 352 00:26:05,320 --> 00:26:07,570 other eye doesn't need to know what he's 353 00:26:07,570 --> 00:26:11,110 doing okay so maybe I add a few more 354 00:26:11,110 --> 00:26:14,440 dots here and so on to to have the 355 00:26:14,440 --> 00:26:17,830 output a bit nicer and you see that if 356 00:26:17,830 --> 00:26:19,990 you now look here in the lower left on 357 00:26:19,990 --> 00:26:23,380 the processors the first run is 358 00:26:23,380 --> 00:26:28,090 currently running sequential so it does 359 00:26:28,090 --> 00:26:30,550 not use up a lot of my processors 360 00:26:30,550 --> 00:26:33,480 actually this conference here takes 361 00:26:33,480 --> 00:26:36,310 takes up consumes up some time and then 362 00:26:36,310 --> 00:26:40,990 it just consumes say in in in some one 363 00:26:40,990 --> 00:26:44,320 processor it took 24 seconds now every 364 00:26:44,320 --> 00:26:47,490 processor is in use you see 365 00:26:47,490 --> 00:26:51,090 and it took five seconds okay so we get 366 00:26:51,090 --> 00:26:55,290 a performance increase of say a factor 367 00:26:55,290 --> 00:26:58,350 of five No roughly four point something 368 00:26:58,350 --> 00:27:01,290 reason is that he does not have all 369 00:27:01,290 --> 00:27:05,190 eight process was and this may be a 370 00:27:05,190 --> 00:27:07,100 small overhead and doing this 371 00:27:07,100 --> 00:27:11,760 colonization okay 372 00:27:11,760 --> 00:27:20,420 so there was a good question in the chat 373 00:27:20,420 --> 00:27:24,510 how does he know how to split up the 374 00:27:24,510 --> 00:27:29,250 stream so how many numbers of compute 375 00:27:29,250 --> 00:27:32,309 tasks is he using and this is actually a 376 00:27:32,309 --> 00:27:36,900 very good question so um he can check 377 00:27:36,900 --> 00:27:38,429 how many course 378 00:27:38,429 --> 00:27:41,550 how many parallel CPUs or how many CPUs 379 00:27:41,550 --> 00:27:46,740 the machine has and he uses 380 00:27:46,740 --> 00:27:52,159 as many worker threats as I have cause 381 00:27:52,159 --> 00:27:57,450 okay so and then he is cutting the 382 00:27:57,450 --> 00:28:02,370 problem into smaller packages but in 383 00:28:02,370 --> 00:28:06,169 this cutting there is maybe an issue 384 00:28:06,169 --> 00:28:09,330 remember the code of the Harton sequence 385 00:28:09,330 --> 00:28:12,210 or the funder corporate sequence if you 386 00:28:12,210 --> 00:28:13,860 look at the code of the front Accord 387 00:28:13,860 --> 00:28:16,380 sequence then you see that he is 388 00:28:16,380 --> 00:28:18,870 calculating the number from the index 389 00:28:18,870 --> 00:28:22,230 and actually this algorithm takes longer 390 00:28:22,230 --> 00:28:28,740 for larger indices so if you now cut 391 00:28:28,740 --> 00:28:33,929 your sequence into smaller packages it 392 00:28:33,929 --> 00:28:37,169 may happen that one package takes a lot 393 00:28:37,169 --> 00:28:40,380 longer than another package which means 394 00:28:40,380 --> 00:28:42,390 that for example if you have eight 395 00:28:42,390 --> 00:28:48,029 packages of work and seven of them just 396 00:28:48,029 --> 00:28:51,960 need one second but the last one needs 397 00:28:51,960 --> 00:28:56,399 10 second then in some if you do it 398 00:28:56,399 --> 00:29:00,690 sequentially you have maybe 17 seconds 7 399 00:29:00,690 --> 00:29:03,720 plus 10 but if you paralyze it you have 400 00:29:03,720 --> 00:29:06,149 10 seconds because it takes the time of 401 00:29:06,149 --> 00:29:09,779 the slowest package and you just have a 402 00:29:09,779 --> 00:29:12,840 factor of 1.7 performance and increase 403 00:29:12,840 --> 00:29:14,700 in this engine example which is a bit 404 00:29:14,700 --> 00:29:17,510 disappointing if you had eight cores 405 00:29:17,510 --> 00:29:21,029 so the question how he is actually 406 00:29:21,029 --> 00:29:25,350 splitting this up is important and this 407 00:29:25,350 --> 00:29:28,679 may be a reason to use another framework 408 00:29:28,679 --> 00:29:32,309 where you have more control on splitting 409 00:29:32,309 --> 00:29:35,010 this up you can of course use more 410 00:29:35,010 --> 00:29:40,450 threads which which somehow also will 411 00:29:40,450 --> 00:29:43,039 [Music] 412 00:29:43,039 --> 00:29:48,029 increase this parallelism you know but I 413 00:29:48,029 --> 00:29:50,880 have another example where we need more 414 00:29:50,880 --> 00:29:55,720 control on how to split these tasks 415 00:29:55,720 --> 00:29:58,440 so actually internally he has some some 416 00:29:58,440 --> 00:30:02,559 logic to split this up I can send you 417 00:30:02,559 --> 00:30:04,570 maybe some some reference where you can 418 00:30:04,570 --> 00:30:07,210 read about this I would like to make 419 00:30:07,210 --> 00:30:10,690 another example let's do the same with 420 00:30:10,690 --> 00:30:16,799 smears and twister for Mersenne twister 421 00:30:16,799 --> 00:30:23,710 I do not have skipped ahead at hand yeah 422 00:30:23,710 --> 00:30:29,009 so for example I would like to do this 423 00:30:29,009 --> 00:30:32,710 technique of starting Mason twister with 424 00:30:32,710 --> 00:30:34,659 different seats with different 425 00:30:34,659 --> 00:30:47,470 multicolored seats 426 00:30:47,470 --> 00:30:51,890 but actually I have a small question why 427 00:30:51,890 --> 00:30:55,940 not just use one Mason twister for all 428 00:30:55,940 --> 00:31:00,170 the parallel calculations and everybody 429 00:31:00,170 --> 00:31:03,890 just picks from the sequence a number 430 00:31:03,890 --> 00:31:06,590 and performs the calculation and then 431 00:31:06,590 --> 00:31:08,900 another one picks a number and performs 432 00:31:08,900 --> 00:31:11,470 the calculation so actually all the 433 00:31:11,470 --> 00:31:14,600 parallel calculations could share the 434 00:31:14,600 --> 00:31:17,150 same method twister and this is actually 435 00:31:17,150 --> 00:31:20,330 OK for Monte Carlo because the order in 436 00:31:20,330 --> 00:31:24,110 which we actually sum these sequences up 437 00:31:24,110 --> 00:31:29,660 does not matter so actually this example 438 00:31:29,660 --> 00:31:32,420 here of splitting a sequence in these 439 00:31:32,420 --> 00:31:35,240 packages it's just one possibility we 440 00:31:35,240 --> 00:31:38,230 could also just have a sequence which is 441 00:31:38,230 --> 00:31:45,860 index 0 1 2 3 4 5 6 7 and then we have 442 00:31:45,860 --> 00:31:49,580 index 8 9 10 and the first thread takes 443 00:31:49,580 --> 00:31:54,170 index 0 and 8 and so on here are 16 and 444 00:31:54,170 --> 00:31:57,560 the next one takes index 1 and even 445 00:31:57,560 --> 00:31:59,960 sometimes they skip one and they just 446 00:31:59,960 --> 00:32:04,520 consume the whole sequence that's sir 447 00:32:04,520 --> 00:32:06,920 possibility which is actually not on 448 00:32:06,920 --> 00:32:10,910 this slide you know just try and use the 449 00:32:10,910 --> 00:32:13,220 sequence in parallel now so maybe I 450 00:32:13,220 --> 00:32:20,210 write here one more alternative if the 451 00:32:20,210 --> 00:32:27,180 sequence allows this 452 00:32:27,180 --> 00:32:35,890 we can make a single sequence generator 453 00:32:35,890 --> 00:32:38,800 so you was a singular sequence generator 454 00:32:38,800 --> 00:32:50,710 from parallel slits 455 00:32:50,710 --> 00:32:54,710 okay so in order to be able to do this 456 00:32:54,710 --> 00:32:58,250 to generator has to be some kind has to 457 00:32:58,250 --> 00:33:01,340 have some kind of property yeah that 458 00:33:01,340 --> 00:33:03,320 allows this this is sometimes called 459 00:33:03,320 --> 00:33:07,190 thread safety so generator has to be 460 00:33:07,190 --> 00:33:10,659 thread safe 461 00:33:10,659 --> 00:33:16,930 you 462 00:33:16,930 --> 00:33:20,240 okay and this is a nice example which I 463 00:33:20,240 --> 00:33:25,280 would like to show you next so next 464 00:33:25,280 --> 00:33:30,740 example is test this integration of P 465 00:33:30,740 --> 00:33:40,360 with the Mezen Twista okay so uh let's 466 00:33:40,360 --> 00:33:50,180 you are quite a new test here let's call 467 00:33:50,180 --> 00:33:54,500 it test Mills end with stream maybe I 468 00:33:54,500 --> 00:34:01,570 also do first the sequential version 469 00:34:01,570 --> 00:34:12,110 okay so the P medicine the approximation 470 00:34:12,110 --> 00:34:20,230 with the Mason twister okay I have some 471 00:34:20,230 --> 00:34:23,330 generator maybe I'd like to use here the 472 00:34:23,330 --> 00:34:26,600 same code as here above so with the 473 00:34:26,600 --> 00:34:28,700 integer stream just to make it more 474 00:34:28,700 --> 00:34:30,500 similar you can also directly generate a 475 00:34:30,500 --> 00:34:33,200 Babel stream with some generator 476 00:34:33,200 --> 00:34:36,770 but to make it more similar let's just 477 00:34:36,770 --> 00:34:40,730 copy this code here so you really see 478 00:34:40,730 --> 00:34:43,070 that it's a fair comparison because we 479 00:34:43,070 --> 00:34:46,419 are doing the same thing again so let's 480 00:34:46,419 --> 00:34:57,650 call this here P MSN and okay and I 481 00:34:57,650 --> 00:35:00,020 remove the parallel so it is sequential 482 00:35:00,020 --> 00:35:03,740 and here inside I would like to create 483 00:35:03,740 --> 00:35:06,470 this random number from a Mersenne 484 00:35:06,470 --> 00:35:08,510 twister and the Mayerson twister 485 00:35:08,510 --> 00:35:10,550 implementation which i would like to use 486 00:35:10,550 --> 00:35:15,740 is that of Apache Commons math so there 487 00:35:15,740 --> 00:35:20,200 is a library that has a Mersenne twister 488 00:35:20,200 --> 00:35:22,550 implementation 489 00:35:22,550 --> 00:35:25,410 so let me write to see explicitly it so 490 00:35:25,410 --> 00:35:26,640 that you see which kind of 491 00:35:26,640 --> 00:35:30,870 implementation I use okay and the 492 00:35:30,870 --> 00:35:33,840 argument here is some seed of the 493 00:35:33,840 --> 00:35:35,640 Mersenne twister so let's use some 494 00:35:35,640 --> 00:35:37,890 number as an initial value of the 495 00:35:37,890 --> 00:35:40,290 generator and we have here a random 496 00:35:40,290 --> 00:35:42,330 number generator and you know that the 497 00:35:42,330 --> 00:35:44,490 Mersenne twister is a soy de vendor 498 00:35:44,490 --> 00:35:46,620 number generator so in order to generate 499 00:35:46,620 --> 00:35:48,900 a two dimensional sequence from a one 500 00:35:48,900 --> 00:35:50,730 dimensional sequence we can make the 501 00:35:50,730 --> 00:35:53,790 click to populate the random vector one 502 00:35:53,790 --> 00:35:57,050 by the other now so the first element is 503 00:35:57,050 --> 00:36:03,900 here the random number from the sequence 504 00:36:03,900 --> 00:36:06,360 generator and then we just pick another 505 00:36:06,360 --> 00:36:11,640 independent so iid simple number for the 506 00:36:11,640 --> 00:36:14,130 second component so very easy this is 507 00:36:14,130 --> 00:36:17,850 the medicine twister implementation okay 508 00:36:17,850 --> 00:36:20,160 you see here the index is actually I is 509 00:36:20,160 --> 00:36:21,600 not used huh 510 00:36:21,600 --> 00:36:24,090 so it's maybe a bit strange to use here 511 00:36:24,090 --> 00:36:26,130 this integer sequence and then transform 512 00:36:26,130 --> 00:36:30,050 it to the dab and never use the I but 513 00:36:30,050 --> 00:36:33,540 that's just to keep the example similar 514 00:36:33,540 --> 00:36:36,870 to the previous one so this here is an 515 00:36:36,870 --> 00:36:42,030 SN it is sequential and as is using the 516 00:36:42,030 --> 00:36:46,560 stream API okay so let's check how this 517 00:36:46,560 --> 00:36:50,160 performs so I have to do this test here 518 00:36:50,160 --> 00:36:59,400 I call this test here with our twenty 519 00:36:59,400 --> 00:37:04,350 million twenty two hundred million 200 520 00:37:04,350 --> 00:37:13,460 million sample points okay so let's try 521 00:37:13,460 --> 00:37:16,950 so maybe I should cut out the first one 522 00:37:16,950 --> 00:37:18,960 because it is always taking twenty 523 00:37:18,960 --> 00:37:21,900 seconds but again you see now the first 524 00:37:21,900 --> 00:37:24,240 approximation is running sequentially 525 00:37:24,240 --> 00:37:27,810 the computer cause are not very busy 526 00:37:27,810 --> 00:37:30,780 yeah you will print out the result and 527 00:37:30,780 --> 00:37:33,450 after that the second simulation run 528 00:37:33,450 --> 00:37:36,400 starts in parallel and the computer core 529 00:37:36,400 --> 00:37:40,810 we'll become very busy now for a few 530 00:37:40,810 --> 00:37:44,080 seconds now and he has the result after 531 00:37:44,080 --> 00:37:47,140 five seconds okay now it's not busy 532 00:37:47,140 --> 00:37:48,940 enough the computer of course are not 533 00:37:48,940 --> 00:37:51,420 busy we are in the example of methane 534 00:37:51,420 --> 00:37:55,090 okay and it takes seven seconds so maybe 535 00:37:55,090 --> 00:37:58,750 I move here a few points away one two or 536 00:37:58,750 --> 00:38:08,980 three okay and you see that mess then I 537 00:38:08,980 --> 00:38:12,060 actually took seven seconds without 538 00:38:12,060 --> 00:38:15,490 paralyzation yeah so it is a factor of 539 00:38:15,490 --> 00:38:19,060 three faster then using the heart in 540 00:38:19,060 --> 00:38:21,190 sequence and this difference will 541 00:38:21,190 --> 00:38:23,170 actually increase because the Horten 542 00:38:23,170 --> 00:38:25,720 sequence becomes slower and slower for 543 00:38:25,720 --> 00:38:29,410 higher indices and also the Horten 544 00:38:29,410 --> 00:38:31,870 sequence is now two dimensional if you 545 00:38:31,870 --> 00:38:34,210 use a three dimensional the difference 546 00:38:34,210 --> 00:38:37,360 will also be even more prominent yeah so 547 00:38:37,360 --> 00:38:40,990 you see now it is not parallel this is 548 00:38:40,990 --> 00:38:42,940 the methane calculation currently 549 00:38:42,940 --> 00:38:46,660 running and it takes seven seconds you 550 00:38:46,660 --> 00:38:49,450 also see here that the pseudo-random 551 00:38:49,450 --> 00:38:57,030 number generator is a bit 552 00:38:57,030 --> 00:39:04,120 less accurate no so if you go back here 553 00:39:04,120 --> 00:39:06,730 you see this is 10 to the minus 6 for 554 00:39:06,730 --> 00:39:13,780 Holton 10 to the minus 4 for the math 555 00:39:13,780 --> 00:39:16,420 and Trista yeah so if we go back here 556 00:39:16,420 --> 00:39:24,400 and I'll make a small remark here so we 557 00:39:24,400 --> 00:39:29,610 perform this simulation with n equal to 558 00:39:29,610 --> 00:39:35,880 2 times 10 to the 8th which means that 559 00:39:35,880 --> 00:39:40,450 Monte Carlo now so Monte Carlo is 1 560 00:39:40,450 --> 00:39:45,060 divided by square root of n which is 561 00:39:45,060 --> 00:39:49,270 approximately one point four so one 562 00:39:49,270 --> 00:39:50,590 divided by one point four which is 563 00:39:50,590 --> 00:39:53,770 approximately 10 to the minus four which 564 00:39:53,770 --> 00:39:57,670 is actually what we see here 10 to the 565 00:39:57,670 --> 00:40:05,310 minus four 566 00:40:05,310 --> 00:40:09,280 so for pseudo-random number generator 567 00:40:09,280 --> 00:40:16,290 and for the Horten sequence we had log n 568 00:40:16,290 --> 00:40:25,180 to the power of T divided by n so 569 00:40:25,180 --> 00:40:30,960 which is approximately 10 to the minus 6 570 00:40:30,960 --> 00:40:39,130 for Alton okay that's nice now so we see 571 00:40:39,130 --> 00:40:42,420 that actually the mathematical result is 572 00:40:42,420 --> 00:40:46,660 coming out exactly from from this so you 573 00:40:46,660 --> 00:40:50,200 see there's a factor of 100 between the 574 00:40:50,200 --> 00:40:54,069 two in accuracy but there's also some 575 00:40:54,069 --> 00:40:58,150 kind of a factor of 3 something in the 576 00:40:58,150 --> 00:41:01,180 performance and this ratio will change 577 00:41:01,180 --> 00:41:04,119 if you have higher dimensional problems 578 00:41:04,119 --> 00:41:06,130 and some at a certain level monitor car 579 00:41:06,130 --> 00:41:09,039 was with pseudo pseudo-random number 580 00:41:09,039 --> 00:41:12,880 generator this is better okay but look 581 00:41:12,880 --> 00:41:15,819 that this is Mayor Zen with our 582 00:41:15,819 --> 00:41:18,609 parallelization so we can become even 583 00:41:18,609 --> 00:41:24,690 faster so let's do mayor send with a 584 00:41:24,690 --> 00:41:29,559 parallel stream okay so I just do a new 585 00:41:29,559 --> 00:41:32,650 test here called it's parallel I call 586 00:41:32,650 --> 00:41:38,550 this test here true 587 00:41:38,550 --> 00:41:43,480 perform the calculation okay and I do 588 00:41:43,480 --> 00:41:47,319 this tiny little change that I call here 589 00:41:47,319 --> 00:41:50,500 please perform all the calculations in 590 00:41:50,500 --> 00:41:52,930 parallel on your computer on your 591 00:41:52,930 --> 00:41:56,619 threads on your course okay so let's run 592 00:41:56,619 --> 00:42:00,760 this little program so in order to save 593 00:42:00,760 --> 00:42:03,849 a little bit time yeah we know by now 594 00:42:03,849 --> 00:42:06,670 that the sequential Halton takes about 595 00:42:06,670 --> 00:42:10,059 20 seconds let's comment this out so I 596 00:42:10,059 --> 00:42:12,790 only run the other examples so he will 597 00:42:12,790 --> 00:42:18,300 start with parallel Halton okay so he's 598 00:42:18,300 --> 00:42:23,540 getting fast 599 00:42:23,540 --> 00:42:26,690 okay so parallel Horton and I get some 600 00:42:26,690 --> 00:42:31,280 kind of ever okay so that's strange 601 00:42:31,280 --> 00:42:34,160 so where does this arrow come from so it 602 00:42:34,160 --> 00:42:36,380 must come from our new part of code and 603 00:42:36,380 --> 00:42:38,750 so let's comment all the others out and 604 00:42:38,750 --> 00:42:46,180 run again so I run again 605 00:42:46,180 --> 00:42:52,789 you 606 00:42:52,789 --> 00:42:54,250 oops I 607 00:42:54,250 --> 00:42:57,370 I made a mistake I did the parallel here 608 00:42:57,370 --> 00:43:00,190 in the sequential and not here and the 609 00:43:00,190 --> 00:43:04,150 parallel okay so here here it should be 610 00:43:04,150 --> 00:43:06,570 this is our court where we want to do 611 00:43:06,570 --> 00:43:09,310 the power because that was just a 612 00:43:09,310 --> 00:43:16,480 sequential okay so if I now run this 613 00:43:16,480 --> 00:43:20,500 part of code here just messin streamers 614 00:43:20,500 --> 00:43:22,390 parallel we get an error 615 00:43:22,390 --> 00:43:28,150 so if I remove the word parallel the 616 00:43:28,150 --> 00:43:31,660 error is gone it's calculating the mess 617 00:43:31,660 --> 00:43:38,290 n sampling in sequential so 6 seconds 10 618 00:43:38,290 --> 00:43:40,300 to the minus 4 if I add the word 619 00:43:40,300 --> 00:43:50,250 parallel 620 00:43:50,250 --> 00:43:53,640 I get some stranger 621 00:43:53,640 --> 00:43:57,480 so let's investigate this error so you 622 00:43:57,480 --> 00:43:59,970 see there is some index out of bounds 623 00:43:59,970 --> 00:44:04,980 exception here a strange number 624 624 00:44:04,980 --> 00:44:07,800 actually this number reminds us of the 625 00:44:07,800 --> 00:44:10,610 dimension in which Mersenne twister is 626 00:44:10,610 --> 00:44:13,890 uniform distributed at most so the 627 00:44:13,890 --> 00:44:15,810 highest dimension we can we can we can 628 00:44:15,810 --> 00:44:20,520 use or we should use and we get this 629 00:44:20,520 --> 00:44:22,950 index out of bounds exception if we run 630 00:44:22,950 --> 00:44:33,090 in parallel and this problem is due to 631 00:44:33,090 --> 00:44:34,890 the fact that actually the Mersenne 632 00:44:34,890 --> 00:44:37,710 twister application implemented is not 633 00:44:37,710 --> 00:44:41,940 set safe so there is an issue called 634 00:44:41,940 --> 00:44:45,840 sketch safety which is important so if 635 00:44:45,840 --> 00:44:48,830 you have multiple threads yeah so 636 00:44:48,830 --> 00:44:53,880 multiple parallel worker threads which 637 00:44:53,880 --> 00:44:57,360 access say a common variable so some 638 00:44:57,360 --> 00:45:00,510 counter was the Montecarlo seed or 639 00:45:00,510 --> 00:45:04,320 whatever and they access this variable 640 00:45:04,320 --> 00:45:07,140 in parallel then this may lead to 641 00:45:07,140 --> 00:45:12,180 inconsistent States for example you have 642 00:45:12,180 --> 00:45:15,150 say some variable where you count 643 00:45:15,150 --> 00:45:19,410 something one sweat is counting up then 644 00:45:19,410 --> 00:45:22,320 the sweat is assuming that this variable 645 00:45:22,320 --> 00:45:28,260 is one larger than the previous value 646 00:45:28,260 --> 00:45:31,050 he's performing some calculations which 647 00:45:31,050 --> 00:45:34,560 used his variable but then another 648 00:45:34,560 --> 00:45:38,730 parallel set is also counting up then it 649 00:45:38,730 --> 00:45:40,590 may happen that the first part of the 650 00:45:40,590 --> 00:45:44,640 first set has used in our I plus 1 and 651 00:45:44,640 --> 00:45:47,730 the second part uses I plus 2 but he or 652 00:45:47,730 --> 00:45:49,830 he assumed that nobody had changed this 653 00:45:49,830 --> 00:45:53,430 variable so there are some basic rules 654 00:45:53,430 --> 00:45:56,430 how you can avoid this but before we can 655 00:45:56,430 --> 00:45:58,290 discuss this I would like to actually 656 00:45:58,290 --> 00:46:01,740 show you what has happened here so you 657 00:46:01,740 --> 00:46:03,700 see there's this index out of town 658 00:46:03,700 --> 00:46:06,520 exception here in the implementation of 659 00:46:06,520 --> 00:46:10,119 the Apache common math math and Twista 660 00:46:10,119 --> 00:46:12,730 and he tells us that this is in unison 661 00:46:12,730 --> 00:46:17,740 twister in line 252 if I click this he 662 00:46:17,740 --> 00:46:19,210 will jump to the court where this 663 00:46:19,210 --> 00:46:23,200 happens okay so this is ugly code that 664 00:46:23,200 --> 00:46:26,369 performs the messin twister calculation 665 00:46:26,369 --> 00:46:30,790 okay and you see that there is some kind 666 00:46:30,790 --> 00:46:34,630 of array this array empty here which 667 00:46:34,630 --> 00:46:37,839 generates the next value and actually 668 00:46:37,839 --> 00:46:39,970 Mason twister works in the following way 669 00:46:39,970 --> 00:46:46,660 that he will generate 624 values in a 670 00:46:46,660 --> 00:46:49,869 single block so he's not generating one 671 00:46:49,869 --> 00:46:52,240 random number from the previous one he's 672 00:46:52,240 --> 00:46:55,300 generating 624 random numbers from the 673 00:46:55,300 --> 00:46:58,510 previous 624 random numbers so actually 674 00:46:58,510 --> 00:47:00,790 it's like linear congruential random 675 00:47:00,790 --> 00:47:04,500 number generator with a very big number 676 00:47:04,500 --> 00:47:10,000 okay so he's storing these 624 numbers 677 00:47:10,000 --> 00:47:13,180 in this array so whenever you ask him 678 00:47:13,180 --> 00:47:16,540 he's checking is your current index 679 00:47:16,540 --> 00:47:19,690 larger than this number so if you over 680 00:47:19,690 --> 00:47:21,579 here over this number you see that this 681 00:47:21,579 --> 00:47:25,869 is 624 yeah so if this number is 682 00:47:25,869 --> 00:47:28,720 actually larger or equal so the he's 683 00:47:28,720 --> 00:47:33,030 beyond the 624 numbers 684 00:47:33,030 --> 00:47:36,300 yes then he will generate the new 685 00:47:36,300 --> 00:47:40,590 numbers and at the end he sets the index 686 00:47:40,590 --> 00:47:43,350 to zero and you can ask for the next 687 00:47:43,350 --> 00:47:46,650 numbers so if you are below this number 688 00:47:46,650 --> 00:47:49,470 you will just get the next number okay 689 00:47:49,470 --> 00:47:51,420 and now you immediately see maybe what 690 00:47:51,420 --> 00:47:54,710 happened here we have parallel threads 691 00:47:54,710 --> 00:47:59,300 in parallel asking for the next number 692 00:47:59,300 --> 00:48:06,210 okay so the first set was here and the 693 00:48:06,210 --> 00:48:13,590 number was may be smaller than the end 694 00:48:13,590 --> 00:48:17,070 now so he was not asking for the next 695 00:48:17,070 --> 00:48:22,140 one so the next thread was here and the 696 00:48:22,140 --> 00:48:25,290 number was also smaller than the end so 697 00:48:25,290 --> 00:48:28,290 he does not generate a new one and he 698 00:48:28,290 --> 00:48:30,390 would like to ask for the next one so 699 00:48:30,390 --> 00:48:33,390 now assume that your index is 606 700 00:48:33,390 --> 00:48:36,930 hundred twenty three and two thread walk 701 00:48:36,930 --> 00:48:40,230 along this in parallel so they will skip 702 00:48:40,230 --> 00:48:43,050 this part here and there are two threads 703 00:48:43,050 --> 00:48:46,530 asking for this number but there's only 704 00:48:46,530 --> 00:48:50,730 one number left now so the next set is 705 00:48:50,730 --> 00:48:53,940 actually seeing here the increment from 706 00:48:53,940 --> 00:48:56,370 the previous thread and he is requesting 707 00:48:56,370 --> 00:49:00,990 a number which did not exist so to avoid 708 00:49:00,990 --> 00:49:04,830 this problem you have to actually allow 709 00:49:04,830 --> 00:49:09,390 only one threat to enter this piece of 710 00:49:09,390 --> 00:49:16,290 code here okay so this can be done with 711 00:49:16,290 --> 00:49:19,470 a verge called synchronize so this is 712 00:49:19,470 --> 00:49:21,780 maybe in a most potent way of fixing 713 00:49:21,780 --> 00:49:28,740 this issue so you can 714 00:49:28,740 --> 00:49:35,400 denies the excess to a threat then the 715 00:49:35,400 --> 00:49:39,810 exit access to to available by using the 716 00:49:39,810 --> 00:49:42,840 word synchronized so this is that only 717 00:49:42,840 --> 00:49:47,400 one thread is allowed to actually 718 00:49:47,400 --> 00:49:50,700 perform this modification and then when 719 00:49:50,700 --> 00:49:54,080 it's done the other set is allowed to 720 00:49:54,080 --> 00:49:58,410 enter this code so if the threads need 721 00:49:58,410 --> 00:50:01,410 to modify a common shared field or 722 00:50:01,410 --> 00:50:03,869 object so in our case this is the 723 00:50:03,869 --> 00:50:13,320 counter inside the SN Twista then the 724 00:50:13,320 --> 00:50:17,880 access has to be synchronized there are 725 00:50:17,880 --> 00:50:20,910 other tips how you can avoid such 726 00:50:20,910 --> 00:50:24,240 problems for example the good thing is 727 00:50:24,240 --> 00:50:28,230 that to use only read-only access if 728 00:50:28,230 --> 00:50:33,420 possible then if you do not modify a 729 00:50:33,420 --> 00:50:35,160 variable it should if you just read it 730 00:50:35,160 --> 00:50:38,460 this problem does not occur or if you 731 00:50:38,460 --> 00:50:41,940 have some kind of counter and it is 732 00:50:41,940 --> 00:50:43,800 possible that every thread has its 733 00:50:43,800 --> 00:50:48,180 individual encounter so the variable 734 00:50:48,180 --> 00:50:54,240 exists in every thread on its own so say 735 00:50:54,240 --> 00:50:56,400 this me this is sometimes called that 736 00:50:56,400 --> 00:50:59,310 the variable is thread-local now then 737 00:50:59,310 --> 00:51:03,240 this problem also does not occur because 738 00:51:03,240 --> 00:51:06,390 every set has its own counter and they 739 00:51:06,390 --> 00:51:10,890 do not interfere I will show you the two 740 00:51:10,890 --> 00:51:13,560 solutions of introducing a thread local 741 00:51:13,560 --> 00:51:19,619 and of synchronization let's try it with 742 00:51:19,619 --> 00:51:27,530 synchronization okay so 743 00:51:27,530 --> 00:51:31,160 maybe I'm a bit reporter and I add here 744 00:51:31,160 --> 00:51:40,400 a synchronized around this block so this 745 00:51:40,400 --> 00:51:42,830 means that actually only one thread is 746 00:51:42,830 --> 00:51:46,490 allowed to enter this block here this 747 00:51:46,490 --> 00:51:49,010 solution is actually a bit stupid but I 748 00:51:49,010 --> 00:51:51,920 just want to show you then it works so 749 00:51:51,920 --> 00:51:55,940 we have a parallel execution but I do 750 00:51:55,940 --> 00:51:59,000 not allow that this critical part here 751 00:51:59,000 --> 00:52:04,370 is calculated by two threads 752 00:52:04,370 --> 00:52:07,640 simultaneously because inside the messin 753 00:52:07,640 --> 00:52:11,390 here next double so there is this 754 00:52:11,390 --> 00:52:14,950 function next called and two threads 755 00:52:14,950 --> 00:52:18,770 modify the counter at the same time not 756 00:52:18,770 --> 00:52:21,470 checking that we are very close to the 757 00:52:21,470 --> 00:52:23,570 bound now because we have already passed 758 00:52:23,570 --> 00:52:30,140 this point here and which then results 759 00:52:30,140 --> 00:52:32,150 that one thread is beyond the size of 760 00:52:32,150 --> 00:52:36,400 this array okay so if I now run this 761 00:52:36,400 --> 00:52:38,870 experiment yeah so maybe I can also 762 00:52:38,870 --> 00:52:42,170 enable all the others here so we maybe 763 00:52:42,170 --> 00:52:48,560 not all xxii Horten sequence sequential 764 00:52:48,560 --> 00:52:50,870 can be may be skipped but the other ones 765 00:52:50,870 --> 00:52:52,880 are may be fast enough so we have fought 766 00:52:52,880 --> 00:52:57,110 in parallel with five point second five 767 00:52:57,110 --> 00:52:59,510 point seven seconds we have may ascend 768 00:52:59,510 --> 00:53:02,830 with seven seconds ten to the minus four 769 00:53:02,830 --> 00:53:05,510 using sequential and now comes to 770 00:53:05,510 --> 00:53:08,120 parallel and what you see here that is 771 00:53:08,120 --> 00:53:11,000 the process was a little bit more busy 772 00:53:11,000 --> 00:53:14,210 but actually he is not using up all the 773 00:53:14,210 --> 00:53:16,400 processors and the reason is that 774 00:53:16,400 --> 00:53:20,570 actually this verge here blocks the 775 00:53:20,570 --> 00:53:24,260 parallelism because this plot here can 776 00:53:24,260 --> 00:53:28,730 be only entered by one thread at a time 777 00:53:28,730 --> 00:53:31,970 and you see that now the time is almost 778 00:53:31,970 --> 00:53:37,950 s verse as sequential halt 779 00:53:37,950 --> 00:53:41,569 it's not only that we blocked the 780 00:53:41,569 --> 00:53:45,180 parallelism so we have became sequential 781 00:53:45,180 --> 00:53:47,849 it's that we have now so much overhead 782 00:53:47,849 --> 00:53:50,099 in parallelizing and then we are 783 00:53:50,099 --> 00:53:52,559 removing the parallelism that it is 784 00:53:52,559 --> 00:53:56,640 actually slower so maybe you say that we 785 00:53:56,640 --> 00:54:00,000 only need to block the generation of the 786 00:54:00,000 --> 00:54:02,970 random numbers so let's pluck only the 787 00:54:02,970 --> 00:54:05,880 generation of the random numbers and the 788 00:54:05,880 --> 00:54:08,369 rest of the calculation can be performed 789 00:54:08,369 --> 00:54:11,069 in parallel which would look like this 790 00:54:11,069 --> 00:54:15,750 so you have a random vector ax the 791 00:54:15,750 --> 00:54:17,490 generation of the random vector is not 792 00:54:17,490 --> 00:54:20,839 allowed to be parallel okay 793 00:54:20,839 --> 00:54:27,059 but each butBut the calculation then if 794 00:54:27,059 --> 00:54:30,420 you are inside the unit cube is allowed 795 00:54:30,420 --> 00:54:33,480 to prepare there so in our example this 796 00:54:33,480 --> 00:54:37,079 doesn't cure a lot because actually the 797 00:54:37,079 --> 00:54:39,299 calculation of are we inside the unit 798 00:54:39,299 --> 00:54:42,180 cube or not is so fast so it does not 799 00:54:42,180 --> 00:54:44,970 matter in other problems this is maybe a 800 00:54:44,970 --> 00:54:51,450 good solution to have a synchronized 801 00:54:51,450 --> 00:54:53,970 around generation of the random numbers 802 00:54:53,970 --> 00:54:57,420 but still it is here not a very good 803 00:54:57,420 --> 00:54:59,400 solution because it takes much longer 804 00:54:59,400 --> 00:55:00,630 you know 805 00:55:00,630 --> 00:55:03,799 then actually the sequential version 806 00:55:03,799 --> 00:55:06,510 this the sequential version is five 807 00:55:06,510 --> 00:55:11,040 seconds here it is still 1970 seconds so 808 00:55:11,040 --> 00:55:14,309 maybe next idea is just to synchronize 809 00:55:14,309 --> 00:55:18,510 each individual s XS which would be like 810 00:55:18,510 --> 00:55:24,770 this okay does this help 811 00:55:24,770 --> 00:55:28,470 so I synchronize the access to the 812 00:55:28,470 --> 00:55:33,059 Mersenne twister yeah but okay 813 00:55:33,059 --> 00:55:36,349 I can I can tell you it does not help 814 00:55:36,349 --> 00:55:40,319 okay so you see that actually with this 815 00:55:40,319 --> 00:55:42,829 implementation of Mayerson 816 00:55:42,829 --> 00:55:47,849 using the stream API it's very difficult 817 00:55:47,849 --> 00:55:50,730 to perform a parallelization 818 00:55:50,730 --> 00:55:52,800 I wouldn't say it is impossible with the 819 00:55:52,800 --> 00:55:54,840 same API the following trick can be also 820 00:55:54,840 --> 00:55:57,450 done with the stream API but I would 821 00:55:57,450 --> 00:56:00,030 like to show you now an alternative 822 00:56:00,030 --> 00:56:06,350 where I perform a parallel execution 823 00:56:06,350 --> 00:56:13,160 make sure there should be parallel here 824 00:56:13,160 --> 00:56:18,780 okay so let's let's go to the cleaner 825 00:56:18,780 --> 00:56:20,940 version of the code here where the 826 00:56:20,940 --> 00:56:23,460 synchronize was around this looks like 827 00:56:23,460 --> 00:56:31,020 bit a bit nicer okay so I would like to 828 00:56:31,020 --> 00:56:36,330 show you an alternative framework so 829 00:56:36,330 --> 00:56:40,119 here on our list 830 00:56:40,119 --> 00:56:43,880 there was distress Java stream API so it 831 00:56:43,880 --> 00:56:49,900 is very easy to parent an eyes with a 832 00:56:49,900 --> 00:56:52,369 stream parallelize the stream by just 833 00:56:52,369 --> 00:56:54,710 adding the word parallel but the code 834 00:56:54,710 --> 00:56:58,640 inside has to be independent of each 835 00:56:58,640 --> 00:56:59,269 other 836 00:56:59,269 --> 00:57:05,719 and has to be set safe okay so the next 837 00:57:05,719 --> 00:57:08,210 thing is an execute the execute of 838 00:57:08,210 --> 00:57:13,670 framework and Java futures so let's 839 00:57:13,670 --> 00:57:20,869 write this so I would like to test my 840 00:57:20,869 --> 00:57:23,900 sin with the so called executor 841 00:57:23,900 --> 00:57:29,950 framework for a given number of samples 842 00:57:29,950 --> 00:57:35,900 okay so in the executor framework you 843 00:57:35,900 --> 00:57:38,809 can get something which is called the 844 00:57:38,809 --> 00:57:42,710 executor and you can submit tasks to the 845 00:57:42,710 --> 00:57:45,380 executor and the executor is performing 846 00:57:45,380 --> 00:57:48,589 all the tasks in parallel okay so I 847 00:57:48,589 --> 00:57:51,049 would like to split now the number of 848 00:57:51,049 --> 00:57:53,900 tasks by hand so let's define some 849 00:57:53,900 --> 00:57:59,839 number of tasks so let's say I have 100 850 00:57:59,839 --> 00:58:00,799 tasks 851 00:58:00,799 --> 00:58:06,499 I would like to split my samples in 100 852 00:58:06,499 --> 00:58:09,710 packages so I have some number of 853 00:58:09,710 --> 00:58:12,619 samples per task so the number of 854 00:58:12,619 --> 00:58:15,229 samples per task is just the number of 855 00:58:15,229 --> 00:58:19,589 samples divided by the number of 856 00:58:19,589 --> 00:58:19,599 samples divided by the number of 857 00:58:19,599 --> 00:58:22,120 samples divided by the number of actually at this point you have to be a 858 00:58:22,120 --> 00:58:22,130 actually at this point you have to be a 859 00:58:22,130 --> 00:58:24,829 bit careful yeah that this integer 860 00:58:24,829 --> 00:58:26,390 division has no remainder 861 00:58:26,390 --> 00:58:28,460 otherwise you have to increase one 862 00:58:28,460 --> 00:58:31,789 package by this size of the remainder or 863 00:58:31,789 --> 00:58:34,430 have a specific package but in our case 864 00:58:34,430 --> 00:58:37,609 I don't care about this now because I 865 00:58:37,609 --> 00:58:40,430 know the integer division is is this is 866 00:58:40,430 --> 00:58:43,700 fine so there is something called an 867 00:58:43,700 --> 00:58:45,890 executor service 868 00:58:45,890 --> 00:58:49,430 and they are different 869 00:58:49,430 --> 00:58:57,770 such executors sorry executors well I 870 00:58:57,770 --> 00:59:04,690 see not liking my my ass okay III need 871 00:59:04,690 --> 00:59:07,340 executors I don't know maybe I write it 872 00:59:07,340 --> 00:59:09,920 by hand anomaly drugs okay so there are 873 00:59:09,920 --> 00:59:14,150 different such as to execute just and 874 00:59:14,150 --> 00:59:17,060 that's for example here the fixed sled 875 00:59:17,060 --> 00:59:21,230 pool executors yeah 876 00:59:21,230 --> 00:59:23,720 and you can specify how many parallel 877 00:59:23,720 --> 00:59:26,870 because how many parallels lets you use 878 00:59:26,870 --> 00:59:29,270 for this calculation you know so let's 879 00:59:29,270 --> 00:59:33,590 have here some number of threats and how 880 00:59:33,590 --> 00:59:37,370 many threats would I like to use maybe a 881 00:59:37,370 --> 00:59:41,090 good value as guidance it's the number 882 00:59:41,090 --> 00:59:44,660 of processors just use as many parallel 883 00:59:44,660 --> 00:59:47,570 workers as you have processors you can 884 00:59:47,570 --> 00:59:52,700 ask for the number of processors by 885 00:59:52,700 --> 00:59:54,890 using runtime get one time available 886 00:59:54,890 --> 01:00:00,110 processors so this is our value of the 887 01:00:00,110 --> 01:00:03,980 number of threads so now you can submit 888 01:00:03,980 --> 01:00:10,280 a working task to this executors so what 889 01:00:10,280 --> 01:00:13,940 do I like to do as a task I would like 890 01:00:13,940 --> 01:00:23,350 to calculate the approximation of P with 891 01:00:23,350 --> 01:00:28,060 Amazon twister but each melon twister 892 01:00:28,060 --> 01:00:31,060 for the task should use a different seat 893 01:00:31,060 --> 01:00:34,720 so actually on our list I used the first 894 01:00:34,720 --> 01:00:38,150 trick which was on our list use 895 01:00:38,150 --> 01:00:40,370 sequences with different seat wasn't 896 01:00:40,370 --> 01:00:42,920 recommended but it is here may be a way 897 01:00:42,920 --> 01:00:46,730 to paralyze the pseudo-random number 898 01:00:46,730 --> 01:00:49,000 generator 899 01:00:49,000 --> 01:01:00,119 so for this let's write a small function 900 01:01:00,119 --> 01:01:06,510 get approximation of P with methane 901 01:01:06,510 --> 01:01:12,310 which uses a given seat and uses a 902 01:01:12,310 --> 01:01:23,460 number of samples okay 903 01:01:23,460 --> 01:01:27,550 and then I submit this function with 904 01:01:27,550 --> 01:01:30,090 different seats to the parallel skirts 905 01:01:30,090 --> 01:01:33,040 so let's first generate the Mersenne 906 01:01:33,040 --> 01:01:37,050 twister so let's use this non set safe 907 01:01:37,050 --> 01:01:40,000 implementation from cut party common 908 01:01:40,000 --> 01:01:55,119 math and let's use this seat which is 909 01:01:55,119 --> 01:01:57,700 here our argument is the seed of this 910 01:01:57,700 --> 01:02:03,670 random number generator okay so now we 911 01:02:03,670 --> 01:02:07,599 have this random number generator let's 912 01:02:07,599 --> 01:02:13,089 calculate P from this maybe I just 913 01:02:13,089 --> 01:02:15,580 copied the cord which we had here and so 914 01:02:15,580 --> 01:02:25,310 it is just fair yeah 915 01:02:25,310 --> 01:02:29,070 okay I generate an in-stream with number 916 01:02:29,070 --> 01:02:33,500 of samples not parallel maybe to double 917 01:02:33,500 --> 01:02:36,750 and I do not need to synchronize here 918 01:02:36,750 --> 01:02:41,220 because every thread has its own local 919 01:02:41,220 --> 01:02:43,320 version of the lesson so they are 920 01:02:43,320 --> 01:02:47,940 independent okay and I returned this 921 01:02:47,940 --> 01:02:53,880 number okay so now I have here a 922 01:02:53,880 --> 01:02:59,690 sequential calculation of the lesson 923 01:02:59,690 --> 01:03:00,870 okay 924 01:03:00,870 --> 01:03:04,260 maybe also you you you can do the 925 01:03:04,260 --> 01:03:06,900 traditional one without the street if 926 01:03:06,900 --> 01:03:09,450 you like to see the court without the 927 01:03:09,450 --> 01:03:18,090 street you would right here Montecarlo 928 01:03:18,090 --> 01:03:29,270 some then you have an integer loop 929 01:03:29,270 --> 01:03:34,130 okay or instead of some I can just count 930 01:03:34,130 --> 01:03:45,200 the number of samples in unit circle 931 01:03:45,200 --> 01:03:51,349 okay then you have an integer loop which 932 01:03:51,349 --> 01:03:54,660 generates the random number and counts 933 01:03:54,660 --> 01:03:59,790 the number of so distance yet zero sorry 934 01:03:59,790 --> 01:04:03,030 that starts at zero and counts up if we 935 01:04:03,030 --> 01:04:08,630 are inside the unit circle this would be 936 01:04:08,630 --> 01:04:19,680 the version without stream and the value 937 01:04:19,680 --> 01:04:23,790 which should be returned is four times 938 01:04:23,790 --> 01:04:26,869 the number of samples in the unit circle 939 01:04:26,869 --> 01:04:34,609 divided by the number of total centers 940 01:04:34,609 --> 01:04:39,240 okay so either way he works 941 01:04:39,240 --> 01:04:41,760 yeah so this is now the approximation of 942 01:04:41,760 --> 01:04:47,550 P using a melon twister pseudo-random 943 01:04:47,550 --> 01:04:51,000 number generator with a given seat so 944 01:04:51,000 --> 01:04:56,400 now I continue here and I would like to 945 01:04:56,400 --> 01:05:03,869 have different such jobs with different 946 01:05:03,869 --> 01:05:06,060 seats so actually I use a random number 947 01:05:06,060 --> 01:05:14,760 generator to generate the seat this 948 01:05:14,760 --> 01:05:16,970 random number generator needs also some 949 01:05:16,970 --> 01:05:23,210 Montecarlo seat some seat and I use the 950 01:05:23,210 --> 01:05:33,810 Java random for this okay so and now 951 01:05:33,810 --> 01:05:38,480 let's submit to the executor our drop 952 01:05:38,480 --> 01:05:41,300 so you write executors dot submit and 953 01:05:41,300 --> 01:05:46,910 then you can just submit some tasks yeah 954 01:05:46,910 --> 01:05:52,609 so this task takes no argument but 955 01:05:52,609 --> 01:05:59,180 returns a value okay so the value is the 956 01:05:59,180 --> 01:06:05,390 value of P approximated so let's get the 957 01:06:05,390 --> 01:06:11,119 seat here see it is random seed dot next 958 01:06:11,119 --> 01:06:14,690 long okay so now it works and for this 959 01:06:14,690 --> 01:06:17,480 seed I would like to submit now a task 960 01:06:17,480 --> 01:06:21,020 ok the task is just a function with no 961 01:06:21,020 --> 01:06:24,290 argument so you do this break it open 962 01:06:24,290 --> 01:06:27,550 pack it close and it returns a value 963 01:06:27,550 --> 01:06:30,980 namely it returns the approximation of P 964 01:06:30,980 --> 01:06:34,430 with methane using this seat and here it 965 01:06:34,430 --> 01:06:43,670 uses number of samples per task ok this 966 01:06:43,670 --> 01:06:47,869 submits one such task to the executor so 967 01:06:47,869 --> 01:06:51,020 actually we would like to submit many 968 01:06:51,020 --> 01:07:01,460 tasks so how many tasks would we like to 969 01:07:01,460 --> 01:07:05,960 submit number of tasks is the number and 970 01:07:05,960 --> 01:07:12,740 I created loop around this which submits 971 01:07:12,740 --> 01:07:15,500 many different tasks to the executor so 972 01:07:15,500 --> 01:07:18,319 now the executor takes these tasks and 973 01:07:18,319 --> 01:07:21,980 works on parallel ok so you see this 974 01:07:21,980 --> 01:07:24,230 doing this parallelism by hand without 975 01:07:24,230 --> 01:07:26,089 the stream is a little bit more work 976 01:07:26,089 --> 01:07:28,250 yeah but actually we have much more 977 01:07:28,250 --> 01:07:31,599 control because we can explicitly 978 01:07:31,599 --> 01:07:35,000 specify how many tasks he should use and 979 01:07:35,000 --> 01:07:37,310 you can also specify how the tasks 980 01:07:37,310 --> 01:07:39,490 should be split now this is like 981 01:07:39,490 --> 01:07:44,119 advantage ok so where do we get the 982 01:07:44,119 --> 01:07:47,660 result ok the result is actually coming 983 01:07:47,660 --> 01:07:51,599 back here so I need some list 984 01:07:51,599 --> 01:08:05,809 of results where I collect all my result 985 01:08:05,809 --> 01:08:10,349 okay so the result from this function is 986 01:08:10,349 --> 01:08:14,279 a double okay 987 01:08:14,279 --> 01:08:17,639 but actually this cannot work because if 988 01:08:17,639 --> 01:08:20,099 I would like write this line he would 989 01:08:20,099 --> 01:08:22,920 have to wait for the executor to finish 990 01:08:22,920 --> 01:08:26,520 to get the result so I cannot get the 991 01:08:26,520 --> 01:08:28,949 result immediately but I can get 992 01:08:28,949 --> 01:08:31,589 something which gets me the result in 993 01:08:31,589 --> 01:08:36,799 the future so I get a handle to it and 994 01:08:36,799 --> 01:08:40,230 later I can ask for the result and 995 01:08:40,230 --> 01:08:43,230 actually this object which I get is 996 01:08:43,230 --> 01:08:43,920 called 997 01:08:43,920 --> 01:08:49,079 a future so a future is a value which 998 01:08:49,079 --> 01:08:51,690 you will get at a future point in time 999 01:08:51,690 --> 01:08:54,089 now so you get a handle on this value 1000 01:08:54,089 --> 01:08:56,819 and you can specify what kind of future 1001 01:08:56,819 --> 01:09:03,109 is it is a future value being a double 1002 01:09:03,109 --> 01:09:06,000 okay so this is what this executor 1003 01:09:06,000 --> 01:09:09,810 returns and I create here now a list of 1004 01:09:09,810 --> 01:09:13,380 such future values which are in the 1005 01:09:13,380 --> 01:09:17,579 future a double okay 1006 01:09:17,579 --> 01:09:24,540 and let's put this result here in this 1007 01:09:24,540 --> 01:09:31,540 list okay so this is 1008 01:09:31,540 --> 01:09:38,130 the first step distribute the vodka task 1009 01:09:38,130 --> 01:09:44,140 so we distribute all the tasks so now 1010 01:09:44,140 --> 01:09:46,569 all the tasks ran in parallel and we can 1011 01:09:46,569 --> 01:09:50,080 ask for the results so the next step is 1012 01:09:50,080 --> 01:10:00,940 collect the results so we can collect 1013 01:10:00,940 --> 01:10:05,650 the results in some so we do a loop over 1014 01:10:05,650 --> 01:10:12,940 all tasks again know and collect the 1015 01:10:12,940 --> 01:10:19,710 results so this is some is some plus 1016 01:10:19,710 --> 01:10:24,720 result besides I would like to have the 1017 01:10:24,720 --> 01:10:30,160 result corresponding to the first second 1018 01:10:30,160 --> 01:10:34,030 third and so on tasks this object here 1019 01:10:34,030 --> 01:10:37,030 is the so-called future and from this 1020 01:10:37,030 --> 01:10:45,150 future I can now get the double value 1021 01:10:45,150 --> 01:10:50,320 okay and here an exception can happen 1022 01:10:50,320 --> 01:10:53,889 yeah so I have to declare that an 1023 01:10:53,889 --> 01:10:55,960 exception can happen if the skate was 1024 01:10:55,960 --> 01:10:58,269 interrupted okay but let's don't care 1025 01:10:58,269 --> 01:11:02,880 about this so then our approximation is 1026 01:11:02,880 --> 01:11:06,610 a sum of different P approximations and 1027 01:11:06,610 --> 01:11:10,960 we just take the average of this so this 1028 01:11:10,960 --> 01:11:13,750 is sum divided by actually now divided 1029 01:11:13,750 --> 01:11:16,960 by the number of tasks so I take the 1030 01:11:16,960 --> 01:11:22,449 average of say 100 approximations each 1031 01:11:22,449 --> 01:11:27,429 approximations being the average of 2 1032 01:11:27,429 --> 01:11:30,960 million simulations 1033 01:11:30,960 --> 01:11:34,000 okay so that was a little bit lengthy 1034 01:11:34,000 --> 01:11:39,510 but it's now an example of calculating 1035 01:11:39,510 --> 01:11:42,790 this approximation in parallel using the 1036 01:11:42,790 --> 01:11:50,410 executor okay let's also calculate the 1037 01:11:50,410 --> 01:11:55,750 time yeah so the time in seconds is the 1038 01:11:55,750 --> 01:11:58,510 start time the end time minus the start 1039 01:11:58,510 --> 01:12:05,580 time so let's calculate here time end is 1040 01:12:05,580 --> 01:12:09,000 the current time in milliseconds and 1041 01:12:09,000 --> 01:12:12,700 also time start before we do all this 1042 01:12:12,700 --> 01:12:16,319 stuff here is 1043 01:12:16,319 --> 01:12:21,329 the current time in milliseconds No so 1044 01:12:21,329 --> 01:12:25,559 let's do it here so we can measure the 1045 01:12:25,559 --> 01:12:27,089 time how long this takes 1046 01:12:27,089 --> 01:12:31,280 so I hope I wrote everything correctly 1047 01:12:31,280 --> 01:12:36,209 maybe again the code is not so long yeah 1048 01:12:36,209 --> 01:12:38,789 it's just these lines and it has two 1049 01:12:38,789 --> 01:12:41,280 parts first we distribute all the 1050 01:12:41,280 --> 01:12:44,280 calculations second we collect all the 1051 01:12:44,280 --> 01:12:48,419 results these calls here run as in 1052 01:12:48,419 --> 01:12:51,329 Chronos Li in parallel so i can submit 1053 01:12:51,329 --> 01:12:54,299 more tasks and again and he's just 1054 01:12:54,299 --> 01:12:56,419 collecting the tasks and working on it 1055 01:12:56,419 --> 01:13:01,199 this call here is synchronizing so this 1056 01:13:01,199 --> 01:13:03,319 call is waiting for the task to finish 1057 01:13:03,319 --> 01:13:06,389 but at this point we can wait for the 1058 01:13:06,389 --> 01:13:09,089 first task to finish because all the 1059 01:13:09,089 --> 01:13:12,419 other tasks are already running and it 1060 01:13:12,419 --> 01:13:14,189 also does not matter if the second task 1061 01:13:14,189 --> 01:13:18,030 ends before the first task before or if 1062 01:13:18,030 --> 01:13:21,299 because he will then wait for the first 1063 01:13:21,299 --> 01:13:23,849 task and then he immediately jumps to 1064 01:13:23,849 --> 01:13:25,409 the second one and sees the result is 1065 01:13:25,409 --> 01:13:27,719 already there now so this loops here 1066 01:13:27,719 --> 01:13:31,260 they it's as long as the slowest task 1067 01:13:31,260 --> 01:13:34,169 which is what we need because we need 1068 01:13:34,169 --> 01:13:42,239 all the values so let's add our last 1069 01:13:42,239 --> 01:13:48,780 test here test messin with executor to 1070 01:13:48,780 --> 01:13:53,489 our list and let's keep fingers crossed 1071 01:13:53,489 --> 01:13:57,439 and run this program 1072 01:13:57,439 --> 01:14:00,689 so now we do a test of the Horten 1073 01:14:00,689 --> 01:14:05,440 sequence approximation of pea 1074 01:14:05,440 --> 01:14:08,410 sequential in parallel using the star 1075 01:14:08,410 --> 01:14:11,800 chalice stream API we do this 1076 01:14:11,800 --> 01:14:13,900 same with absurdo random number 1077 01:14:13,900 --> 01:14:17,110 generation later with Zen twister 1078 01:14:17,110 --> 01:14:19,719 sequential and parallel and then we use 1079 01:14:19,719 --> 01:14:22,060 the executor framework you see the first 1080 01:14:22,060 --> 01:14:24,040 sequential one did not use all the 1081 01:14:24,040 --> 01:14:26,469 processors now we are going in parallel 1082 01:14:26,469 --> 01:14:28,810 we were faster we using all the 1083 01:14:28,810 --> 01:14:31,210 processors now we are going sequential 1084 01:14:31,210 --> 01:14:33,370 we are not using all the processors 1085 01:14:33,370 --> 01:14:36,610 partners and was quite fast now we have 1086 01:14:36,610 --> 01:14:40,270 this buggy implementation of using the 1087 01:14:40,270 --> 01:14:43,239 screen stream API where we synchronize 1088 01:14:43,239 --> 01:14:46,380 inside and we lose all the performance 1089 01:14:46,380 --> 01:14:49,390 okay and the last one is messing with 1090 01:14:49,390 --> 01:14:53,620 executor so now he should maybe use all 1091 01:14:53,620 --> 01:14:59,199 processors and it is very fast okay it's 1092 01:14:59,199 --> 01:15:02,500 actually the fastest one 1.6 second you 1093 01:15:02,500 --> 01:15:07,770 know okay so this one is even faster 1094 01:15:07,770 --> 01:15:14,739 then the Horten parallelization yeah so 1095 01:15:14,739 --> 01:15:17,739 you see again the factor of 4 or 5 1096 01:15:17,739 --> 01:15:23,890 between Mason and Orton so Mason is 1097 01:15:23,890 --> 01:15:27,219 factor of 3 or 4 faster and now the 1098 01:15:27,219 --> 01:15:29,080 fastest thing we have is the power 1099 01:15:29,080 --> 01:15:33,640 they're messing you see that the result 1100 01:15:33,640 --> 01:15:36,130 here is different from this result here 1101 01:15:36,130 --> 01:15:41,320 why is that okay so maybe I add here on 1102 01:15:41,320 --> 01:15:48,840 two more digits to two more dots to be 1103 01:15:48,840 --> 01:15:55,329 bit nicer 1104 01:15:55,329 --> 01:15:59,110 so why is the result of methane using 1105 01:15:59,110 --> 01:16:02,139 the executor framework different from 1106 01:16:02,139 --> 01:16:08,619 the stream API the reason is that we are 1107 01:16:08,619 --> 01:16:11,440 using completely different sequences so 1108 01:16:11,440 --> 01:16:17,889 here we used one sequence with a single 1109 01:16:17,889 --> 01:16:24,159 seat okay here we are using many 1110 01:16:24,159 --> 01:16:26,290 different sequences with different seeds 1111 01:16:26,290 --> 01:16:30,400 the seeds are generated from another 1112 01:16:30,400 --> 01:16:32,860 sequence but actually the methane 1113 01:16:32,860 --> 01:16:35,050 sequences have different Montecarlo 1114 01:16:35,050 --> 01:16:36,849 seeds so they are completely different 1115 01:16:36,849 --> 01:16:40,119 sequences and you see I believe the 1116 01:16:40,119 --> 01:16:43,719 result was 7 times 10 to the minus 5 so 1117 01:16:43,719 --> 01:16:46,809 actually this sequence was by chance a 1118 01:16:46,809 --> 01:16:48,460 little bit better but you know that 1119 01:16:48,460 --> 01:16:50,550 Monte Carlo is actually just in 1120 01:16:50,550 --> 01:16:52,960 probability oh no it's a little bit 1121 01:16:52,960 --> 01:16:56,409 worse I see it's a little bit worse ok 1122 01:16:56,409 --> 01:16:58,329 so it was by chance a little bit worse 1123 01:16:58,329 --> 01:17:02,500 you can use a different number here and 1124 01:17:02,500 --> 01:17:11,739 maybe you get another you can also use 1125 01:17:11,739 --> 01:17:15,659 the executor framework to parallelize 1126 01:17:15,659 --> 01:17:20,290 the horton sequence and maybe I like to 1127 01:17:20,290 --> 01:17:22,389 do this because it's just copy and paste 1128 01:17:22,389 --> 01:17:25,900 it's not so much work yeah so get 1129 01:17:25,900 --> 01:17:29,679 approximation of P with me then I copy 1130 01:17:29,679 --> 01:17:34,119 this get approximation of P with Horton 1131 01:17:34,119 --> 01:17:39,250 and now I do not want to split using the 1132 01:17:39,250 --> 01:17:42,670 seed I want to split using the start 1133 01:17:42,670 --> 01:17:45,909 index of the secrets of the start index 1134 01:17:45,909 --> 01:17:49,719 of the sequence so this loop here goes 1135 01:17:49,719 --> 01:17:53,139 from start index to start index plus 1136 01:17:53,139 --> 01:17:56,559 number of samples and inside it uses a 1137 01:17:56,559 --> 01:17:59,020 horton sequence so I can get rid of this 1138 01:17:59,020 --> 01:18:01,630 mess in here 1139 01:18:01,630 --> 01:18:04,510 and inside are you is a heart in 1140 01:18:04,510 --> 01:18:09,250 sequence so Halton sequence dot get halt 1141 01:18:09,250 --> 01:18:13,360 a number four given base I and true for 1142 01:18:13,360 --> 01:18:17,110 the first component and I and three for 1143 01:18:17,110 --> 01:18:21,370 the second component okay so now I have 1144 01:18:21,370 --> 01:18:24,370 an approximation of P using the Horton 1145 01:18:24,370 --> 01:18:26,230 sequence with a different start index 1146 01:18:26,230 --> 01:18:30,430 and I would like to know now create as 1147 01:18:30,430 --> 01:18:33,310 we did here wisdom as n different 1148 01:18:33,310 --> 01:18:36,460 batches you know using the Horton 1149 01:18:36,460 --> 01:18:46,320 sequence okay so I copied this here test 1150 01:18:46,320 --> 01:18:50,200 Carlton and this is my last test now 1151 01:18:50,200 --> 01:18:54,250 Horton with executors so I'd like to run 1152 01:18:54,250 --> 01:18:58,690 this test also here test 1153 01:18:58,690 --> 01:19:03,250 Alton all this executors maybe we maybe 1154 01:19:03,250 --> 01:19:06,000 move it here so we have Horton and 1155 01:19:06,000 --> 01:19:12,760 messin tests okay and I need to modify 1156 01:19:12,760 --> 01:19:16,720 this code here so in this code we have 1157 01:19:16,720 --> 01:19:22,630 here 100 different tasks we create the 1158 01:19:22,630 --> 01:19:25,270 executor service this was needed to 1159 01:19:25,270 --> 01:19:28,600 generate random seats now this is not 1160 01:19:28,600 --> 01:19:32,140 needed because we do not use the gate 1161 01:19:32,140 --> 01:19:34,390 approximation business and we used to 1162 01:19:34,390 --> 01:19:37,360 get approximation with Halton for a 1163 01:19:37,360 --> 01:19:40,780 given start index so and what is the 1164 01:19:40,780 --> 01:19:43,650 start index the start index is just 1165 01:19:43,650 --> 01:19:50,260 which task am i working on 1166 01:19:50,260 --> 01:19:53,240 multiplied with number of samples per 1167 01:19:53,240 --> 01:19:55,430 task and also the first task goes from 1168 01:19:55,430 --> 01:20:00,080 zero to two million minus one and the 1169 01:20:00,080 --> 01:20:02,330 next one goes from two million to four 1170 01:20:02,330 --> 01:20:03,880 million minus one and so on 1171 01:20:03,880 --> 01:20:06,710 okay so you see this will be for the 1172 01:20:06,710 --> 01:20:08,930 first one two million but the two 1173 01:20:08,930 --> 01:20:10,580 million is not included in the loop now 1174 01:20:10,580 --> 01:20:12,890 because we start in zero okay so I have 1175 01:20:12,890 --> 01:20:16,130 created now a parallelization using the 1176 01:20:16,130 --> 01:20:20,770 execute of framework based e hot 1177 01:20:20,770 --> 01:20:26,440 sequence and let's scan this experiment 1178 01:20:26,440 --> 01:20:27,620 okay 1179 01:20:27,620 --> 01:20:30,140 so unfortunately running all the guys 1180 01:20:30,140 --> 01:20:32,750 takes so long yeah we have 20 seconds 1181 01:20:32,750 --> 01:20:36,140 for the first one you see processors are 1182 01:20:36,140 --> 01:20:44,030 not used up yeah it's sequential so I 1183 01:20:44,030 --> 01:20:47,690 hope I did everything all right then 1184 01:20:47,690 --> 01:20:50,690 what's the next one the next one is the 1185 01:20:50,690 --> 01:20:54,830 parallel Horton no which gives us the 1186 01:20:54,830 --> 01:20:58,580 same number much faster the next one is 1187 01:20:58,580 --> 01:21:01,210 hot and with executors which gives us 1188 01:21:01,210 --> 01:21:04,820 the same member almost there's a small 1189 01:21:04,820 --> 01:21:07,160 difference maybe you can think about 1190 01:21:07,160 --> 01:21:10,400 where the difference comes from and it 1191 01:21:10,400 --> 01:21:12,710 is faster it's actually one second 1192 01:21:12,710 --> 01:21:17,630 faster which is 20% you have faster 1193 01:21:17,630 --> 01:21:22,280 you're almost 20% then comes me as n in 1194 01:21:22,280 --> 01:21:26,030 sequential order which is similar to the 1195 01:21:26,030 --> 01:21:27,250 parallel yeah 1196 01:21:27,250 --> 01:21:30,410 the parallel Mezen using three was buggy 1197 01:21:30,410 --> 01:21:36,230 was slow no and we have here the very 1198 01:21:36,230 --> 01:21:38,780 fast messin with the executor framework 1199 01:21:38,780 --> 01:21:42,040 okay there is a surprising thing 1200 01:21:42,040 --> 01:21:44,570 parallelization usually the executor is 1201 01:21:44,570 --> 01:21:47,270 faster than using the stream and the 1202 01:21:47,270 --> 01:21:50,120 reason here is that actually very much 1203 01:21:50,120 --> 01:21:51,890 related to the question which we had in 1204 01:21:51,890 --> 01:21:54,260 the chat which we had from you how does 1205 01:21:54,260 --> 01:21:58,070 he know how to split the task so 1206 01:21:58,070 --> 01:22:00,230 actually he doesn't know a lot of our 1207 01:22:00,230 --> 01:22:03,170 problem so he uses some reasonable split 1208 01:22:03,170 --> 01:22:05,900 but the Horton sequence has the 1209 01:22:05,900 --> 01:22:09,440 unpleasant effect that later blocks take 1210 01:22:09,440 --> 01:22:13,970 longer than earlier blocks so actually 1211 01:22:13,970 --> 01:22:17,270 we should split it up into many small 1212 01:22:17,270 --> 01:22:19,970 blocks but I still would like to use 1213 01:22:19,970 --> 01:22:21,470 eights heads because if I use more 1214 01:22:21,470 --> 01:22:24,380 threads he will switch the threads 1215 01:22:24,380 --> 01:22:26,390 between the processors which takes time 1216 01:22:26,390 --> 01:22:28,640 actually I would like to use once that 1217 01:22:28,640 --> 01:22:31,730 pair processor and every thread works on 1218 01:22:31,730 --> 01:22:34,700 some task and the task should be small 1219 01:22:34,700 --> 01:22:37,730 enough such that he can distribute it 1220 01:22:37,730 --> 01:22:41,290 more evenly so if you just have here 1221 01:22:41,290 --> 01:22:45,020 actually the example is here eight 1222 01:22:45,020 --> 01:22:48,260 blocks he has a very large number of 1223 01:22:48,260 --> 01:22:53,270 blocks and one block is much slower than 1224 01:22:53,270 --> 01:22:55,640 the other one and the calculation takes 1225 01:22:55,640 --> 01:23:00,530 as long as the slowest block if you have 1226 01:23:00,530 --> 01:23:03,290 two small blocks it may be that the 1227 01:23:03,290 --> 01:23:07,160 overhead to schedule the task is too 1228 01:23:07,160 --> 01:23:09,770 large so here in this executor framework 1229 01:23:09,770 --> 01:23:12,470 you have much more control about the 1230 01:23:12,470 --> 01:23:15,170 size of this block and you can choose a 1231 01:23:15,170 --> 01:23:17,810 number which gives you then an even 1232 01:23:17,810 --> 01:23:18,580 faster 1233 01:23:18,580 --> 01:23:20,930 calculation now so if I have here an 1234 01:23:20,930 --> 01:23:28,370 eight it's maybe worse so let's let's 1235 01:23:28,370 --> 01:23:31,460 stop the calculation and just remove 1236 01:23:31,460 --> 01:23:41,840 this first one because it takes so long 1237 01:23:41,840 --> 01:23:46,080 okay so Alton parallel using stream so 1238 01:23:46,080 --> 01:23:47,760 now the difference is a little bit 1239 01:23:47,760 --> 01:23:50,790 smaller yeah it's only it's it's it's 1240 01:23:50,790 --> 01:23:52,860 not one second it's a little bit smaller 1241 01:23:52,860 --> 01:23:55,650 because we used a small number of blocks 1242 01:23:55,650 --> 01:24:00,320 to summarize I let it run all together 1243 01:24:00,320 --> 01:24:04,080 and we have seen many aspects in this 1244 01:24:04,080 --> 01:24:05,699 little experiment yeah for example 1245 01:24:05,699 --> 01:24:08,130 you've seen that you cannot paralyze 1246 01:24:08,130 --> 01:24:10,830 every code because it's not sweat safe 1247 01:24:10,830 --> 01:24:13,880 if you have code which is not set safe 1248 01:24:13,880 --> 01:24:18,239 you need to synchronize things but if 1249 01:24:18,239 --> 01:24:20,190 you use the stream API and just 1250 01:24:20,190 --> 01:24:22,290 synchronize the whole block you you you 1251 01:24:22,290 --> 01:24:26,130 you lose all the advantages okay this is 1252 01:24:26,130 --> 01:24:28,710 maybe trivia but maybe you should have 1253 01:24:28,710 --> 01:24:32,760 seen it once and if you'd like to have 1254 01:24:32,760 --> 01:24:38,580 finer control on splitting the worker 1255 01:24:38,580 --> 01:24:41,760 tasks then you can do this with this 1256 01:24:41,760 --> 01:24:45,300 executor framework I will commit this 1257 01:24:45,300 --> 01:24:48,570 code to our repo so you can use it maybe 1258 01:24:48,570 --> 01:24:55,230 as a blueprint for your examples or when 1259 01:24:55,230 --> 01:24:56,790 you read you like to do something in 1260 01:24:56,790 --> 01:24:59,310 parallel either use the stream API or 1261 01:24:59,310 --> 01:25:07,590 just use the executor brush and you see 1262 01:25:07,590 --> 01:25:11,289 in this two dimension Halton is actually 1263 01:25:11,289 --> 01:25:15,610 still better yeah it takes here a little 1264 01:25:15,610 --> 01:25:17,739 bit longer but it is a factor of 100 1265 01:25:17,739 --> 01:25:21,840 better but in higher dimension this 1266 01:25:21,840 --> 01:25:25,840 difference is shrinking and having a 1267 01:25:25,840 --> 01:25:28,530 fast pseudo-random number generator 1268 01:25:28,530 --> 01:25:31,630 parallelized is maybe then still the 1269 01:25:31,630 --> 01:25:35,860 best way to go okay so actually this was 1270 01:25:35,860 --> 01:25:38,699 quite exhaustive code session you know 1271 01:25:38,699 --> 01:25:42,400 and maybe there were many details inside 1272 01:25:42,400 --> 01:25:45,369 yeah that you do not need for the theory 1273 01:25:45,369 --> 01:25:49,690 for the next sections on theoretical 1274 01:25:49,690 --> 01:25:53,380 sections but I believe it's may be nice 1275 01:25:53,380 --> 01:25:57,070 to have seen this at least once and you 1276 01:25:57,070 --> 01:25:59,679 have also seen that the code you need to 1277 01:25:59,679 --> 01:26:02,260 pile and I something you know there's a 1278 01:26:02,260 --> 01:26:07,050 stream API or using the executor API 1279 01:26:07,050 --> 01:26:09,820 it's not so long and also executor is a 1280 01:26:09,820 --> 01:26:11,679 little bit longer you define your 1281 01:26:11,679 --> 01:26:14,499 executor you distribute your tasks you 1282 01:26:14,499 --> 01:26:17,139 collect the result yeah so maybe it's 1283 01:26:17,139 --> 01:26:23,159 bit longer but still fits on one page 1284 01:26:23,159 --> 01:26:25,539 okay so thanks for today 1285 01:26:25,539 --> 01:26:28,900 yeah so you have seen already the script 1286 01:26:28,900 --> 01:26:32,999 we will continue with generation of 1287 01:26:32,999 --> 01:26:38,400 random numbers with other distributions 1288 01:26:38,400 --> 01:26:42,570 so so far we have just looked at uniform 1289 01:26:42,570 --> 01:26:45,849 distributed random numbers so next 1290 01:26:45,849 --> 01:26:49,869 session will be on random number 1291 01:26:49,869 --> 01:26:52,539 sequences which have other distributions 1292 01:26:52,539 --> 01:26:54,309 like normal distribution exponential 1293 01:26:54,309 --> 01:26:57,280 distribution and so on for this we need 1294 01:26:57,280 --> 01:27:00,449 to prove some some small results and 1295 01:27:00,449 --> 01:27:02,979 have their nice applications in 1296 01:27:02,979 --> 01:27:05,729 mathematical finance