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I have written a simulation of a rotating molecule that I was hoping to get help to optimize. Below you see the main function in the simulation. The function FlatPart calculates the angle θ and the angular velocity ω from an obtained analytical solution to a Langevin equation. The function OtherPart just changes the angle and angular velocity to discontinuously, i.e., just add a number to the existing value.

\[Delta] = 0;

prob[\[Gamma]_] := 
  ProbabilityDistribution[(\[Gamma] Exp[-\[Gamma] t]), {t, 
    0, \[Infinity]}];

\[CapitalOmega]0 = Sqrt[1 - (\[Chi]/2)^2];

\[Theta][t_, \[Theta]0_, \[Omega]0_] := (\[Theta]0 Cos[
       t \[CapitalOmega]0] + ((\[Theta]0 \[Chi] + 2 \[Omega]0) Sin[
       t \[CapitalOmega]0])/(2 \[CapitalOmega]0)) E^(-((t \[Chi])/2));

\[Omega][t_, \[Theta]0_, \[Omega]0_] := (
  E^(-((t \[Chi])/
    2)) (4 \[Omega]0 \[CapitalOmega]0 Cos[
       t \[CapitalOmega]0] - (2 \[Chi] \[Omega]0 + \[Theta]0 \
(\[Chi]^2 + 4 \[CapitalOmega]0^2)) Sin[t \[CapitalOmega]0]))/(
  4 \[CapitalOmega]0);

FlatPart[\[Theta]0_, \[Omega]0_] := Module[{},
  \[Theta]1 = \[Theta][t1, \[Theta]0, \[Omega]0];
  \[Omega]1 = \[Omega][t1, \[Theta]0, \[Omega]0];
  par = {\[Theta]1, \[Omega]1}
  ]

OtherPart[\[Theta]0_, \[Omega]0_] := Module[{},
  \[Omega]1 = \[Omega]0 + \[CapitalDelta]\[CapitalOmega];
  \[Theta]1 = \[Theta]0 + \[CapitalDelta]\[CapitalOmega] \
\[CapitalDelta]t;
  par = {\[Theta]1, \[Omega]1}
  ]

mod[\[Theta]_, \[Omega]_, \[Gamma]_, \
\[CapitalDelta]\[CapitalOmega]1_, \[Chi]h_] := Module[{},
  \[CapitalDelta]ts = RandomVariate[prob[\[Gamma]], 50];
  T = Accumulate[\[CapitalDelta]ts];
  
  \[Chi] = \[Chi]h;
  \[CapitalDelta]\[CapitalOmega] = \[CapitalDelta]\[CapitalOmega]1;
  \[Theta]0 = \[Theta];
  \[Omega]0 = \[Omega];
  t0 = 0;
  \[CapitalTheta] = {\[Theta]0};
  \[CapitalOmega] = {\[Omega]0};
  x = {t0};
  i = 2;
  t1 = T[[1]] - \[Delta];
  
  While[\[Theta]0 < \[Pi] && i < (Length[T] - 2),
   p = FlatPart[\[Theta]0, \[Omega]0];
   \[Theta]0 = p[[1]];
   \[Omega]0 = p[[2]];
   AppendTo[\[CapitalTheta], \[Theta]0];
   AppendTo[\[CapitalOmega], \[Omega]0];
   t0 = t1;
   AppendTo[x, t0];
   t1 = t1 + 2 \[Delta];
   \[CapitalDelta]t = t1 - t0;
   p = OtherPart[\[Theta]0, \[Omega]0];
   \[Theta]0 = p[[1]];
   \[Omega]0 = p[[2]];
   AppendTo[\[CapitalTheta], \[Theta]0];
   AppendTo[\[CapitalOmega], \[Omega]0];
   t0 = t1;
   AppendTo[x, t0];
   t1 = T[[i]] - \[Delta];
   i = i + 1;
   ];

   ListLinePlot[Transpose[{x[[1 ;;]], \[CapitalTheta][[1 ;;]]}], 
   AxesLabel -> {"t", "\[Theta]"}, PlotLegends -> {i}, PlotRange -> All]]

Manipulate[
 mod[\[Theta]0, \[Omega]0, \[Gamma], \[CapitalDelta]\[CapitalOmega], \
\[Chi]], {\[Theta]0, 0, 1}, {\[Omega]0, 0, 1}, {{\[Gamma], 16}, 0.01, 
  50}, {{\[CapitalDelta]\[CapitalOmega], 0.3}, 0, 1}, {{\[Chi], 0.01},
   0, 0.05}]

My problem arises when I want to run this simulation many times so that I can do some statistics on the simulation (since the problem is stochastic it has a different outcome every time I run the code). I do this in a simple Do loop (as seen below) but maybe this can be done in a more efficient way? With this code I can only run the simulation around 130 times but I would like to be able to run it at least 1000 times.

sampl[\[Theta]0_, \[Omega]0_, \[Gamma]_, \[CapitalDelta]\
\[CapitalOmega]_, \[Chi]_, n_] := Module[{},
  Tend = {};
  Tnot = 0;
  Do[
   mod[\[Theta]0, \[Omega]0, \[Gamma], \
\[CapitalDelta]\[CapitalOmega], \[Chi]];
   If[i < (Length[T] - 2),
    AppendTo[Tend, t0], Tnot += 1
    ]
   
   , n];
  tmean = Mean[Tend];
  Histogram[Tend, {.025}]
  ]

sampl[0, 0, 16, 0.8, 0.01, 1000]

Thank you!

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  • 1
    $\begingroup$ It would be helpful if you could provide a full working example. Now OtherPart[] is not defined. There is also no input to the function (sampl?) that we can try. $\endgroup$
    – a20
    Nov 16 '21 at 8:25
  • $\begingroup$ When optimizing, it is good to take out each part by itself and wrap it with AbsoluteTiming[] to see where the bottlenecks are. One problem with your code is that you use AppendTo[], this is very slow since it has no assumption on the datastructure and rebuilds the complete list every time. One option is to use Reap and Sow instead. Another is that you e.g. initalize a list x = Table[0,{1,xmax}] and then set x[[i]] = t, or even better if you could work on a complete list at one time such as x = listOfTresults $\endgroup$
    – a20
    Nov 16 '21 at 8:28
  • $\begingroup$ Thank you so much for the suggestions a20. I have edited the post so that the full code is now included, I just thought it would be too crowded if I included it all, I am sorry. $\endgroup$
    – flg
    Nov 16 '21 at 9:05
  • $\begingroup$ Thanks for the update. mod is currently not returning anything. What are we supposed to look at? Also, when I run mod[] now, it takes me about 0.008 seconds. This could certainly be made faster, but just running this 1000 times should not be a major concern? Can you please show how you call the sampl function? $\endgroup$
    – a20
    Nov 16 '21 at 9:08
  • $\begingroup$ Sorry, I forgot that I had removed that part as well, it is included now. It just generates a plot, but that is not really important. The main thing is to get the sampl function, that runs the mod function many times, to run faster. Using Sow instead of AppendTo makes the AbsoluteTiming of the mod function to be 0.0388 instead of 0.04336, so that a start! :) The length of the lists can vary depending on the parameters of the system, otherwise I would definitely just define a list from the beginning. $\endgroup$
    – flg
    Nov 16 '21 at 9:21
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Okay, so there were two main bottlenecks in your code. First of all, the LineListPlot should be removed unless you actually want the plots for every iteration.

Secondly, you iterate over \[CapitalDelta]ts = RandomVariate[prob[\[Gamma]], 50];

which is very slow. It is better to generate all the random numbers you need first, and then iterate. I thus generate a list that is large enough for all iterations in the sampl[] function, and then I send a subset of 50 values to the mod[] function for every call.

I have also improved some other things

  • Reap and Sow instead of AppendTo*
  • changing constant numbers to non-constant
  • avoid assigning values to variables when not necessary

however the performance gain from these changes is very small in comparison. Here is a faster solution that should do the same thing (please cross-check the calculations):

\[Delta] = 0;
prob[\[Gamma]_] := 
  ProbabilityDistribution[(\[Gamma] Exp[-\[Gamma] t]), {t, 
    0, \[Infinity]}];
\[CapitalOmega]0 = Sqrt[1 - (\[Chi]/2.)^2];
\[Theta][t_, \[Theta]0_, \[Omega]0_] := (\[Theta]0 Cos[
       t \[CapitalOmega]0] + ((\[Theta]0 \[Chi] + 2. \[Omega]0) Sin[
         t \[CapitalOmega]0])/(2. \[CapitalOmega]0)) E^(-((t \[Chi])/
        2.));
\[Omega][t_, \[Theta]0_, \[Omega]0_] := (E^(-((t \[Chi])/
          2.)) (4. \[Omega]0 \[CapitalOmega]0 Cos[
         t \[CapitalOmega]0] - (2. \[Chi] \[Omega]0 + \[Theta]0 (\
\[Chi]^2 + 4. \[CapitalOmega]0^2)) Sin[
         t \[CapitalOmega]0]))/(4. \[CapitalOmega]0);
FlatPart[\[Theta]0_, \[Omega]0_] := Module[{},
  Return[{\[Theta][t1, \[Theta]0, \[Omega]0], \[Omega][
     t1, \[Theta]0, \[Omega]0]}]]
OtherPart[\[Theta]0_, \[Omega]0_] := Module[{},
  Return[{\[Theta]0 + \[CapitalDelta]\[CapitalOmega] \
\[CapitalDelta]t, \[Omega]0 + \[CapitalDelta]\[CapitalOmega]}]]

mod[\[Theta]_, \[Omega]_, \[Gamma]_, \
\[CapitalDelta]\[CapitalOmega]1_, \[Chi]h_, dts_] := Module[{},
  T = Accumulate[dts];
  \[Chi] = \[Chi]h;
  \[CapitalDelta]\[CapitalOmega] = \[CapitalDelta]\[CapitalOmega]1;
  p = {\[Theta], \[Omega]};
  t0 = 0;
  i = 2;
  t1 = T[[1]] - \[Delta];
  res = Reap[
    While[p[[1]] < \[Pi] && i < (Length[T] - 2),
      p = FlatPart[p[[1]], p[[2]]];
      t0 = t1;
      Sow[{p[[1]], p[[2]], t0}];
      t1 = t1 + 2. \[Delta];
      \[CapitalDelta]t = t1 - t0;
      p = OtherPart[p[[1]], p[[2]]];
      t0 = t1;
      Sow[{p[[1]], p[[2]], t0}];
      t1 = T[[i]] - \[Delta];
      i = i + 1;];
    ];
  Return[res[[2, 1]]];
  ]

sampl[\[Theta]0_, \[Omega]0_, \[Gamma]_, \[CapitalDelta]\
\[CapitalOmega]_, \[Chi]_, n_] := Module[{}, Tend = {};
  Tnot = 0;
  dts = RandomVariate[prob[\[Gamma]], 50*n];
  Tend = Reap[
    Do[mod[\[Theta]0, \[Omega]0, \[Gamma], \[CapitalDelta]\
\[CapitalOmega], \[Chi], dts[[(j - 1)*50 + 1 ;; j*50]]];
      If[i < (Length[T] - 2), Sow[t0], Tnot += 1], {j, 1, n}];
    ];
  Tend = Tend[[2, 1]];
  tmean = Mean[Tend];
  Histogram[Tend, {.025}, 
   PlotLabel -> {": mean time to reach \[Theta]=\[Pi]" tmean}]
  ]

When calling sampl using the following code:

AbsoluteTiming[sampl[0, 0, 16, 0.8, 0.01, 100]] 

I had 2.7 seconds before removing ListLinePlot, 0.7 seconds before extracting the RandomVariate out from the mod[] function and finally ending at 0.04 seconds.

The code still requires quite a bit of cleaning I think. One of my main difficulties with it was that you are mixing global and local variables everywhere, and the path is not clear what depends on what and where the things are changed. I would suggest to really only define globally what should be global, outside of the loops, and then call functions with local variables. I did not attempt to sort this out because I am afraid that I would destroy some of the calculations, so I leave that to you.

* reap and sow might not be faster or only negligibly faster than AppendTo for small and simple lists. This seems to be the case here, but if you want to scale up, reap and sow is really the way to go.

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1
  • $\begingroup$ Wow this is really great, thank you so much! I am quite new to Mathematica, so local and global variables are definitely a thing I am confused about, but I will look into this. $\endgroup$
    – flg
    Nov 18 '21 at 7:16
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The main culprit is not AppendTo", but the superfluous "ListLinePlot". Here is the improved version with "Reap" that take approx. 9 sec.

\[Delta] = 0;

prob[\[Gamma]_] := 
  ProbabilityDistribution[(\[Gamma] Exp[-\[Gamma] t]), {t, 
    0, \[Infinity]}];

\[CapitalOmega]0 = Sqrt[1 - (\[Chi]/2)^2];

\[Theta][t_, \[Theta]0_, \[Omega]0_] := (\[Theta]0 Cos[
       t \[CapitalOmega]0] + ((\[Theta]0 \[Chi] + 2 \[Omega]0) Sin[
         t \[CapitalOmega]0])/(2 \[CapitalOmega]0)) E^(-((t \[Chi])/
        2));

\[Omega][t_, \[Theta]0_, \[Omega]0_] := (E^(-((t \[Chi])/
          2)) (4 \[Omega]0 \[CapitalOmega]0 Cos[
         t \[CapitalOmega]0] - (2 \[Chi] \[Omega]0 + \[Theta]0 \
(\[Chi]^2 + 4 \[CapitalOmega]0^2)) Sin[
         t \[CapitalOmega]0]))/(4 \[CapitalOmega]0);

FlatPart[\[Theta]0_, \[Omega]0_] := 
 Module[{}, \[Theta]1 = \[Theta][t1, \[Theta]0, \[Omega]0];
  \[Omega]1 = \[Omega][t1, \[Theta]0, \[Omega]0];
  par = {\[Theta]1, \[Omega]1}]

OtherPart[\[Theta]0_, \[Omega]0_] := 
 Module[{}, \[Omega]1 = \[Omega]0 + \[CapitalDelta]\[CapitalOmega];
  \[Theta]1 = \[Theta]0 + \[CapitalDelta]\[CapitalOmega] \
\[CapitalDelta]t;
  par = {\[Theta]1, \[Omega]1}]

mod[\[Theta]_, \[Omega]_, \[Gamma]_, \
\[CapitalDelta]\[CapitalOmega]1_, \[Chi]h_] := 
 Module[{}, \[CapitalDelta]ts = RandomVariate[prob[\[Gamma]], 50];
  T = Accumulate[\[CapitalDelta]ts];
  \[Chi] = \[Chi]h;
  \[CapitalDelta]\[CapitalOmega] = \[CapitalDelta]\[CapitalOmega]1;
  \[Theta]0 = \[Theta];
  \[Omega]0 = \[Omega];
  t0 = 0;
  i = 2;
  t1 = T[[1]] - \[Delta];
  {\[CapitalTheta], \[CapitalOmega], x} = Reap[
     Sow[\[Theta]0, 1];
     Sow[\[Omega]0, 2];
     Sow[t0, 3];
     While[\[Theta]0 < \[Pi] && i < (Length[T] - 2), 
      p = FlatPart[\[Theta]0, \[Omega]0];
      \[Theta]0 = p[[1]];
      \[Omega]0 = p[[2]];
      Sow[\[Theta]0, 1];
      Sow[\[Omega]0, 2];
      t0 = t1;
      Sow[t0, 3];
      t1 = t1 + 2 \[Delta];
      \[CapitalDelta]t = t1 - t0;
      p = OtherPart[\[Theta]0, \[Omega]0];
      \[Theta]0 = p[[1]];
      \[Omega]0 = p[[2]];
      Sow[\[Theta]0, 1];
      Sow[\[Omega]0, 2];
      t0 = t1;
      Sow[t0, 3];
      t1 = T[[i]] - \[Delta];
      i = i + 1;];
     ][[2]];
  ]

sampl[\[Theta]0_, \[Omega]0_, \[Gamma]_, \[CapitalDelta]\[CapitalOmega]_, \[Chi]_, n_] := Module[{}, Tend = {};
  Tnot = 0;
  Tend = Reap[
     Do[mod[\[Theta]0, \[Omega]0, \[Gamma], \[CapitalDelta]\[CapitalOmega], \[Chi]];
       If[i < (Length[T] - 2), Sow[t0], Tnot += 1], n];
     ][[2]];
  tmean = Mean[Tend];
  Histogram[Tend, {.025}]];

sampl[0, 0, 16, 0.8, 0.01, 1000] // Timing

enter image description here

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3
  • $\begingroup$ You missed that he is iterating over RandomVariate. My solution only takes 0.24 seconds for n=1000. $\endgroup$
    – a20
    Nov 16 '21 at 11:31
  • $\begingroup$ You are right, thank you. $\endgroup$ Nov 16 '21 at 11:37
  • $\begingroup$ She is iterating over RandomVariete ;) Thank you so much both of you! $\endgroup$
    – flg
    Nov 18 '21 at 8:59

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