Skip to main content

differential equations - When does NDSolve parallelize ODE system solving?


I've long believed that NDSolve cannot make use of multiple cores to solve ODE system, but things seem to be different at least since v12. Consider the following toy example:


$Version
(* "12.0.0 for Microsoft Windows (64-bit) (May 19, 2019)" *)

(* Tested on a 8-core machine. *)

eq = 2 y''[x] == y'[x] - 3 y[x] - 4;
ic = {y[0] == 5, y'[0] == 7};

With[{n = 4096}, sys = Table[{eq, ic} /. y -> y@i, {i, n}];
                 NDSolve[sys, y /@ Range@n, {x, 0, 10^3}]]

enter image description here


Seems that when n >= 4096, NDSolve automatically parallelizes. I'm pretty sure this isn't the case in v9.



So my question is:




  1. In which version was this optimization introduced?




  2. In what situation does the parallelization happen? More tests show the threshold depends on the specific system being solved. In certain situations (e.g. this) parallelization seems to never happen.




  3. Can we make NDSolve parallel for n < 4096? Is the parallelization controlled by any option?






Answer



The number 8192 is an autoparallelization threshold (see SystemOptions["ParallelOptions"]). There are 2*4096 = 8192 variables, y[k] and y'[k], in the system, perhaps a coincidence. The threshold can be lowered with SetSystemOptions, and one can see that when it is run, autoparallelization kicks in.


With[{ops = SystemOptions["ParallelOptions"]},
 Internal`WithLocalSettings[
  SetSystemOptions[
   "ParallelOptions" -> 
    "VectorParallelLengthThresholds" -> {8192, 8192, 8192, 8192, 
       8192}/2

   ],
  With[{n = 4096/2}, sys = Table[{eq, ic} /. y -> y@i, {i, n}];
   NDSolve[sys, y /@ Range@n, {x, 0, 10^3}]],
  SetSystemOptions[ops]
  ]]

This is a system arithmetic feature (vectorization), not an NDSolve feature per se.


Comments

Post a Comment

Popular posts from this blog

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...
Post a Comment
Read more

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...
Post a Comment
Read more

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...
Post a Comment
Read more