Evaluating values from a indexed ParametricNDSolve solution in parallel

Question

I have a ParametricNDsolve solution for a differential equation for a huge system given by s1. I have no problem in evaluating the value using the solution. It's, in fact, speeding up the process using Parallel cores.

I have represented the variables by a, b, c and d, some of which are indexed by j, and the parameters are g, n1and n2. My attempt is to create a table of all the required variables and replace s1 to them as in Tab1and finally evaluate t shown by Tab2

Tab1 = Table[ Table[Table[{a[j, t][g, n1, n2], b[j, t][g, n1, n2], 
   c[g, n1, n2][t], d[g, n1, n2][t]}, {j, 1, 2}], {g, 1, 3, 1}], {n1, 1, 4}, {n2, 1, 4}] /. s1; (*first step*)

Tab2=Tab1/.t->12 (say)

So,Tab1 is only a replacement of the solution s1 to the given variables a, b, c and d, which takes me more time to evaluate than Tab2. Is there any way I can parallel map or parallelEvaluate Tab1 using all cores available such that s1 is mapped to the table of variables simultaneously in parallel

Answer 1

Since your Table is 5 levels deep to an element we can do this:

Tab0 = Table[{a[j, t][g, n1, n2], b[j, t][g, n1, n2], c[g, n1, n2][t], d[g, n1, n2][t]},
 {n1, 1, 4}, {n2, 1, 4}, {g, 1, 3, 1}, {j, 1, 2}];

Tab1 = ParallelMap[# /. s1 &, Tab0, {5}];

It might also work at 4 levels deep too - you'd want to experiment to choose the right level that offers the best performance.

I am assuming s1 is a proper rule in a list of the form {LHS -> RHS}. Of course there is no guarantee this will be faster.

We could also do the replacement inside the Table during construction without needing the Map at all, and this would save memory:

Tab1 = ParallelTable[{
 a[j, t][g, n1, n2],
 b[j, t][g, n1, n2],
 c[g, n1, n2][t],
 d[g, n1, n2][t]} /. s1,
 {n1, 1, 4}, {n2, 1, 4}, {g, 1, 3, 1}, {j, 1, 2}];