Added reduction monte carlo

[elevien]

Playing around with ReductionMonteCarloSimulation Type and dispatch for monte_carlo_simulation. prob_func is now a keyword argument.

[ChrisRackauckas]

Does it need to be a separate function? I think the original MonteCarloSimulation could be modified to have the fieldname changed to solution_data, and have an argument for output_func(sol). I don't think there's a need to make it only work on last(sol), since the user can turn off all intermediate saving by passing save_timeseries=false (since the splatted kwargs are sent to the solver). That would get better performance than creating a full solver and then discarding all but the last point anyways, and would make the output_func more general.

As for making prob_func a kwarg: did you check the @code_warntype to see if there's any type-stability issues? You always have to be careful about changing the type of the argument in a keyword argument because kwargs don't dispatch (at least until JuliaLang/julia#16580 gets merged). I think it may be okay here, but I think it should be checked for stability and whether there's a performance hit first.

If prob_func is fine as a keyword argument performance-wise, then it all could collapse back into one function with output_func=identity as a kwarg.

[ChrisRackauckas]

Also, you need to make sure you update the tests when you make changes. The CI tests are failing because it's still using prob_func as an optional arg in the 4th test.

[elevien]

Ok thanks. Do you mean get rid of ReductionMonteCarloSimulation, or just the dispatch of monte_carlo_simulation? When running monte_carlo_simulation with output_func, it would be nice if one could run simulations until the sample standard deviation is below some desired tolerance. Does it makes sense to implement this within pmap?

[coveralls]

Coverage remained the same at 55.172% when pulling 72ceb0a on elevien:levien-reduction into e6e553a on JuliaDiffEq:master.

[codecov-io]

Codecov Report

Merging #5 into master will not impact coverage.

@@           Coverage Diff           @@
##           master       #5   +/-   ##
=======================================
  Coverage   55.17%   55.17%           
=======================================
  Files           1        1           
  Lines          29       29           
=======================================
  Hits           16       16           
  Misses         13       13

Impacted Files	Coverage Δ
src/DiffEqMonteCarlo.jl	55.17% <81.25%> (ø)	✅

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[ChrisRackauckas]

Thanks!