DifferentialEquations.jl provides a benchmarking suite to be able to test the difference in error, speed, and efficiency between algorithms. DifferentialEquations.jl includes current benchmarking notebooks to help users understand the performance of the methods. These benchmarking notebooks use the included benchmarking suite. There are two parts to the benchmarking suite: shootouts and work-precision. The
Shootout tests methods head-to-head for timing and error on the same problem. A
WorkPrecision draws a work-precision diagram for the algorithms in question on the chosen problem.
To use the benchmarking notebooks, IJulia is required. The commands are as follows:
]add "www.github.com/JuliaDiffEq/DiffEqBenchmarks" using IJulia notebook(dir = Pkg.dir("DiffEqBenchmarks"))
A shootout is where you compare between algorithms. For example, to see how different Runge-Kutta algorithms fair against each other, one can define a setup which is a dictionary of Symbols to Any, where the symbol is the keyword argument. Then you call
Shootout on that setup. The code is as follows:
using OrdinaryDiffEq,DiffEqProblemLibrary.ODEProblemLibrary,DiffEqDevTools,ODE,ODEInterface,ODEInterfaceDiffEq ODEProblemLibrary.importodeproblems() prob = ODEProblemLibrary.prob_ode_2Dlinear setups = [Dict(:alg=>DP5()) Dict(:abstol=>1e-3,:reltol=>1e-6,:alg=>ode45()) # Fix ODE to be normal Dict(:alg=>dopri5())] names = ["DifferentialEquations";"ODE";"ODEInterface"] shoot = Shootout(prob,setups;dt=1/2^(10),names=names)
Note that keyword arguments applied to
Shootout are applied to every run, so in this example every run has the same starting timestep. Here we explicitly chose names. If you don't, then the algorithm name is the default. This returns a Shootout type which holds the times it took for each algorithm and the errors. Using these, it calculates the efficiency defined as 1/(error*time), i.e. if the error is low or the run was quick then it's efficient.
print(shoot) will show all of this information, and
plot(shoot) will show the efficiencies of the algorithms in comparison to each other.
For every benchmark function there is a special keyword
numruns which controls the number of runs used in the time estimate. To be more precise, these functions by default run the algorithm 20 times on the problem and take the average time. This amount can be increased and decreased as needed.
appxsol allows for specifying a reference against which the error is computed. The method of error computation can be specified by the keyword
error_estimate with values
:L2 for the L2 error over the solution time interval,
:l2 calculates the l2 error at the actual steps and the default
:final only compares the endpoints.
A ShootoutSet is a where you define a vector of probs and tspans and run a shootout on each of these values.
A WorkPrecision calculates the necessary componnets of a work-precision plot. This shows how time scales with the user chosen tolerances on a given problem. To make a WorkPrecision, you give it a vector of absolute and relative tolerances:
abstols = 1 ./10 .^ (3:10) reltols = 1 ./10 .^ (3:10) wp = WorkPrecision(prob,DP5(),abstols,reltols;name="Dormand-Prince 4/5")
If we want to plot many WorkPrecisions together in order to compare between algorithms, you can make a WorkPrecisionSet. To do so, you pass the setups into the function as well:
wp_set = WorkPrecisionSet(prob,tspan,abstols,reltols,setups;numruns=2) setups = [Dict(:alg=>RK4());Dict(:alg=>Euler());Dict(:alg=>BS3()); Dict(:alg=>Midpoint());Dict(:alg=>BS5());Dict(:alg=>DP5())] wp_set = WorkPrecisionSet(prob,abstols,reltols,setups;dt=1/2^4,numruns=2)
Both of these types have a plot recipe to produce a work-precision diagram, and a print which will show some relevant information.