DiffEqBenchmarks.jl holds Jupyter notebooks showing the benchmarks for the JuliaDiffEq ecosystem.
Viewing the Notebooks Locally
To view the notebooks locally and interact with the contents, use the following commands (requires IJulia):
Pkg.clone("https://github.com/JuliaDiffEq/DiffEqBenchmarks.jl") using IJulia notebook(dir=Pkg.dir("DiffEqBenchmarks"))
Table of Contents
The notebooks can be viewed remotely on Github or via nbviewer
- Non-stiff ODEs
- Stiff ODEs
- Dynamical ODEs
- Nonstiff SDEs
- Stiff SDEs
- Nonstiff DDEs
- Stiff DDEs
- Parameter Estimation
- Lorenz Equation Parameter Estimation by Optimization Methods
- Bayesian Lotka-Volterra Parameter Estimation
- Bayesian Lorenz Equation Estimation
- Bayesian FitzHugh-Nagumo Equation Estimation
- Lotka Volterra Equation Parameter Estimation by Optimization Methods
- FitzHugh-Nagumo Equation Parameter Estimation by Optimization Methods
The following tests were developed for the paper Adaptive Methods for Stochastic Differential Equations via Natural Embeddings and Rejection Sampling with Memory. These notebooks track their latest developments.
The following is a quick summary of the benchmarks. These paint broad strokes over the set of tested equations and some specific examples may differ.
- OrdinaryDiffEq.jl's methods are the most efficient by a good amount
Vernmethods tend to do the best in every benchmark of this category
- At lower tolerances,
Tsit5does well consistently.
- ARKODE and Hairer's
dop853perform very similarly, but are both far less efficient than the
- The multistep methods,
lsoda, tend to not do very well.
- The ODEInterface multistep method
ddeabmdoes not do as well as the other multistep methods.
- ODE.jl's methods are not able to consistently solve the problems.
- Fixed time step methods are less efficient than the adaptive methods.
- In this category, the best methods are much more problem dependent.
- For smaller problems:
TRBDF2tend to be the most efficient at high
Rodas5tend to be the most efficient at low tolerances.
- For larger problems (Filament PDE):
CVODE_BDFdoes the best at all tolerances.
- The ESDIRK methods like
KenCarp4can come close.
radauis always the most efficient when tolerances go to the low extreme (
- Fixed time step methods tend to diverge on every tested problem because the high stiffness results in divergence of the Newton solvers.
- ARKODE is very inconsistent and requires a lot of tweaking in order to not
diverge on many of the tested problems. When it doesn't diverge, the similar
algorithms in OrdinaryDiffEq.jl (
KenCarp4) are much more efficient in most cases.
- ODE.jl and GeometricIntegrators.jl fail to converge on any of the tested problems.
- Higher order (generally order >=6) symplectic integrators are much more efficient than the lower order counterparts.
- For high accuracy, using a symplectic integrator is not preferred. Their extra cost is not necessary since the other integrators are able to not drift simply due to having low enough error.
- In this class, the
DPRKNmethods are by far the most efficient. The
Vernmethods do well for not being specific to the domain.
- For simple 1-dimensional SDEs at low accuracy, the
RKMilmethods can do well. Beyond that, they are simply outclassed.
SRImethods both are very similar within-class on the simple SDEs.
SRA3is the most efficient when applicable and the tolerances are low.
- Generally, only low accuracy is necessary to get to sampling error of the mean.
- The adaptive method is very conservative with error estimates.
- The high order adaptive methods (
SRIW1) generally do well on stiff problems.
- The "standard" low-order implicit methods,
ImplicitRK, do not do well on all stiff problems. Some exceptions apply to well-behaved problems like the Stochastic Heat Equation.
- The efficiency ranking tends to match the ODE Tests, but the cutoff from low to high tolerance is lower.
Tsit5does well in a large class of problems here.
Vernmethods do well in low tolerance cases.
- The Rosenbrock methods, specifically
Rodas5, perform well.
- Broadly two different approaches have been used, Bayesian Inference and Optimisation algorithms.
- In general it seems that the optimisation algorithms perform more accurately but that can be attributed to the larger number of data points being used in the optimisation cases, Bayesian approach tends to be slower of the two and hence lesser data points are used, accuracy can increase if proper data is used.
- Within the different available optimisation algorithms, BBO from the BlackBoxOptim package and GN_CRS2_LM for the global case while LD_SLSQP,LN_BOBYQA and LN_NELDERMEAD for the local case from the NLopt package perform the best.
- Another algorithm being used is the QuadDIRECT algorithm, it gives very good results in the shorter problem case but doesn't do very well in the case of the longer problems.
- The choice of global versus local optimization make a huge difference in the timings. BBO tends to find the correct solution for a global optimization setup. For local optimization, most methods in NLopt, like :LN_BOBYQA, solve the problem very fast but require a good initial condition.
- The different backends options available for Bayesian method offer some tradeoffs beteween time, accuracy and control. It is observed that sufficiently high accuracy can be observed with any of the backends with the fine tuning of stepsize, constraints on the parameters, tightness of the priors and number of iterations being passed.