MATLABDiffEq.jl

MATLABDiffEq.jl is a common interface binding for the MATLAB ordinary differential equation solvers. It uses the MATLAB.jl interop in order to send the differential equation over to MATLAB and solve it. Note that this package requires the differential equation function to be defined using ParameterizedFunctions.jl.

Note that this package isn't for production use and is mostly just for benchmarking and helping new users migrate models over to Julia. For more efficient solvers, see the DifferentialEquations.jl documentation.

Installation

To install MATLABDiffEq.jl, use the following:

julia using Pkg Pkg.add("MATLABDiffEq")

Using MATLABDiffEq.jl

MATLABDiffEq.jl is simply a solver on the DiffEq common interface, so for details see the DifferentialEquations.jl documentation. However, the only options implemented are those for error calculations (timeseries_error), saveat and tolerances.

Note that the algorithms are defined to have the same name as the MATLAB algorithms, but are not exported. Thus to use ode45, you would specify the algorithm as MATLABDiffEq.ode45().

Example

```julia using MATLABDiffEq, ParameterizedFunctions

f = @ode_def LotkaVolterra begin dx = 1.5x - xy dy = -3y + xy end

tspan = (0.0, 10.0) u0 = [1.0, 1.0] prob = ODEProblem(f, u0, tspan) sol = solve(prob, MATLABDiffEq.ode45())

function lorenz(du, u, p, t) du[1] = 10.0(u[2]-u[1]) du[2] = u[1](28.0-u[3]) - u[2] du[3] = u[1]u[2] - (8/3)*u[3] end u0 = [1.0; 0.0; 0.0] tspan = (0.0, 100.0) prob = ODEProblem(lorenz, u0, tspan) sol = solve(prob, MATLABDiffEq.ode45()) ```

Measuring Overhead

To measure the overhead of over the wrapper, note that the variables from the session will be still stored in MATLAB after the computation is done. Thus you can simply call the same ODE function and time it directly. This is done by:

julia @time MATLABDiffEq.eval_string("[t,u] = $(algstr)(diffeqf,tspan,u0,options);")

To be even more pedantic, you can play around in the actual MATLAB session by using

MATLABDiffEq.show_msession()

Overhead Amount

Generally, for long enough problems the overhead is minimal. Example:

julia using DiffEqBase, ParameterizedFunctions, MATLABDiffEq f = @ode_def_bare RigidBodyBench begin dy1 = -2*y2*y3 dy2 = 1.25*y1*y3 dy3 = -0.5*y1*y2 + 0.25*sin(t)^2 end prob = ODEProblem(f, [1.0; 0.0; 0.9], (0.0, 100.0)) alg = MATLABDiffEq.ode45() algstr = string(typeof(alg).name.name)

For this, we get the following:

```julia julia> @time sol = solve(prob, alg); 0.063918 seconds (38.84 k allocations: 1.556 MB)

julia> @time sol = solve(prob, alg); 0.062600 seconds (38.84 k allocations: 1.556 MB)

julia> @time sol = solve(prob, alg); 0.061438 seconds (38.84 k allocations: 1.556 MB)

julia> @time sol = solve(prob, alg); 0.065460 seconds (38.84 k allocations: 1.556 MB)

julia> @time MATLABDiffEq.eval_string("[t,u] = $(algstr)(diffeqf,tspan,u0,options);") 0.058249 seconds (11 allocations: 528 bytes)

julia> @time MATLABDiffEq.eval_string("[t,u] = $(algstr)(diffeqf,tspan,u0,options);") 0.060367 seconds (11 allocations: 528 bytes)

julia> @time MATLABDiffEq.eval_string("[t,u] = $(algstr)(diffeqf,tspan,u0,options);") 0.060171 seconds (11 allocations: 528 bytes)

julia> @time MATLABDiffEq.eval_string("[t,u] = $(algstr)(diffeqf,tspan,u0,options);") 0.058928 seconds (11 allocations: 528 bytes) ```

Benchmark

The following benchmarks demonstrate a 100x performance advantage for the pure-Julia methods over the MATLAB ODE solvers across a range of stiff and non-stiff ODEs. These were ran with Julia 1.2, MATLAB 2019B, deSolve 1.2.5, and SciPy 1.3.1 after verifying negligible overhead on interop. Note that the MATLAB solvers do outperform that of Python and R.

Non-Stiff Problem 1: Lotka-Volterra

```julia f = @odedefbare LotkaVolterra begin dx = ax - bxy dy = -cy + dxy end a b c d p = [1.5, 1, 3, 1] tspan = (0.0, 10.0) u0 = [1.0, 1.0] prob = ODEProblem(f, u0, tspan, p) sol = solve(prob, Vern7(), abstol = 1/10^14, reltol = 1/10^14) test_sol = TestSolution(sol)

setups = [Dict(:alg=>DP5()) Dict(:alg=>dopri5()) Dict(:alg=>Tsit5()) Dict(:alg=>Vern7()) Dict(:alg=>MATLABDiffEq.ode45()) Dict(:alg=>MATLABDiffEq.ode113()) Dict(:alg=>SciPyDiffEq.RK45()) Dict(:alg=>SciPyDiffEq.LSODA()) Dict(:alg=>SciPyDiffEq.odeint()) Dict(:alg=>deSolveDiffEq.lsoda()) Dict(:alg=>deSolveDiffEq.ode45()) Dict(:alg=>CVODE_Adams())]

names = ["Julia: DP5" "Hairer: dopri5" "Julia: Tsit5" "Julia: Vern7" "MATLAB: ode45" "MATLAB: ode113" "SciPy: RK45" "SciPy: LSODA" "SciPy: odeint" "deSolve: lsoda" "deSolve: ode45" "Sundials: Adams"]

abstols = 1.0 ./ 10.0 .^ (6:13) reltols = 1.0 ./ 10.0 .^ (3:10) wp = WorkPrecisionSet(prob, abstols, reltols, setups; names = names, appxsol = testsol, dense = false, saveeverystep = false, numruns = 100, maxiters = 10000000, timeseries_errors = false, verbose = false) plot(wp, title = "Non-stiff 1: Lotka-Volterra") ```

Non-Stiff Problem 2: Rigid Body

```julia f = @odedefbare RigidBodyBench begin dy1 = -2y2y3 dy2 = 1.25y1y3 dy3 = -0.5y1y2 + 0.25*sin(t)^2 end prob = ODEProblem(f, [1.0; 0.0; 0.9], (0.0, 100.0)) sol = solve(prob, Vern7(), abstol = 1/10^14, reltol = 1/10^14) test_sol = TestSolution(sol)

setups = [Dict(:alg=>DP5()) Dict(:alg=>dopri5()) Dict(:alg=>Tsit5()) Dict(:alg=>Vern7()) Dict(:alg=>MATLABDiffEq.ode45()) Dict(:alg=>MATLABDiffEq.ode113()) Dict(:alg=>SciPyDiffEq.RK45()) Dict(:alg=>SciPyDiffEq.LSODA()) Dict(:alg=>SciPyDiffEq.odeint()) Dict(:alg=>deSolveDiffEq.lsoda()) Dict(:alg=>deSolveDiffEq.ode45()) Dict(:alg=>CVODE_Adams())]

names = ["Julia: DP5" "Hairer: dopri5" "Julia: Tsit5" "Julia: Vern7" "MATLAB: ode45" "MATLAB: ode113" "SciPy: RK45" "SciPy: LSODA" "SciPy: odeint" "deSolve: lsoda" "deSolve: ode45" "Sundials: Adams"]

abstols = 1.0 ./ 10.0 .^ (6:13) reltols = 1.0 ./ 10.0 .^ (3:10) wp = WorkPrecisionSet(prob, abstols, reltols, setups; names = names, appxsol = testsol, dense = false, saveeverystep = false, numruns = 100, maxiters = 10000000, timeseries_errors = false, verbose = false) plot(wp, title = "Non-stiff 2: Rigid-Body") ```

Stiff Problem 1: ROBER

```julia rober = @odedef begin dy₁ = -k₁y₁+k₃y₂y₃ dy₂ = k₁y₁-k₂y₂^2-k₃y₂y₃ dy₃ = k₂y₂^2 end k₁ k₂ k₃ prob = ODEProblem(rober, [1.0, 0.0, 0.0], (0.0, 1e5), [0.04, 3e7, 1e4]) sol = solve(prob, CVODEBDF(), abstol = 1/10^14, reltol = 1/10^14) test_sol = TestSolution(sol)

abstols = 1.0 ./ 10.0 .^ (7:8) reltols = 1.0 ./ 10.0 .^ (3:4);

setups = [Dict(:alg=>Rosenbrock23()) Dict(:alg=>TRBDF2()) Dict(:alg=>RadauIIA5()) Dict(:alg=>rodas()) Dict(:alg=>radau()) Dict(:alg=>MATLABDiffEq.ode23s()) Dict(:alg=>MATLABDiffEq.ode15s()) Dict(:alg=>SciPyDiffEq.LSODA()) Dict(:alg=>SciPyDiffEq.BDF()) Dict(:alg=>SciPyDiffEq.odeint()) Dict(:alg=>deSolveDiffEq.lsoda()) Dict(:alg=>CVODE_BDF())]

names = ["Julia: Rosenbrock23" "Julia: TRBDF2" "Julia: radau" "Hairer: rodas" "Hairer: radau" "MATLAB: ode23s" "MATLAB: ode15s" "SciPy: LSODA" "SciPy: BDF" "SciPy: odeint" "deSolve: lsoda" "Sundials: CVODE"]

wp = WorkPrecisionSet(prob, abstols, reltols, setups; names = names, printnames = true, dense = false, verbose = false, saveeverystep = false, appxsol = test_sol, maxiters = Int(1e5)) plot(wp, title = "Stiff 1: ROBER", legend = :topleft) ```

Stiff Problem 2: HIRES

```julia f = @ode_def Hires begin dy1 = -1.71y1 + 0.43y2 + 8.32y3 + 0.0007 dy2 = 1.71y1 - 8.75y2 dy3 = -10.03y3 + 0.43y4 + 0.035y5 dy4 = 8.32y2 + 1.71y3 - 1.12y4 dy5 = -1.745y5 + 0.43y6 + 0.43y7 dy6 = -280.0y6y8 + 0.69y4 + 1.71y5 - 0.43y6 + 0.69y7 dy7 = 280.0y6y8 - 1.81y7 dy8 = -280.0y6y8 + 1.81y7 end

u0 = zeros(8) u0[1] = 1 u0[8] = 0.0057 prob = ODEProblem(f, u0, (0.0, 321.8122))

sol = solve(prob, Rodas5(), abstol = 1/10^14, reltol = 1/10^14) test_sol = TestSolution(sol)

abstols = 1.0 ./ 10.0 .^ (5:8) reltols = 1.0 ./ 10.0 .^ (1:4);

setups = [Dict(:alg=>Rosenbrock23()) Dict(:alg=>TRBDF2()) Dict(:alg=>RadauIIA5()) Dict(:alg=>rodas()) Dict(:alg=>radau()) Dict(:alg=>MATLABDiffEq.ode23s()) Dict(:alg=>MATLABDiffEq.ode15s()) Dict(:alg=>SciPyDiffEq.LSODA()) Dict(:alg=>SciPyDiffEq.BDF()) Dict(:alg=>SciPyDiffEq.odeint()) Dict(:alg=>deSolveDiffEq.lsoda()) Dict(:alg=>CVODE_BDF())]

names = ["Julia: Rosenbrock23" "Julia: TRBDF2" "Julia: radau" "Hairer: rodas" "Hairer: radau" "MATLAB: ode23s" "MATLAB: ode15s" "SciPy: LSODA" "SciPy: BDF" "SciPy: odeint" "deSolve: lsoda" "Sundials: CVODE"]

wp = WorkPrecisionSet(prob, abstols, reltols, setups; names = names, printnames = true, saveeverystep = false, appxsol = test_sol, maxiters = Int(1e5), numruns = 100) plot(wp, title = "Stiff 2: Hires", legend = :topleft) ```