SDE with diagonal noise

General Usage
1

I am trying to solve the following SDE:

du=f(u,p,t)dt+\sigma dW,

where f(u,p,t) can be written as following:

function f(du,u,p,t)

#calculate the derivative of the vector at time j
p=0.0

for i=1:Nx
    if i==2
        du[i]=-(u[i+1]^2-0)/(4*Δx)-p*(u[i+1]-0)/(2*Δx)-(u[i+1]-2*u[i]+0)/(Δx^2)-(u[i]-0+6*u[i]-4*u[i+1]+u[i+2])/(Δx^4)
    end

    if i==3   
        du[i]=-(u[i+1]^2-u[i-1]^2)/(4*Δx)-p*(u[i+1]-u[i-1])/(2*Δx)-(u[i+1]-2*u[i]+u[i-1])/(Δx^2) -(0-4*u[i-1]+6*u[i]-4*u[i+1]+u[i+2])/(Δx^4)
    end


    if i>=4 && i<=Nx-3   # Inner scheme
        du[i]=-(u[i+1]^2-u[i-1]^2)/(4*Δx)-p*(u[i+1]-u[i-1])/(2*Δx)-(u[i+1]-2*u[i]+u[i-1])/(Δx^2) -(u[i-2]-4*u[i-1]+6*u[i]-4*u[i+1]+u[i+2])/(Δx^4)
    end

    if i==Nx-2  # u[i+2]=0
        du[i]=-(u[i+1]^2-u[i-1]^2)/(4*Δx)-p*(u[i+1]-u[i-1])/(2*Δx)-(u[i+1]-2*u[i]+u[i-1])/(Δx^2) -(u[i-2]-4*u[i-1]+6*u[i]-4*u[i+1]+0)/(Δx^4)
    end

    if i==Nx-1

        du[i]=-(0-u[i-1]^2)/(4*Δx)-p*(0-u[i-1])/(2*Δx)-(0-2*u[i]+u[i-1])/(Δx^2)-(u[i-2]-4*u[i-1]+6*u[i]-0+u[i])/(Δx^4)
    end

end

    return du
end

and \sigma=3.0. In order to do that, I wrote the following code:

p = [c]          # parameter
Random.seed!(SEED)
u0 =vec(-0.5 * ones(N, 1) + (0.5 + 0.5) * rand(N, 1))
u0[1] = 0.0
u0[end] = 0.0
tspan = (0.0, Tspan)

###############################################################################

function g(du1,u,p,t)
    du1=ones(1,N).*3.0
    return du1
end

###############################################################################

prob_sde= SDEProblem(f,g, u0, tspan, p)
sol = solve(prob_sde,saveat=Δt)
U = convert(Array, sol)

Now I expect that every time I solve this problem without changing anything, I should get a different solution U due to the noise. However, I always get the same solution. Why is that?

2

Try

(you are not modifying du1, so the solver doesn’t see your choice of dispersion coefficient)

3

Are you sure you want diagonal not scalar noise? By your definition it looks scalar. But for diagonal @mschauer’s response is the correct thing to do.

4

Thanks for your answer. I was doing something similar to here. I thought I could add the diagonal noise this way as with lorenz system. Did I misunderstood the example?

5

Yes, if you want diagonal noise it’s just du . = 0.3 for the general sized form, because you need to respect mutation.