SDAE Problems
Mathematical Specification of a Stochastic Differential-Algebraic Equation (SDAE) Problem
To define an SDAE, you simply define an SDE Problem with the forcing function f
, the noise function g
, a mass matrix M
and the initial condition u₀
which define the SDAE in mass matrix form:
\[M du = f(u,p,t)dt + Σgᵢ(u,p,t)dWⁱ\]
f
and g
should be specified as f(u,p,t)
and g(u,p,t)
respectively, and u₀
should be an AbstractArray whose geometry matches the desired geometry of u
. Note that we are not limited to numbers or vectors for u₀
; one is allowed to provide u₀
as arbitrary matrices / higher dimension tensors as well. A vector of g
s can also be defined to determine an SDE of higher Ito dimension.
Nonsingular mass matrices correspond to constraint equations and thus a stochastic DAE.
Example
const mm_A = [-2.0 1 4
4 -2 1
0 0 0]
function f!(du,u,p,t)
du[1] = u[1]
du[2] = u[2]
du[3] = u[1] + u[2] + u[3] - 1
end
function g!(du,u,p,t)
@. du = 0.1
end
prob = SDEProblem(SDEFunction(f!,g!;mass_matrix=mm_A),g!,
ones(3),(0.0,1.0))