design: Simple gMAM

[reykboerner]

After giving it some thought, here's what I would suggest for sgMAM:

• I would rename sgmam to simple_geometric_min_action_method following the style conventions of JuliaDynamics (alternative: we drop the _method in all functions we have defined)
• I would rename SgmamSystem(H_x, H_p) to HamiltonsTerms(H_x, H_p) or similar (can't think of a great name but it's not really Hamilton's equations and not the Hamiltonian either)
• It makes sense to write sgmam as a function of SgmamSystem, which I would interpret as a mid-level function. It would be great to have a function HamiltonsTerms(CoupledSDEs) that extracts the terms H_x and H_p from a suitable CoupledSDEs system, and then one could write a high-level function:

simple_geometric_min_action_method(CoupledSDEs) = simple_geometric_min_action_method(HamiltonsTerms(CoupledSDEs))

I suppose it would be some work to write the function HamiltonsTerms(CoupledSDEs) generally, so maybe we can leave that as an open issue and for now just

• rename the type/function
• explain in the docs that HamiltonsTerms is related to the underlying SDE via

Happy to hear your thoughts @oameye :)

[oameye]

• I am happy to rename sgmam to simple_geometric_min_action

• For SgmamSystem(H_x, H_p) we can do ExtendedPhaseSpace

It would be great to have a function HamiltonsTerms(CoupledSDEs) that extracts the terms H_x and H_p from a suitable CoupledSDEs system, and then one could write a high-level function:

• performance wise this will be terrible. The code will have to take the derivative every time it evaluates the phase space function. We could instead have a symbolic interface. Then the derivative has to be only computed once.

[reykboerner]

I like the symbolic idea, would be great for many aspects of the package but for now it's fine to leave it then.

Not sure about ExtendedPhaseSpace because the object is not a phase space, more like a gradient of the Hamiltonian in extended coordinates, right? How about ExtendedHamiltonEq?

[oameye]

Ok lol turns out I already wrote SgmamSystem(ds::ContinuousTimeDynamicalSystem) method

CriticalTransitions.jl/src/largedeviations/sgMAM.jl

Lines 12 to 40 in 4ccae50

	function SgmamSystem(ds::ContinuousTimeDynamicalSystem)
	if ds isa CoupledSDEs
	proper_sgMAM_system(ds)
	end
	f = dynamic_rule(ds)
	jac = jacobian(ds)
	function H_x(x, p) # ℜ² → ℜ²
	Hx = similar(x)
	for idx in 1:size(x, 2)
	jax = jac(x[:, idx], (), 0.0)
	for idc in 1:size(x, 1)
	Hx[idc, idx] = dot(jax[:, idc], p[:, idx])
	end
	end
	return Hx
	end
	function H_p(x, p) # ℜ² → ℜ²
	Hp = similar(x)
	for idx in 1:size(x, 2)
	Hp[:, idx] = p[:, idx] + f(x[:, idx], ds.p0, 0.0)
	end
	return Hp
	end
	return new{isinplace(ds),dimension(ds),typeof(H_x),typeof(H_p)}(H_x, H_p)
	end

[oameye]

Not sure about ExtendedPhaseSpace because the object is not a phase space, more like a gradient of the Hamiltonian in extended coordinates, right? How about ExtendedHamiltonEq?

I like ExtendedHamiltonianSystem to make it that it in the future can carry more then equations. You agree?

[oameye]

I actually disagree with you. The extended Hamiltonian defined a new phase space in the classical dynamics sense. You embed the (dissiaptive) vectorflow in an energy conserving one. The vectorflow will be the contours of the extended hamiltonian. The new conjugate momenta and coordinates do define a phase space.

I largely think ExtendedPhaseSpace is better, because we not really define a Hamiltonian, so ExtendedHamiltonianSystem conflicts.