more detailed explanation about relax and minimize function

[moo eun je]

I am using mumax to find energy minimum state of ferromagnetic system.

What I am doing doesn't include thermal fluctuation, any current, spin-orbit torque.

just applying in-plane field and getting energy minimum state is everything I want.

regarding energy minimum state, there are two function, relax() and minimize()

I already read mumax github but the explanation is not detailed enough. 

what is the difference between relax and minimize?

I can't understand that there is difference between relax(damping without precession) and minimize (conjugate gradient method)

doesn't gradient of energy equal to torque?

[Josh Lauzier]

Hello,

They are both energy minimization techniques. The state with zero (or minimal) torque is a stationary state, and as such is an energy minima.

Relax() fully solves the LLG equation, but with precession disabled. With precession disabled, the effective field points towards decreasing energy, and you can thus minimize the energy by solving LLG. (and in actuality, relax() actually ends up looking at both energy and torque. It first minimizes the energy until it hits a noise floor. Then it minimizes the torque, which happens to be robust against numerical noise)

Minimize instead minimizes the energy via CG method. (for more detailed documentation, if you search 'minimization' on this board, the author of that function, Ahmad Syukri bin Abdollah has a fairly detailed post on how minimize works. The paper by Abert that he references also compares it to what in mumax is called RK45F, a version of LLG solver. You might find it, and any references/citations, useful for further technical details)

In theory, they are in some sense equivalent ways which should give the same ground state. In practice, sometimes the methods do not converge to the same result. For example, minimize has some known issues- with getting stuck in local minima, failing on high energy states etc. Relax can also get stuck in e.g. a saddle point in the energy landscape, but it tends to be more robust in some situations. CG method has the advantage of often being much much faster (potentially orders of magnitude), computationally, so it is very attractive.

In general, the recommendation is to use relax() for high energy states, like random magnetization. Minimize() does best for in situations where you're already close to the ground state, like hysteresis problems.

One often never really knows for sure that a numerical solution is a 'true' ground state.  If the methods disagree, deciding which result is better can be tricky. One often needs some intuition about the problem, or compare to another solver ( like OOMMF), or perhaps using a mix of both methods (ideally relax might shake minimize out of a local minima, say). In general, from what i've seen people tend to default to the LLG method being more reliable, if forced to pick between the two.

Hopefully that helps. The differences are rather technical, and mainly on the numerical algorithm side of things. For many cases, the distinctions do not matter as long as you follow the advice about high energy states and the like.

Best,
Josh L.