Deformation operators

[kohr-h]

We need a framework for handling deformations defined by certain actions of deformation groups.

• PR 488 Currently, we have the "standard" group action defined by right composition

phi.f = f o phi = ( x --> f(phi(x)) )

• Issue 505 and the mass-preserving group action using

phi.f = |det(Dphi)| f o phi = ( x --> |det(Dphi(x))| f(phi(x)) )

where Dphi(x) is the Jacobian matrix of phi in x.

[adler-j]

edited the issue to add checklist, checked of non mass preserving and added link to the mass preserving case.

[chongchenmath]

For the mass-preserving or standard deformation operator, the case is a little bit complicated with regard to the derivative and its adjoint. Sometimes we need to compute the Eulerian derivative rather than the standard derivative. So we can first just implement the deformation operator itself.

[kohr-h]

For us, operator.derivative is always the standard derivative. If the Eulerian derivative is required somewhere, it needs to be a separate operator.

[chongchenmath]

Yes. But the operator.derivative might be useless. The another problem is the group actions like: phi.f = f o phi^{-1} or phi.f = |det(Dphi^{-1})| f o phi^{-1} where the phi^{-1} and |det(Dphi^{-1})| are given as arguments directly, rather computed from phi.

[kohr-h]

Yes. But the operator.derivative might be useless.

If it's useless, we don't implement it.

The another problem is the group actions like: phi.f = f o phi^{-1} or phi.f = |det(Dphi^{-1})| f o phi^{-1} where the phi^{-1} and |det(Dphi^{-1})| are given as arguments directly, rather computed from phi.

Makes sense, I don't see a problem as long as we don't need phi as well.

[chongchenmath]

Agree. The phi^{-1} can be seen as a general deformation, and the f o deformation and |det(D deformation)| f o deformation is always useful.

[adler-j]

Closing this in favor of #505 which covers whats left.