On a new operator acting on the right side

[albertomercurio]

The current defined operators act only on the left.

For example

A = MatrixOperator(rand(10,10))
u = rand(10, 10)

the action A(u, p, t) is equivalent to A.A * u, acting on the left.

It would be very helpful to define an operator that allows to act the embedded operators from the right instead. Something like

O = RightActionOperator(A)

O(u, p, t) == u * A.A # true

Do you think it is something possible?

[ChrisRackauckas]

That would be the adjoint operator and it's defined via reverse-mode AD of the normal operator.

[albertomercurio]

In practice, given a AbstractSciMLOperator, how can I compute the action of it but acting on the right?

[ChrisRackauckas]

In general you cannot, but if the operator is linear in it's action on v, then it is given by the push forward of u and reverse mode AD pullback by v