Clarification: JacVecOperator interface

[lindnemi]

Hi, I'm trying to implement a custom JacVecOperator.

In the docs it sounds like L(u,p,t) should be multiplication of L with a vector u. If L = Jac(x) is some Jacobian at location x that would be the directional derivative Jac(x) * u.

On the other hand here it seems that the function call L(u,p,t) will always evaluate to Jac(u) * u. Meaning that the direction and the point at which Jac is evaluated will always be the same.

In this case, how am I supposed to implement the general directional derivative Jac(u) * z where z is an arbitrary direction?

[ChrisRackauckas]

* should not update the Jacobian, as you say that would allow Jac(x) * u. Using L(u,p,t) as a function inside of the ODE defines the ODE Jac(u)*u which is a representation of the original nonlinear operator and is thus useful in things like local sensitivity analysis, hence the default.

[lindnemi]

Thank you for the quick answer. So there are basically three things to implement for a JacVecOperator:

• L(u,p,t) for Jac(u) * u
• * for Jac(u) * x
• update_coefficients

Each of which should be given in mutating and oop form. I would love to see a brief section specifically about JacVecOperator in the docs.

[ChrisRackauckas]

Yes, http://diffeqoperators.sciml.ai/dev/operators/jacobian_vector_product/ needs a bit more.