> On Jul 1, 2015, at 1:19 AM, Michael White <mrwh...@umn.edu> wrote:
>
> Hello,
>
> I am not quite sure how to use the L-BFGS implementation in PETSc with a
> preconditioner that approximates the Jacobian.
The documentation in the manual pages is terrible for this.
> As far as I understand it, L-BFGS is like Newton's Method but doesn't require
> the user to input the Jacobian matrix.
Correct. BFGS builds a rank-k approximation to the (inverse) of the
Jacobian at each iteration k. By default in PETSc for the zero iteration it
uses the identity as the initial approximation of the (inverse) of the Jacobian
which is generally not so good. However instead of using the identity it can
use a matrix (which approximates the Jacobian) for the initial approximation
of the (inverse) of the Jacobian and then build a rank k update to that
approximation. Essentially you can think of this as running the basic BFGS
algorithm on the new problem B*F(u) where B is the preconditioner for the
Jacobian you provided at the first iteration. (Note that if you provide the
true Jacobian as your "approximate" Jacobian then the first iteration (but not
the later iterations) is exactly the same as the first iteration with Newtons'
method).
> However, for my application, I do have a good preconditioner that
> approximates the Jacobian matrix.
You don't mean this. A "good preconditioner" cannot approximate the
Jacobian matrix. A good preconditioner needs to approximate the inverse of the
Jacobian.
So either
1) you have a "good preconditioner" that is, an operator that approximates the
inverse of the Jacobian or
2) or a sparse matrix that approximates the Jacobian from which you hope a
reasonable preconditioner can be constructed.
Based on your other text I am guessing you have 2.
> My question is where to input such a preconditioner. I have 3 possible
> guesses about the correct way to do it.
>
> 1. Use SNESSetJacobian to provide a function which computes the
> preconditioner. I'm unsure if this information is even used though, since
> L-BFGS builds its own approximation to the inverse Jacobian as it goes.
>
> 2. Use SNESQNSetScaleType with SNES_QN_SCALE_JACOBIAN to set my
> preconditioner as an initial approximation to the Jacobian somehow?
You must use 1) and 2) together. What happens is QN computes the
preconditioner B at the first iteration and uses that in combination with the
rank k "update" to approximate the future Jacobians.
Barry
>
> 3. Use an SNESSetNPC with my preconditioner as a linear preconditioner?
>
> Anyone that could point me in the right direction would be greatly
> appreciated.
>
> Best regards,
> Michael White
>
>
