[petsc-users] Pre-check before each line search function evaluation

[Alexander Lindsay]

Hi Matt, your derivation is spot on. However, the problem is not linear,
which is why I am using SNES. So you need to replace

U = A^{-1} f - A^{-1} B L

with

dU = A^{-1} f - A^{-1} B dL



On Thu, Nov 30, 2023 at 1:47 PM Matthew Knepley <knepley at gmail.com> wrote:

> On Thu, Nov 30, 2023 at 4:23 PM Alexander Lindsay <
> alexlindsay239 at gmail.com> wrote:
>
>> If someone passes me just L, where L represents the "global" degrees of
>> freedom, in this case they represent unknowns on the trace of the mesh,
>> this is insufficient information for me to evaluate my function. Because in
>> truth my degrees of freedom are the sum of the trace unknowns (the unknowns
>> in the global solution vector) and the eliminated unknowns which are
>> entirely local to each element. So I will say my dofs are L + U.
>>
>
> I want to try and reduce this to the simplest possible thing so that I can
> understand. We have some system which has two parts to the solution, L and
> U. If this problem is linear, we have
>
>   / A  B \ / U \ = / f \
>   \ C D / \ L /   \ g /
>
> and we assume that A is easily invertible, so that
>
>   U + A^{-1} B L = f
>   U = f - A^{-1} B L
>
>   C U + D L = g
>   C (f - A^{-1} B L) + D L = g
>   (D - C A^{-1} B) L = g - C f
>
> where I have reproduced the Schur complement derivation. Here, given any
> L, I can construct the corresponding U by inverting A. I know your system
> may be different, but if you are only solving for L,
> it should have this property I think.
>
> Thus, if the line search generates a new L, say L_1, I should be able to
> get U_1 by just plugging in. If this is not so, can you write out the
> equations so we can see why this is not true?
>
>   Thanks,
>
>      Matt
>
>
>> I start with some initial guess L0 and U0. I perform a finite element
>> assembly procedure on each element which gives me things like K_LL, K_UL,
>> K_LU, K_UU, F_U, and F_L. I can do some math:
>>
>> K_LL = -K_LU * K_UU^-1 * K_UL + K_LL
>> F_L = -K_LU * K_UU^-1 * F_U + F_L
>>
>> And then I feed K_LL and F_L into the global system matrix and vector
>> respectively. I do something (like a linear solve) which gives me an
>> increment to L, I'll call it dL. I loop back through and do a finite
>> element assembly again using **L0 and U0** (or one could in theory save off
>> the previous assemblies) to once again obtain the same K_LL, K_UL, K_LU,
>> K_UU, F_U, F_L. And now I can compute the increment for U, dU, according to
>>
>> dU = K_UU^-1 * (-F_U - K_UL * dL)
>>
>> Armed now with both dL and dU, I am ready to perform a new residual
>> evaluation with (L0 + dL, U0 + dU) = (L1, U1).
>>
>> The key part is that I cannot get U1 (or more generally an arbitrary U)
>> just given L1 (or more generally an arbitrary L). In order to get U1, I
>> must know both L0 and dL (and U0 of course). This is because at its core U
>> is not some auxiliary vector; it represents true degrees of freedom.
>>
>> On Thu, Nov 30, 2023 at 12:32 PM Barry Smith <bsmith at petsc.dev> wrote:
>>
>>>
>>>   Why is this all not part of the function evaluation?
>>>
>>>
>>> > On Nov 30, 2023, at 3:25 PM, Alexander Lindsay <
>>> alexlindsay239 at gmail.com> wrote:
>>> >
>>> > Hi I'm looking at the sources, and I believe the answer is no, but is
>>> there a dedicated callback that is akin to SNESLineSearchPrecheck but is
>>> called before *each* function evaluation in a line search method? I am
>>> using a Hybridized Discontinuous Galerkin method in which most of the
>>> degrees of freedom are eliminated from the global system. However, an
>>> accurate function evaluation requires that an update to the "global" dofs
>>> also trigger an update to the eliminated dofs.
>>> >
>>> > I can almost certainly do this manually but I believe it would be more
>>> prone to error than a dedicated callback.
>>>
>>>
>
> --
> What most experimenters take for granted before they begin their
> experiments is infinitely more interesting than any results to which their
> experiments lead.
> -- Norbert Wiener
>
> https://www.cse.buffalo.edu/~knepley/
> <http://www.cse.buffalo.edu/~knepley/>
>
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