[petsc-users] Tao iterations

[Jed Brown]

Justin Chang <jychang48 at gmail.com> writes:

> Jed,
>
> Thank you for your response. I agree completely with all that you said. I
> just wonder what would happen if I attempted to use the TAO routines after
> forming the Jacobian J and residual r arising from the advection diffusion
> equation.
>
> In my current (linear) diffusion framework, i have the following objective
> function and gradient vector:
>
> f = \frac{1}{2} x \cdot J*x + x\cdot[r - J*x^(0)]

If J is non-symmetric, then you're no longer looking for a minimum of
this functional.  Let's take a prototypical example in 2 variables

  J = [0 1; -1 0]

Now the J-"inner product"

  conj(x) \cdot J x = 0

for all real-valued x.  (I'll assume you're working over the reals.)
Similarly, the J-"inner product" with

  J = [1 1; -1 1]

is identical to that with J = eye(2), but obviously you want the
anti-symmetric part to affect your solution.

In short, none of this makes mathematical sense in the way you intend if
J is nonsymmetric.
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