       • To: mathgroup at smc.vnet.net
• Subject: [mg91239] Adjoint Frechet Derivative in Mathematica
• From: "Euler@DAMTP" <anthony_ashton1 at hotmail.com>
• Date: Mon, 11 Aug 2008 06:05:43 -0400 (EDT)

```Hi all,

I'm currently trying to implement some code to compute the Frechet derivative, and the associated adjoint, of a system of PDEs. To find the Frechet derivative, I've been using:

FrechetD[support_List, dependVar_List,
independVar_List, testfunction_List] :=
Block[{indep, frechet, deriv, \[Epsilon], r0, x1, x2},
r0 = Function[indep, x1 + \[Epsilon] x2];
frechet = {}; Do[deriv = {};
Do[AppendTo[deriv, \!\(
\*SubscriptBox[\(\[PartialD]\), \(\[Epsilon]\)]\ \((support[[
j]] /. \[IndentingNewLine]dependVar[[
i]] -> \((r0 /. \[IndentingNewLine]{indep ->
independVar, \[IndentingNewLine]x1 ->
dependVar[[i]] @@ independVar, \[IndentingNewLine]x2 ->
testfunction[[i]] @@
independVar})\))\)\) /. \[Epsilon] -> 0],
{i, 1, Length[support]}];
AppendTo[frechet, deriv],
{j, 1, Length[support]}];
frechet]

This works an absolute charm. Then to compute the associated adjoint equation, I use:

independVar_List, testfunction_List] :=
Block[{subrule, \$testf, frechet, n, b},
subrule = b_. (\$testf^(n__)) @@ independVar :>
(-1)^Plus @@ {n} \!\(
0]\)]\((b\ \$testf @@ independVar)\)\);
frechet = FrechetD[support, dependVar,
independVar , testfunction];
Do[frechet =
frechet /.
(subrule /. \$testf -> testfunction[[i]]),
{i, 1, Length[testfunction]}];
frechet = Transpose[frechet]]

This doesn't work, sadly. All it seems to do is give me the transpose of the original Frechet derivative matrix. The code originally came from a book I've been looking at, so it can't be too far wrong. Sadly, my mathematica skills aren't good enough to spot why it's going wrong. If anyone could help if would be much appreciated!

:)

```

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