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Sander, Oliver
dune-gfe
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ad8fc1f9
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ad8fc1f9
authored
13 years ago
by
Oliver Sander
Committed by
sander@FU-BERLIN.DE
13 years ago
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Document how the Hessian fd approximation is computed
[[Imported from SVN: r7390]]
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dune/gfe/localgeodesicfestiffness.hh
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@@ -146,7 +146,12 @@ public:
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@@ -146,7 +146,12 @@ public:
This default implementation used finite-difference approximations to compute the second derivatives
This default implementation used finite-difference approximations to compute the second derivatives
The formula for the Riemannian Hessian has been taken from Absil, Mahony, Sepulchre:
The formula for the Riemannian Hessian has been taken from Absil, Mahony, Sepulchre:
'Optimization algorithms on matrix manifolds', page 107
'Optimization algorithms on matrix manifolds', page 107. There it says that
\f[
\langle Hess f(x)[\xi], \eta \rangle
= \frac 12 \frac{d^2}{dt^2} \Big(f(\exp_x(t(\xi + \eta))) - f(\exp_x(t\xi)) - f(\exp_x(t\eta))\Big)\Big|_{t=0}.
\f]
We compute that using a finite difference approximation.
*/
*/
virtual
void
assembleHessian
(
const
Entity
&
e
,
virtual
void
assembleHessian
(
const
Entity
&
e
,
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