From ad8fc1f9a8a285b27c3a64d4ba3f4a0808a93f60 Mon Sep 17 00:00:00 2001
From: Oliver Sander <sander@igpm.rwth-aachen.de>
Date: Tue, 7 Jun 2011 10:27:37 +0000
Subject: [PATCH] Document how the Hessian fd approximation is computed
[[Imported from SVN: r7390]]
---
dune/gfe/localgeodesicfestiffness.hh | 7 ++++++-
1 file changed, 6 insertions(+), 1 deletion(-)
diff --git a/dune/gfe/localgeodesicfestiffness.hh b/dune/gfe/localgeodesicfestiffness.hh
index b91bcc20..7e332e90 100644
--- a/dune/gfe/localgeodesicfestiffness.hh
+++ b/dune/gfe/localgeodesicfestiffness.hh
@@ -146,7 +146,12 @@ public:
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:
- '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,
--
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