(→PDE based valley seeking) |
(→PDE based valley seeking) |
||
| Line 12: | Line 12: | ||
* Simple Example | * Simple Example | ||
| + | |||
| + | Consider 1-D curves | ||
| + | |||
| + | <math>u(x): [0,1] \rightarrow \Re</math> with fixed end points u(0)=a, u(1)=b (3-1) | ||
| + | |||
| + | <<Picture>> | ||
| + | |||
| + | Suppose Energy of curve is | ||
| + | <math>E(u)=\int _0 ^1 F(u, u')dx</math> for some function <math>F: \Re ^2 \rightarrow \Re</math> | ||
| + | |||
| + | e.g.) <math>F(u,u')=|u'|^2</math> (3-2) | ||
| + | |||
| + | The curve that minimizes (or maximizes) E(u) satisfies Euler Equation | ||
| + | |||
| + | <math>\frac{\partial F} {\partial u} -\frac{d}{dx}(\frac{\partial F}{\partial u'})=0</math> (3-3) | ||
| + | |||
| + | sometimes written as <math>E'=0 \Rightarrow \frac{\partial E}{\partial u}=0</math> (3-4) | ||
| + | |||
| + | Similarly if <math>E=\int _0 ^1 F(u,u',u'')dx</math> (3-5) | ||
| + | |||
| + | e.g.) <math>F(u,u',u'')=|u''|^2</math> (3-6) | ||
| + | |||
| + | Then Euler equation is <math>\frac{\partial F}{\partial u} - \frac{d}{dx}(\frac{\partial F}{\partial u'}) + \frac{d}{dx^2}(\frac{\partial F}{\partial u''})=0</math> (3-7) | ||
| + | |||
| + | Similarly, for surface in <math>\Re ^2</math> | ||
| + | |||
| + | <math>u(x,y): [0,1] \times [0,1] \rightarrow \Re</math> (3-8) | ||
| + | |||
| + | Suppose energy is given by | ||
| + | |||
| + | <math>E(u)=\int _{surface} F(u,u_x, u_y, u_{xx}, u_{xy},u_{yy})dxdy</math> (3-9) | ||
| + | |||
| + | e.g.) <math>F={u_x}^2+{u_y}^2</math> (3-10) | ||
| + | |||
| + | Then Euler equation is | ||
Revision as of 10:44, 22 April 2008
Clustering by finding valleys of densities
Graph based implementation
PDE based valley seeking
PDE: Partial Differential Equation
PDE's can be used to minimize energy functionals
- Simple Example
Consider 1-D curves
$ u(x): [0,1] \rightarrow \Re $ with fixed end points u(0)=a, u(1)=b (3-1)
<<Picture>>
Suppose Energy of curve is $ E(u)=\int _0 ^1 F(u, u')dx $ for some function $ F: \Re ^2 \rightarrow \Re $
e.g.) $ F(u,u')=|u'|^2 $ (3-2)
The curve that minimizes (or maximizes) E(u) satisfies Euler Equation
$ \frac{\partial F} {\partial u} -\frac{d}{dx}(\frac{\partial F}{\partial u'})=0 $ (3-3)
sometimes written as $ E'=0 \Rightarrow \frac{\partial E}{\partial u}=0 $ (3-4)
Similarly if $ E=\int _0 ^1 F(u,u',u'')dx $ (3-5)
e.g.) $ F(u,u',u'')=|u''|^2 $ (3-6)
Then Euler equation is $ \frac{\partial F}{\partial u} - \frac{d}{dx}(\frac{\partial F}{\partial u'}) + \frac{d}{dx^2}(\frac{\partial F}{\partial u''})=0 $ (3-7)
Similarly, for surface in $ \Re ^2 $
$ u(x,y): [0,1] \times [0,1] \rightarrow \Re $ (3-8)
Suppose energy is given by
$ E(u)=\int _{surface} F(u,u_x, u_y, u_{xx}, u_{xy},u_{yy})dxdy $ (3-9)
e.g.) $ F={u_x}^2+{u_y}^2 $ (3-10)
Then Euler equation is
