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revision 3838 by gross, Fri Feb 24 06:23:45 2012 UTC revision 3840 by gross, Sun Feb 26 22:37:39 2012 UTC
# Line 39  dx Line 39  dx
39  \end{equation}  \end{equation}
40  A more general from is given as  A more general from is given as
41  \begin{equation}\label{APP FIT EQU 4}  \begin{equation}\label{APP FIT EQU 4}
42  J(u,p) = \frac{1}{2} \int_{\Omega} H^2\; dx + \frac{1}{2} \int_{\partial \Omega} h^2 \; ds  J(u,p) = \int_{\Omega} H\; dx +  \int_{\partial \Omega} h \; ds
43  \end{equation}  \end{equation}
44  where $H$ is a scalar, possibly spatially variable  function  where $H$ is a scalar, possibly spatially variable  function
45  of the solution $u_i$ and the parameter $p_i$ and their gradients  of the solution $u_i$ and the parameter $p_i$ and their gradients
# Line 47  and  $H$ is a scalar, possibly spatially Line 47  and  $H$ is a scalar, possibly spatially
47  of the solution $u_i$ and the parameter $p_i$. For example~(\ref{APP FIT EQU 2a})  of the solution $u_i$ and the parameter $p_i$. For example~(\ref{APP FIT EQU 2a})
48  one has  one has
49  \begin{equation}\label{APP FIT EQU 2a}  \begin{equation}\label{APP FIT EQU 2a}
50  H = \sqrt{ \chi \cdot (d_i u_{,i} - \hat{g})^2  H = \frac{1}{2} \chi \cdot (d_i u_{,i} - \hat{g})^2 +  \frac{1}{2} (p_{,i}p_{,i})^{2}
 + (p_{,i}p_{,i})^{2} }  
51  \end{equation}  \end{equation}
52  So task is to minimize the cost function $J$ over $u$ and $p$    So task is to minimize the cost function $J$ over $u$ and $p$  
53  subject to the PDE~(\ref{APP FIT EQU 1a} connection $p$ and $u$. The secondary condition  subject to the PDE~(\ref{APP FIT EQU 1a} connection $p$ and $u$. The secondary condition
54  is mixed into the cost function using a Lagrangean multiplier before the variation is calculated:  is mixed into the cost function using a Lagrangean multiplier before the variation is calculated:
55  \begin{equation}\label{APP FIT EQU 5}  \begin{equation}\label{APP FIT EQU 5}
56  J(u,p,\lambda) = \frac{1}{2} \int_{\Omega} H^2 \; dx + \frac{1}{2} \int_{\partial \Omega} h^2 \; ds  J(u,p,\lambda) = \int_{\Omega} H \; dx + \int_{\partial \Omega} h \; ds
57  + \int_{\Omega} \lambda_{i,j} \cdot X_{ij} + \lambda_{i} \cdot Y_{i} \; dx  + \int_{\Omega} \lambda_{i,j} \cdot X_{ij} + \lambda_{i} \cdot Y_{i} \; dx
58  + \int_{\partial \Omega}  \lambda_{i} \cdot y_{i} \; ds  + \int_{\partial \Omega}  \lambda_{i} \cdot y_{i} \; ds
59  \end{equation}  \end{equation}
# Line 70  J(u,p,\lambda) = \int_{\Omega} Z \; dx Line 69  J(u,p,\lambda) = \int_{\Omega} Z \; dx
69  \end{equation}  \end{equation}
70  with  with
71  \begin{align}\label{APP FIT EQU 6}  \begin{align}\label{APP FIT EQU 6}
72   Z = \frac{1}{2}  H^2 + \lambda_{i,j} \cdot X_{ij} +  \lambda_{i} \cdot Y_{i} \\   Z =  H+ \lambda_{i,j} \cdot X_{ij} +  \lambda_{i} \cdot Y_{i} \\
73  z=\frac{1}{2} h^2 + \lambda_{i} \cdot y_{i}  z= h  + \lambda_{i} \cdot y_{i}
74  \end{align}  \end{align}
75    
76  We are taking variation along $p$:  We are taking variation along $p$:

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