 # Diff of /trunk/doc/inversion/CostFunctions.tex

revision 4146 by gross, Tue Jan 15 09:06:06 2013 UTC revision 4147 by gross, Fri Jan 18 03:35:15 2013 UTC
# Line 204  J^{reg}(m) = \int_{\Omega} K^{reg} \; dx Line 204  J^{reg}(m) = \int_{\Omega} K^{reg} \; dx
204  where $K^{reg}$ is a given function of the  where $K^{reg}$ is a given function of the
205  level set function $m_k$ and its spatial derivative $m_{k,i}$. If $n$ is an increment to the level set function  level set function $m_k$ and its spatial derivative $m_{k,i}$. If $n$ is an increment to the level set function
206  then the directional derivative of $J^{ref}$ in the direction of $n$ is given as  then the directional derivative of $J^{ref}$ in the direction of $n$ is given as
207  \begin{equation}\label{REF:EQU:INTRO 2a}  \begin{equation}\label{REF:EQU:INTRO 2aa}
208  <n, \nabla J^{reg}(m)> = \int_{\Omega} \frac{ \partial K^{reg}}{\partial m_k} n_k + \frac{ \partial K^{reg}}{\partial m_{k,i}} n_{k,i} \; dx  <n, \nabla J^{reg}(m)> = \int_{\Omega} \frac{ \partial K^{reg}}{\partial m_k} n_k + \frac{ \partial K^{reg}}{\partial m_{k,i}} n_{k,i} \; dx
209  \end{equation}  \end{equation}
210  where $<.,.>$ denotes the dual product, see Chapter~\ref{chapter:ref:Minimization}. Consequently, the gradient $\nabla J^{reg}$  where $<.,.>$ denotes the dual product, see Chapter~\ref{chapter:ref:Minimization}. Consequently, the gradient $\nabla J^{reg}$
# Line 216  can be represented by a pair of values $Line 216 can be represented by a pair of values$
216  \end{array}  \end{array}
217  \end{equation}  \end{equation}
218  while the dual product $<.,.>$ of a level set increment $n$ and a gradient increment $g=(Y,X)$ is given as  while the dual product $<.,.>$ of a level set increment $n$ and a gradient increment $g=(Y,X)$ is given as
219  \begin{equation}\label{REF:EQU:INTRO 2a}  \begin{equation}\label{REF:EQU:INTRO 2aaa}
220  <n,g> = \int_{\Omega} Y_k n_k + X_{ki} n_{k,i} \; dx  <n,g> = \int_{\Omega} Y_k n_k + X_{ki} n_{k,i} \; dx
221  \end{equation}  \end{equation}
222  We also need to provide (an approximation of) the value $p$ of the inverse of the Hessian operator $\nabla \nabla J$  We also need to provide (an approximation of) the value $p$ of the inverse of the Hessian operator $\nabla \nabla J$
# Line 246  which is solved using \escript \class{Li Line 246  which is solved using \escript \class{Li
246  \end{equation}  \end{equation}
247  The calculation of the gradient of the forward model component is more complicated:  The calculation of the gradient of the forward model component is more complicated:
248  the data defect $J^{f}$ for forward model $f$ is expressed use a cost function kernel $K^{f}$  the data defect $J^{f}$ for forward model $f$ is expressed use a cost function kernel $K^{f}$
249  \begin{equation}\label{REF:EQU:INTRO 2b}  \begin{equation}\label{REF:EQU:INTRO 2bb}
250  J^{f}(p^f) = \int_{\Omega} K^{f} \; dx  J^{f}(p^f) = \int_{\Omega} K^{f} \; dx
251  \end{equation}  \end{equation}
252  In this case the cost function kernel $K^{f}$ is a function of the  In this case the cost function kernel $K^{f}$ is a function of the
# Line 283  for all increments $r$ to the stat $u$. Line 283  for all increments $r$ to the stat $u$.
283  model. They depend of the state variable $u_{k}$ and its gradient $u_{k,i}$ and the physical parameter $p$. The change  model. They depend of the state variable $u_{k}$ and its gradient $u_{k,i}$ and the physical parameter $p$. The change
284  $d_k$  of the state  $d_k$  of the state
285  $u_f$ for physical parameter $p$ in the direction of $q$ is given from the equation  $u_f$ for physical parameter $p$ in the direction of $q$ is given from the equation
286  \begin{equation}\label{REF:EQU:costfunction 100b}  \begin{equation}\label{REF:EQU:costfunction 100bb}
287   \int_{\Omega}   \int_{\Omega}
288  \displaystyle{\frac{\partial F_k }{\partial u_l }  } d_l r_k +  \displaystyle{\frac{\partial F_k }{\partial u_l }  } d_l r_k +
289  \displaystyle{\frac{\partial F_k }{\partial u_{l,j}} } d_{l,j} r_k +  \displaystyle{\frac{\partial F_k }{\partial u_{l,j}} } d_{l,j} r_k +
# Line 295  $u_f$ for physical parameter $p$ in the Line 295  $u_f$ for physical parameter $p$ in the
295  \end{equation}  \end{equation}
296  to be fulfilled for all functions $r$. Now let $d^*_k$ be the solution of the  to be fulfilled for all functions $r$. Now let $d^*_k$ be the solution of the
297  variational equation  variational equation
298  \begin{equation}\label{REF:EQU:costfunction 100d}  \begin{equation}\label{REF:EQU:costfunction 100dd}
299   \int_{\Omega}   \int_{\Omega}
300  \displaystyle{\frac{\partial F_k }{\partial u_l }  } h_l d^*_k +  \displaystyle{\frac{\partial F_k }{\partial u_l }  } h_l d^*_k +
301  \displaystyle{\frac{\partial F_k }{\partial u_{l,j}} } h_{l,j} d^*_k +  \displaystyle{\frac{\partial F_k }{\partial u_{l,j}} } h_{l,j} d^*_k +
# Line 308  variational equation Line 308  variational equation
308  \end{equation}  \end{equation}
309  for all increments $h_k$ to the physical property $p$. This problem  for all increments $h_k$ to the physical property $p$. This problem
310  is solved using \escript \class{LinearPDE} class. In the \escript notation we need to provide  is solved using \escript \class{LinearPDE} class. In the \escript notation we need to provide
311  \begin{equation}\label{ref:EQU:REG:600}  \begin{equation}\label{ref:EQU:REG:600b}
312  \begin{array}{rcl}  \begin{array}{rcl}
313   A_{kilj} & = &  \displaystyle{\frac{\partial G_{lj}}{\partial u_{k,i}} }  \\   A_{kilj} & = &  \displaystyle{\frac{\partial G_{lj}}{\partial u_{k,i}} }  \\
314   B_{kil} & = &  \displaystyle{\frac{\partial F_l }{\partial u_{k,i}} }   \\   B_{kil} & = &  \displaystyle{\frac{\partial F_l }{\partial u_{k,i}} }   \\
# Line 323  state variables in equation~\ref{REF:EQU Line 323  state variables in equation~\ref{REF:EQU
323
324  Setting $h_l=d_l$ in equation~\ref{REF:EQU:costfunction 100d} and  Setting $h_l=d_l$ in equation~\ref{REF:EQU:costfunction 100d} and
325  $r_k=d^*_k$ in equation~\ref{REF:EQU:costfunction 100b} one gets  $r_k=d^*_k$ in equation~\ref{REF:EQU:costfunction 100b} one gets
326  \begin{equation}\label{ref:EQU:REG:601}  \begin{equation}\label{ref:EQU:costfunction:601}
327  \int_{\Omega}  \int_{\Omega}
328  \displaystyle{\frac{\partial K }{\partial u_k }  } d_k +  \displaystyle{\frac{\partial K }{\partial u_k }  } d_k +
329  \displaystyle{\frac{\partial K }{\partial u_{k,i} }  } d_{k,i}+  \displaystyle{\frac{\partial K }{\partial u_{k,i} }  } d_{k,i}+
# Line 345  We need in fact the gradient of $J^f$ wi Line 345  We need in fact the gradient of $J^f$ wi
345  \cdot \displaystyle{\frac{\partial p^f }{\partial m_l} } n_l \; dx  \cdot \displaystyle{\frac{\partial p^f }{\partial m_l} } n_l \; dx
346  \end{equation}  \end{equation}
347  for any increment $n$ to the level set function. So in summary we get    for any increment $n$ to the level set function. So in summary we get
348  \begin{equation}\label{ref:EQU:CS:101}  \begin{equation}\label{ref:EQU:CS:101b}
349  \begin{array}{rcl}  \begin{array}{rcl}
350    Y_k & = & \displaystyle{\frac{\partial K^{reg}}{\partial m_k}} +    Y_k & = & \displaystyle{\frac{\partial K^{reg}}{\partial m_k}} +
351   \sum_{f} \mu^{data}_{f} \left(   \sum_{f} \mu^{data}_{f} \left(

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