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

revision 4437 by gross, Mon Apr 22 08:44:45 2013 UTC revision 4438 by jfenwick, Tue Jun 4 09:00:50 2013 UTC
# Line 4  The general form of the cost function mi Line 4  The general form of the cost function mi
4  J(m) = J^{reg}(m) + \sum_{f} \mu^{data}_{f} \cdot J^{f}(p^f)  J(m) = J^{reg}(m) + \sum_{f} \mu^{data}_{f} \cdot J^{f}(p^f)
5  \end{equation}  \end{equation}
6  where $m$ represents the level set function, $J^{reg}$ is the regularization term, see Chapter~\ref{Chp:ref:regularization},  where $m$ represents the level set function, $J^{reg}$ is the regularization term, see Chapter~\ref{Chp:ref:regularization},
7  and $J^{f}$ are a set of forward problems, see Chapter~\ref{Chp:ref:forward models} depending of  and $J^{f}$ are a set of cost functions for forward models, (see Chapter~\ref{Chp:ref:forward models}) depending on
8  physical parameters $p^f$.  The physical parameters $p^f$ are known functions  physical parameters $p^f$.  The physical parameters $p^f$ are known functions
9  of the  level set function $m$ which is the unknown to be calculated by the optimization process.  of the  level set function $m$ which is the unknown to be calculated by the optimization process.
10  $\mu^{data}_{f}$ are trade-off factors. It is pointed out that the regularization term includes additional trade-off factors  $\mu^{data}_{f}$ are trade-off factors. It is pointed out that the regularization term includes additional trade-off factors
# Line 255  which is solved using \escript \class{Li Line 255  which is solved using \escript \class{Li
255  \end{array}  \end{array}
256  \end{equation}  \end{equation}
257  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:
258  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 using a cost function kernel $K^{f}$
259  \begin{equation}\label{REF:EQU:INTRO 2bb}  \begin{equation}\label{REF:EQU:INTRO 2bb}
260  J^{f}(p^f) = \int_{\Omega} K^{f} \; dx  J^{f}(p^f) = \int_{\Omega} K^{f} \; dx
261  \end{equation}  \end{equation}

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