# Contents of /trunk/doc/inversion/Regularization.tex

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Tue Dec 11 04:04:47 2012 UTC (7 years, 1 month ago) by gross
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Clarification of terminology and scaling.
Joint inversion is not working yet.


 1 \chapter{Regularization}\label{Chp:ref:regularization} 2 3 The general cost function $J^{total}$ to be minimized has some of the cost 4 function $J^f$ measuring the defect of the result from the 5 forward model with the data, and the cost function $J^{reg}$ introducing the 6 regularization into the problem and makes sure that a unique answer exists. 7 The regularization term is a function of, possibly vector-valued, level set 8 function $m$ which represents the physical properties to be represented and is, 9 from a mathematical point of view, the unknown of the inversion problem. 10 It is the intention that the values of $m$ are between zero and one and that 11 actual physical values are created from a mapping before being fed into a 12 forward model. In general the cost function $J^{reg}$ is defined as 13 \begin{equation}\label{EQU:REG:1} 14 J^{reg}(m) = \frac{1}{2} \int_{\Omega} \left( 15 \sum_{k} \mu_k \cdot ( \omega^{(0)}_k \cdot m_k^2 + \omega^{(1)}_{ki}m_{k,i}^2 ) 16 + \sum_{l = [p, r] 143 \end{equation} 144 for all $p$. This problem can be solved using \escript \class{LinearPDE} class by setting 145 \begin{equation}\label{EQU:REG:8b} 146 \begin{array}{rcl} 147 A_{ij} & =& \mu \cdot \omega^{(1)}_i \cdot \delta_{ij} \\ 148 D & = & \mu \cdot \omega^{(0)} 149 \end{array} 150 \end{equation} 151 and $X$ and $Y$ as defined by $r$ for the case of a single-valued level set function. 152 For a vector-valued level-set function one sets: 153 \begin{equation}\label{EQU:REG:8c} 154 \begin{array}{rcl} 155 A_{kilj} & = & \mu_k \cdot \omega^{(1)}_{ki} \cdot \delta_{ij} \cdot \delta_{kl} \\ 156 D_{kl} & = & \mu_k \cdot \omega^{(0)}_k \cdot \delta_{kl} 157 \end{array} 158 \end{equation} 159