# Contents of /branches/doubleplusgood/doc/inversion/Regularization.tex

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 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