# Contents of /trunk/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 $J_{forward}$ 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} 15 \sum_{k} \mu^{(0)}_k \cdot s^{(0)}_k \cdot m_k^2 + \mu^{(1)}_{ki} \cdot s^{(1)}_{ki} \cdot L_i^2 \cdot m_{k,i} \cdot m_{k,i} 16 + \sum_{l$is an inner product. If the level set function$m$has several components$m_j$the inner product$<.>$is given 86 in the form 87$\omega^{(k)}$and$\omega^{(k)}_i$are fixed non-negative weighting factors and$\mu^{reg}_k$are weighting factors 88 which may be modified during the inversion.$L_i$is the length of the domain in$x_i$direction. In the special case that 89 the level set function$m$has a single component the inner product takes the form 90 \begin{equation}\label{EQU:REG:2b} 91 = 92 \mu^{reg} \int_{\Omega} \omega \cdot p \cdot q + \omega_i \cdot L_i^2 \cdot p_{,i} \cdot q_{,i} dx 93 \end{equation} 94 In practice it is assumed that the level set function is known to be zero in certain regions in the domain. Typically these regions 95 corresponds to region above the surface or regions explored by drilling. 96 97 We need to provide the derivative of the cost function$J_{reg}$with respect to a given direction$q$which equals zero at locations 98 where$m$is assumed to be zero. For a single-valued 99 level set function th is takes the form 100 \begin{equation}\label{EQU:REG:3} 101 \frac{ \partial J_{reg}}{\partial q}(m) = 102 \mu^{reg} \int_{\Omega} \omega \cdot m \cdot q + \omega_i \cdot L_i^2 \cdot m_{,i} \cdot q_{,i} dx 103 \end{equation} 104 So we can represent the gradient$\nabla J_{reg}$of the cost function$J_{reg}$by the pair of values$(Y,X)$where we set 105 \begin{equation}\label{EQU:REG:3b} 106 Y=\mu^{reg} \cdot \omega \cdot m \mbox{ and } X_i = \mu^{reg} \cdot \omega_i \cdot L_i^2 \cdot m_{,i} 107 \end{equation} 108 and 109 \begin{equation}\label{EQU:REG:3c} 110 \frac{ \partial J_{reg}}{\partial q}(m) = [ \nabla J_{reg}(m), q ] = 111 \int_{\Omega} Y \cdot q + X_i \cdot q_{,i} dx 112 \end{equation} 113 where$[.,.]$is called the dual product. 114 115 For a multi-valued level set function an additional correlation term is introduced into the cost function$J_{total}$: 116 \begin{equation}\label{EQU:REG:1c} 117 J_{reg}(m) = \frac{1}{2} < m, m > + \frac{1}{2} \sum_{k,l} \mu_{kl}^{sec} \cdot \int_{\Omega} \sigma(m_k,m_l) dx 118 \end{equation} 119 where$sigma$is a given symmetric, non-negative correlation function, and$\mu_{kl}^{sec}$are symmetric, weighting factors 120 ($\mu_{kl}^{sec} = \mu_{lk}^{sec}$,$\mu_{kk}^{sec}=0$) which may 121 be altered during the inversion. We use the correlation function 122 \begin{equation}\label{EQU:REG:4} 123 \sigma(a,b) = \frac{L^2}{2} \cdot ( ( a_{,i} \cdot a_{,i}) \cdot ( b_{,i} \cdot b_{,i}) - ( a_{,i} \cdot b_{,i})^2 ) 124 \end{equation} 125 with$L=L_i \cdot L_i$. Minimizing$J_{reg}(m)$is minimizing the angle between the surface normals of the contours formed by 126 two level set function. the derivative of the cost function$J_{reg}$with respect to a given direction$q$which equals zero at locations 127 where$m$is assumed to be zero: 128 \begin{equation}\label{EQU:REG:5} 129 \begin{array}{ll} 130 \displaystyle{\frac{ \partial J_{reg}}{\partial q}(m)} = 131 \displaystyle{\sum_{k} \mu^{reg}_k \int_{\Omega} \omega^{(k)} \cdot m_k \cdot q_k + \omega^{(k)}_i \cdot L_i^2 \cdot m_{k,i} \cdot q_{k,i} dx } \\ 132 + \displaystyle{\sum_{k,l} \mu_{kl}^{sec} \cdot {L^2} \int_{\Omega} ( m_{k,i} \cdot q_{k,i}) \cdot ( m_{l,j} \cdot m_{l,j}) - ( m_{k,j} \cdot m_{l,j}) \cdot ( q_{l,i} \cdot m_{k,i}) } dx 133 \end{array} 134 \end{equation} 135 Similar to the single-case we can represent 136 the gradient$\nabla J_{reg}$of the cost function$J_{reg}$by the pair of values$(Y,X)$where we set 137 \begin{equation}\label{EQU:REG:6} 138 Y_k= \mu^{reg}_k \cdot \omega^{(k)} \cdot m_k 139 \end{equation} 140 and 141 \begin{equation}\label{EQU:REG:6b} 142 X_{ki} = \mu^{reg}_k \cdot \omega^{(k)} \cdot L_i^2 \cdot m_{k,i} + 143 \sum_{l} \mu_{kl}^{sec} \cdot {L^2} \cdot ( ( m_{l,j} \cdot m_{l,j}) \cdot m_{k,i} - ( m_{l,j} \cdot m_{k,j}) \cdot m_{l,i} ) 144 \end{equation} 145 and 146 \begin{equation}\label{EQU:REG:7} 147 \frac{ \partial J_{reg}}{\partial q}(m) = [ \nabla J_{reg}(m), q ] = 148 \int_{\Omega} Y_j \cdot q_j + X_{ki} \cdot q_{k,i} dx 149 \end{equation} 150 where$[.,.]$is the dual product. 151 152 We also need to provide an approximation of the inverse of the Hessian operator which provides a 153 level set function$h$for a given value$r$represented by the pair of values$(Y,X)$. If one ignores the correlation function 154 the inner product defines the Hessian operator of the cost function. In this approach we set 155 \begin{equation}\label{EQU:REG:8} 156 < p, h > = [p, r] 157 \end{equation} 158 for all$p$. This problem can be solved using \escript \class{LinearPDE} class by setting 159 \begin{equation}\label{EQU:REG:8b} 160 \begin{array}{rcl} 161 A_{ij} & =& (\omega_i \cdot L_i^2) \cdot \delta_{ij} \\ 162 D & = & \mu^{reg} \cdot \omega 163 \end{array} 164 \end{equation} 165 and$X$and$Y$as defined by$r\$ for the case of a single-valued level set function. 166 For a vector-valued level-set function one sets: 167 \begin{equation}\label{EQU:REG:8c} 168 \begin{array}{rcl} 169 A_{kilj} & = & (\mu^{reg}_l \omega^{(l)}_i L_i^2) \cdot \delta_{kl} \cdot \delta_{ij} \\ 170 D_{kl} & = & \mu^{reg}_l \cdot \omega^{(l)} \delta_{kl} \omega 171 \end{array} 172 \end{equation} 173 ==================================================== 174 \begin{classdesc}{Regularization}{domain 175 \optional{, s0=\None} 176 \optional{, s1=\None} 177 \optional{, sc=\None} 178 \optional{, location_of_set_m=Data()} 179 \optional{, numLevelSets=1} 180 \optional{, useDiagonalHessianApproximation=\True} 181 \optional{, tol=1e-8}} 182 opens a linear, steady, second order PDE on the \member{domain}. 183 The parameters \member{numEquations} and \member{numSolutions} give the number 184 of equations and the number of solution components. 185 If \member{numEquations} and \member{numSolutions} are non-positive, then the 186 number of equations and the number of solutions, respectively, stay undefined 187 until a coefficient is defined. 188 \end{classdesc} 189 190 191 192 \section{The general regularization class} 193 \begin{classdesc}{RegularizationBase}{} 194 195 \end{classdesc}