# Contents of /trunk/doc/user/fitter.tex

Revision 4052 - (show annotations)
Wed Oct 31 05:50:35 2012 UTC (6 years, 10 months ago) by gross
File MIME type: application/x-tex
File size: 5823 byte(s)
missing references added (mantis id=683)

 1 \chapter{The Data Inversion Problem} 2 \label{APP: FITTER} 3 4 We solve the following nonlinear PDE for the unknown solution $u_i$: 5 \begin{equation} \label{APP FIT EQU 1a} 6 \int_{\Omega} v_{i,j} \cdot X_{ij} + v_{i} \cdot Y_{i} \; dx 7 + \int_{\partial \Omega} v_{i} \cdot y_{i} \; ds = 0 8 \end{equation} 9 for all smooth $v_i$ with $v_i=0$ where $q_i>0$ and 10 \begin{equation} \label{APP FIT EQU 1b} 11 u_i=r_i \mbox{ where } q_i>0 12 \end{equation} 13 where $X_{ij}$ and $Y_i$ are non-linear functions of the solution $u_k$ and its gradient $u_{k,l}$ 14 and $y_i$ is a function of solution $u_k$. 15 16 The equation may depend on set of parameters $p_i$. It is the task to 17 give values of the parameters $p_i$ such that the solution gives the best 18 approximation of given measurements. In mathematical terms we need to minimize 19 a so-called cost function $J$ which measures the distance of the solution 20 to the data over the set of valid parameters. A typical example 21 for a cost function measuring the gradient of a scalar solution $u$ 22 in a given direction $d_i$ against measured data $\hat{g}$ is given as 23 \begin{equation}\label{APP FIT EQU 2a} 24 J_{data}(u) = \frac{1}{2}\int_{\Omega} \chi \cdot ( d_i u_{,i} - \hat{g})^2 dx 25 \end{equation} 26 where $\chi$ is a weighting function which has a non-negative value where data are available. 27 Typically, $\chi$ is to be the inverse of the square of the deviation of the measurements or zero. 28 If the parameter $p_i$ is spatially variable a regularization term needs to be 29 added into the cost function in order to get a unique solution. Typically, 30 for a scalar parameter $p$ the regularization term takes the from 31 \begin{equation}\label{APP FIT EQU 3} 32 J_{reg}(p) = \int_{\Omega} \frac{1}{2} \cdot (p_{,i}p_{,i})^{2} dx 33 \end{equation} 34 The cost function to be minimized then takes the from 35 \begin{equation}\label{APP FIT EQU 2a} 36 J(p,u) = \frac{1}{2}\int_{\Omega} \chi \cdot ( d_i u_{,i} - \hat{g})^2 37 + (p_{,i}p_{,i})^{2} \; 38 dx 39 \end{equation} 40 A more general from is given as 41 \begin{equation}\label{APP FIT EQU 4} 42 J(u,p) = \int_{\Omega} H\; dx + \int_{\partial \Omega} h \; ds 43 \end{equation} 44 where $H$ is a scalar, possibly spatially variable function 45 of the solution $u_i$ and the parameter $p_i$ and their gradients 46 and $H$ is a scalar, possibly spatially variable function 47 of the solution $u_i$ and the parameter $p_i$. For example~(\ref{APP FIT EQU 2a}) 48 one has 49 \begin{equation}\label{APP FIT EQU 4a} 50 H = \frac{1}{2} \chi \cdot (d_i u_{,i} - \hat{g})^2 + \frac{1}{2} (p_{,i}p_{,i})^{2} 51 \end{equation} 52 So task is to minimize the cost function $J$ over $u$ and $p$ 53 subject to the PDE~(\ref{APP FIT EQU 1a} connection $p$ and $u$. The secondary condition 54 is mixed into the cost function using a Lagrangean multiplier before the variation is calculated: 55 \begin{equation}\label{APP FIT EQU 5} 56 J(u,p,\lambda) = \int_{\Omega} H \; dx + \int_{\partial \Omega} h \; ds 57 + \int_{\Omega} \lambda_{i,j} \cdot X_{ij} + \lambda_{i} \cdot Y_{i} \; dx 58 + \int_{\partial \Omega} \lambda_{i} \cdot y_{i} \; ds 59 \end{equation} 60 Notice that the Lagrangean multiplier needs to fullfull the constraint 61 \begin{equation} \label{APP FIT EQU 5b} 62 \lambda_{i}=0 \mbox{ where } q_i>0 63 \end{equation} 64 65 We can rearrange $J$ to 66 \begin{equation}\label{APP FIT EQU 5} 67 J(u,p,\lambda) = \int_{\Omega} Z \; dx 68 + \int_{\partial \Omega} z \; ds 69 \end{equation} 70 with 71 \begin{align}\label{APP FIT EQU 6} 72 Z = H+ \lambda_{i,j} \cdot X_{ij} + \lambda_{i} \cdot Y_{i} \\ 73 z= h + \lambda_{i} \cdot y_{i} 74 \end{align} 75 76 We are taking variation along $p$: 77 \begin{equation}\label{APP FIT EQU 10} 78 \int_{\Omega} \fracp{Z}{p_{i,j}} \cdot (\delta p)_{i,j} + \fracp{Z}{p_{i}} \cdot (\delta p)_{i} 79 \; dx + \int_{\partial \Omega} \fracp{z}{p_{i}} \cdot (\delta p)_{i} \; ds =0 80 \end{equation} 81 along $u$: 82 \begin{align}\label{APP FIT EQU 11} 83 \int_{\Omega} \fracp{Z}{u_{i,j}} \cdot (\delta u)_{i,j} + \fracp{Z}{u_{i}} \cdot (\delta u)_{i} 84 \; dx + \int_{\partial \Omega} \fracp{z}{u_{i}} \cdot (\delta u)_{i} \; ds =0 85 \end{align} 86 and $\lambda$: 87 \begin{equation}\label{APP FIT EQU 12} 88 \int_{\Omega} X_{ij} \cdot (\delta \lambda)_{i,j} + Y_{i} \cdot (\delta \lambda)_{i} \; dx 89 + \int_{\partial \Omega} y_{i} \cdot (\delta \lambda)_{i} \; ds = 0 90 \end{equation} 91 This defines a system of non-linear PDEs for the unknown solution $\widehat{u} = (p,u,\lambda)$. With 92 $\widehat{v} = (\delta p,\delta \lambda, \delta u)$\footnote{Notice that in comparison to the solution 93 the corresponding components for $u$ and $\lambda$ are swapped in order to bring strong couplings into the main-diagonal.} 94 we can write equations~(\ref{APP FIT EQU 10})-(\ref{APP FIT EQU 12}) in the form: 95 \begin{equation} \label{APP FIT EQU 13} 96 \int_{\Omega} \widehat{v}_{i,j} \cdot \widehat{X}_{ij} + \widehat{v}_{i} \cdot \widehat{Y}_{i} \; dx 97 + \int_{\partial \Omega} \widehat{v}_{i} \cdot \widehat{y}_{i} \; ds = 0 98 \end{equation} 99 with 100 \begin{align}\label{APP FIT EQU 15} 101 \widehat{X}_{:j} = \left[ \fracp{Z}{p_{:,j}}, X_{:j}, \fracp{Z}{u_{:,j}} \right] \\ 102 \widehat{Y}_{:} = \left[ \fracp{Z}{p_{:}}, Y_{:}, \fracp{Z}{u_{:}} \right] \\ 103 \widehat{y}_{:} = \left[ \fracp{z}{p_{:}}, y_{:}, \fracp{z}{u_{:}} \right] 104 \end{align} 105 In some cases values for the parameter are known. So similar to the constraint~(\ref{APP FIT EQU 1b}) fro the solution 106 we need to observe a constraint for the parameter $p_i$: 107 \begin{equation} \label{APP FIT EQU 12a} 108 p_i=rp_i \mbox{ where } qp_i>0 109 \end{equation} 110 So for the composed solution $\widehat{u} = (p,u,\lambda)$ we need to observe the constraint 111 \begin{equation} \label{APP FIT EQU 12b} 112 \widehat{u}_i=\widehat{r}_i \mbox{ where } \widehat{q}_i>0 113 \end{equation} 114 with 115 \begin{align}\label{APP FIT EQU 12c} 116 \widehat{q}_{:j} = \left[ qp_{:}, q_{:}, q_{:} \right] \\ 117 \widehat{r}_{:} = \left[ rp_{:}, r_{:}, 0\right] \\ 118 \end{align}