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some documenatation on newton-raphson scheme and variatonal problems added
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) = \frac{1}{2} \int_{\Omega} H^2\; dx + \frac{1}{2} \int_{\partial \Omega} h^2 \; 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 2a}
50 H = \sqrt{ \chi \cdot (d_i u_{,i} - \hat{g})^2
51 + (p_{,i}p_{,i})^{2} }
52 \end{equation}
53 So task is to minimize the cost function $J$ over $u$ and $p$
54 subject to the PDE~(\ref{APP FIT EQU 1a} connection $p$ and $u$. The secondary condition
55 is mixed into the cost function using a Lagrangean multiplier before the variation is calculated:
56 \begin{equation}\label{APP FIT EQU 5}
57 J(u,p,\lambda) = \frac{1}{2} \int_{\Omega} H^2 \; dx + \frac{1}{2} \int_{\partial \Omega} h^2 \; ds
58 + \int_{\Omega} \lambda_{i,j} \cdot X_{ij} + \lambda_{i} \cdot Y_{i} \; dx
59 + \int_{\partial \Omega} \lambda_{i} \cdot y_{i} \; ds
60 \end{equation}
61 Notice that the Lagrangean multiplier needs to fullfull the constraint
62 \begin{equation} \label{APP FIT EQU 1b}
63 \lambda_{i}=0 \mbox{ where } q_i>0
64 \end{equation}
65
66 We can rearrange $J$ to
67 \begin{equation}\label{APP FIT EQU 5}
68 J(u,p,\lambda) = \int_{\Omega} Z \; dx
69 + \int_{\partial \Omega} z \; ds
70 \end{equation}
71 with
72 \begin{align}\label{APP FIT EQU 6}
73 Z = \frac{1}{2} H^2 + \lambda_{i,j} \cdot X_{ij} + \lambda_{i} \cdot Y_{i} \\
74 z=\frac{1}{2} h^2 + \lambda_{i} \cdot y_{i}
75 \end{align}
76
77 We are taking variation along $p$:
78 \begin{equation}\label{APP FIT EQU 10}
79 \int_{\Omega} \fracp{Z}{p_{i,j}} \cdot (\delta p)_{i,j} + \fracp{Z}{p_{i}} \cdot (\delta p)_{i}
80 \; dx + \int_{\partial \Omega} \fracp{z}{p_{i}} \cdot (\delta p)_{i} \; ds =0
81 \end{equation}
82 along $u$:
83 \begin{align}\label{APP FIT EQU 11}
84 \int_{\Omega} \fracp{Z}{u_{i,j}} \cdot (\delta u)_{i,j} + \fracp{Z}{u_{i}} \cdot (\delta u)_{i}
85 \; dx + \int_{\partial \Omega} \fracp{z}{u_{i}} \cdot (\delta u)_{i} \; ds =0
86 \end{align}
87 and $\lambda$:
88 \begin{equation}\label{APP FIT EQU 12}
89 \int_{\Omega} X_{ij} \cdot (\delta \lambda)_{i,j} + Y_{i} \cdot (\delta \lambda)_{i} \; dx
90 + \int_{\partial \Omega} y_{i} \cdot (\delta \lambda)_{i} \; ds = 0
91 \end{equation}
92 This defines a system of non-linear PDEs for the unknown solution $\widehat{u} = (p,u,\lambda)$. With
93 $\widehat{v} = (\delta p,\delta \lambda, \delta u)$\footnote{Notice that in comparison to the solution
94 the corresponding components for $u$ and $\lambda$ are swapped in order to bring strong couplings into the main-diagonal.}
95 we can write equations~(\ref{APP FIT EQU 10})-(\ref{APP FIT EQU 12}) in the form:
96 \begin{equation} \label{APP FIT EQU 13}
97 \int_{\Omega} \widehat{v}_{i,j} \cdot \widehat{X}_{ij} + \widehat{v}_{i} \cdot \widehat{Y}_{i} \; dx
98 + \int_{\partial \Omega} \widehat{v}_{i} \cdot \widehat{y}_{i} \; ds = 0
99 \end{equation}
100 with
101 \begin{align}\label{APP FIT EQU 15}
102 \widehat{X}_{:j} = \left[ \fracp{Z}{p_{:,j}}, X_{:j}, \fracp{Z}{u_{:,j}} \right] \\
103 \widehat{Y}_{:} = \left[ \fracp{Z}{p_{:}}, Y_{:}, \fracp{Z}{u_{:}} \right] \\
104 \widehat{y}_{:} = \left[ \fracp{z}{p_{:}}, y_{:}, \fracp{z}{u_{:}} \right]
105 \end{align}
106 In some cases values for the parameter are known. So similar to the constraint~(\ref{APP FIT EQU 1b}) fro the solution
107 we need to observe a constraint for the parameter $p_i$:
108 \begin{equation} \label{APP FIT EQU 12a}
109 p_i=rp_i \mbox{ where } qp_i>0
110 \end{equation}
111 So for the composed solution $\widehat{u} = (p,u,\lambda)$ we need to observe the constraint
112 \begin{equation} \label{APP FIT EQU 12b}
113 \widehat{u}_i=\widehat{r}_i \mbox{ where } \widehat{q}_i>0
114 \end{equation}
115 with
116 \begin{align}\label{APP FIT EQU 12c}
117 \widehat{q}_{:j} = \left[ qp_{:}, q_{:}, q_{:} \right] \\
118 \widehat{r}_{:} = \left[ rp_{:}, r_{:}, 0\right] \\
119 \end{align}

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