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1 gross 3838 \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 gross 3840 J(u,p) = \int_{\Omega} H\; dx + \int_{\partial \Omega} h \; ds
43 gross 3838 \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 gross 4052 \begin{equation}\label{APP FIT EQU 4a}
50 gross 3840 H = \frac{1}{2} \chi \cdot (d_i u_{,i} - \hat{g})^2 + \frac{1}{2} (p_{,i}p_{,i})^{2}
51 gross 3838 \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 gross 3840 J(u,p,\lambda) = \int_{\Omega} H \; dx + \int_{\partial \Omega} h \; ds
57 gross 3838 + \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 gross 4052 \begin{equation} \label{APP FIT EQU 5b}
62 gross 3838 \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 gross 3840 Z = H+ \lambda_{i,j} \cdot X_{ij} + \lambda_{i} \cdot Y_{i} \\
73     z= h + \lambda_{i} \cdot y_{i}
74 gross 3838 \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}

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