/[escript]/release/3.4.2/doc/inversion/Regularization.tex
ViewVC logotype

Contents of /release/3.4.2/doc/inversion/Regularization.tex

Parent Directory Parent Directory | Revision Log Revision Log


Revision 4919 - (show annotations)
Wed Apr 30 06:25:55 2014 UTC (5 years, 3 months ago) by jfenwick
File MIME type: application/x-tex
File size: 8637 byte(s)
Because we've got to!

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<k} \mu^{(c)}_{lk} \cdot \omega^{(c)}_{lk} \cdot \chi(m_l,m_k) \right) \; dx
17 \end{equation}
18 where summation over $i$ is performed. The additional trade--off factors
19 $\mu_k$ and $\mu^{(c)}_{lk}$ ($l<k$) are between zero and one and constant across the
20 domain. They are potentially modified during the inversion in order to improve the balance between the different terms
21 in the cost function.
22
23 $\chi$ is a given symmetric, non-negative cross-gradient function\index{cross-gradient
24 }. We use
25 \begin{equation}\label{EQU:REG:4}
26 \chi(a,b) = ( a_{,i} a_{,i}) \cdot ( b_{,j} b_{,j}) - ( a_{,i} b_{,i})^2
27 \end{equation}
28 where summations over $i$ and $j$ are performed, see~\cite{GALLARDO2005a}. Notice that cross-gradient function
29 is measuring the angle between the surface normals of contours of level set functions. So
30 minimizing the cost function will align the surface normals of the contours.
31
32 The coefficients $\omega^{(0)}_k$, $\omega^{(1)}_{ki}$ and $\omega^{(c)}_{lk}$ define weighting factors which
33 may depend on their location within the domain. We assume that for given level set function $k$ the
34 weighting factors $\omega^{(0)}_k$, $\omega^{(1)}_{ki}$ are scaled such that
35 \begin{equation}\label{ref:EQU:REG:5}
36 \int_{\Omega} ( \omega^{(0)}_k + \frac{\omega^{(1)}_{ki}}{L_i^2} ) \; dx = \alpha_k
37 \end{equation}
38 where $\alpha_k$ defines the scale which is typically set to one. $L_i$ is the width of the domain in $x_i$ direction.
39 Similarly we set for $l<k$ we set
40 \begin{equation}\label{ref:EQU:REG:6}
41 \int_{\Omega} \frac{\omega^{(c)}_{lk}}{L^4} \; dx = \alpha^{(c)}_{lk}
42 \end{equation}
43 where $\alpha^{(c)}_{lk}$ defines the scale which is typically set to one and
44 \begin{equation}\label{ref:EQU:REG:6b}
45 \frac{1}{L^2} = \sum_i \frac{1}{L_i^2} \;.
46 \end{equation}
47
48 In some cases values for the level set functions are known to be zero at certain regions in the domain. Typically this is the region
49 above the surface of the Earths. This expressed using a
50 a characteristic function $q$ which varies with its location within the domain. The function $q$ is set to zero except for those
51 locations $x$ within the domain where the values of the level set functions is known to be zero. For these locations $x$
52 $q$ takes a positive value. for a single level set function one has
53 \begin{equation}\label{ref:EQU:REG:7}
54 q(x) = \left\{
55 \begin{array}{rl}
56 1 & \mbox{ if } m \mbox{ is set to zero at location } x \\
57 0 & \mbox{ otherwise }
58 \end{array}
59 \right.
60 \end{equation}
61 For multi-valued level set function the characteristic function is set componentwise:
62 \begin{equation}\label{ref:EQU:REG:7b}
63 q_k(x) = \left\{
64 \begin{array}{rl}
65 1 & \mbox{ if component } m_k \mbox{ is set to zero at location } x \\
66 0 & \mbox{ otherwise }
67 \end{array}
68 \right.
69 \end{equation}
70
71
72 \section{Usage}
73
74 \begin{classdesc}{Regularization}{domain
75 \optional{, w0=\None}
76 \optional{, w1=\None}
77 \optional{, wc=\None}
78 \optional{, location_of_set_m=Data()}
79 \optional{, numLevelSets=1}
80 \optional{, useDiagonalHessianApproximation=\False}
81 \optional{, tol=1e-8}
82 \optional{, scale=\None}
83 \optional{, scale_c=\None}
84 }
85
86
87 initializes a regularization component of the cost function for inversion.
88 \member{domain} defines the domain of the inversion. \member{numLevelSets}
89 sets the number of level set functions to be found during the inversion.
90 \member{w0}, \member{w1} and \member{wc} define the weighting factors
91 $\omega^{(0)}$,
92 $\omega^{(1)}$ and
93 $\omega^{(c)}$, respectively. A value for \member{w0} or \member{w1} or both must be given.
94 If more than one level set function is involved \member{wc} must be given.
95 \member{location_of_set_m} sets the characteristic function $q$
96 to define locations where the level set function is set to zero, see equation~(\ref{ref:EQU:REG:7}).
97 \member{scale} and
98 \member{scale_c} set the scales $\alpha_k$ in equation~(\ref{ref:EQU:REG:5}) and
99 $\alpha^{(c)}_{lk}$ in equation~(\ref{ref:EQU:REG:6}), respectively. By default, their values are set to one.
100 Notice that weighting factors are rescaled to meet the scaling conditions. \member{tol} sets the
101 tolerance for the calculation of the Hessian approximation. \member{useDiagonalHessianApproximation}
102 indicates to ignore coupling in the Hessian approximation produced by the
103 cross-gradient term. This can speed-up an individual iteration step in the inversion but typically leads to more
104 inversion steps.
105 \end{classdesc}
106
107 \section{Gradient Calculation}
108
109
110 The cost function kernel\index{cost function!kernel} is given as
111 \begin{equation}\label{ref:EQU:REG:100}
112 K^{reg}(m) = \frac{1}{2}
113 \sum_{k} \mu_k \cdot ( \omega^{(0)}_k \cdot m_k^2 + \omega^{(1)}_{ki}m_{k,i}^2 )
114 + \sum_{l<k} \mu^{(c)}_{lk} \cdot \omega^{(c)}_{lk} \cdot \chi(m_l,m_k)
115 \end{equation}
116 We need to provide the gradient of the cost function $J^{reg}$ with respect to the level set functions $m$.
117 The gradient is represented by two functions $Y$ and $X$ which define the
118 derivative of the cost function kernel with respect to $m$ and to the gradient $m_{,i}$, respectively:
119 \begin{equation}\label{ref:EQU:REG:101}
120 \begin{array}{rcl}
121 Y_k & = & \displaystyle{\frac{\partial K^{reg}}{\partial m_k}} \\
122 X_{ki} & = & \displaystyle{\frac{\partial K^{reg}}{\partial m_{k,i}}}
123 \end{array}
124 \end{equation}
125 For the case of a single valued level set function $m$ we get
126 \begin{equation}\label{ref:EQU:REG:202}
127 Y = \mu \cdot \omega^{(0)} \cdot m
128 \end{equation}
129 and
130 \begin{equation}\label{ref:EQU:REG:203}
131 X_{i} = \mu \cdot \omega^{(1)}_{i} \cdot m_{,i}
132 \end{equation}
133 For a two-valued level set function $(m_0,m_1)$ we have
134 \begin{equation}\label{ref:EQU:REG:302}
135 Y_k = \mu_k \cdot \omega^{(0)}_k \cdot m_k \mbox{ for } k=0,1
136 \end{equation}
137 and for $X$
138 \begin{equation}\label{ref:EQU:REG:303}
139 \begin{array}{rcl}
140 X_{0i} & = & \mu_0 \cdot \omega^{(1)}_{0i} \cdot m_{0,i} + \mu^{(c)}_{01} \cdot \omega^{(c)}_{01} \cdot
141 \left( (m_{1,j}m_{1,j} ) \cdot m_{0,i} - (m_{1,j}m_{0,j} ) \cdot m_{1,i} \right) \\
142 X_{1i} & = & \mu_1 \cdot \omega^{(1)}_{1i} \cdot m_{1,i} + \mu^{(c)}_{01} \cdot \omega^{(c)}_{01} \cdot
143 \left( (m_{0,j}m_{0,j} ) \cdot m_{1,i} - (m_{1,j}m_{0,j} ) \cdot m_{0,i} \right)
144 \\
145 \end{array}
146 \end{equation}
147 We also need to provide an approximation of the inverse of the Hessian operator as discussed in section~\ref{chapter:ref:inversion cost function:gradient}.
148 For the case of a single valued level set function $m$ we get
149 \begin{equation}\label{ref:EQU:REG:601}
150 \begin{array}{rcl}
151 A_{ij} & =& \mu \cdot \omega^{(1)}_i \cdot \delta_{ij} \\
152 D & = & \mu \cdot \omega^{(0)}
153 \end{array}
154 \end{equation}
155 For a two-valued level set function $(m_0,m_1)$ we have
156 \begin{equation}\label{ref:EQU:REG:602}
157 D_{kl} = \mu_k \cdot \omega^{(0)}_k \cdot \delta_{kl}
158 \end{equation}
159 and
160 \begin{equation}\label{ref:EQU:REG:603}
161 \begin{array}{rcl}
162 A_{0i0j} & = & \mu_0 \cdot \omega^{(1)}_{0i} \cdot \delta_{ij} + \mu^{(c)}_{01} \cdot \omega^{(c)}_{01} \cdot
163 \left( (m_{1,j'}m_{1,j'} )\cdot \delta_{ij} - m_{1,i} \cdot m_{1,j} \right) \\
164 A_{0i1j} & = & \mu^{(c)}_{01} \cdot \omega^{(c)}_{01} \cdot \left( 2 \cdot m_{0,i} \cdot m_{1,j}
165 - m_{1,i} \cdot m_{0,j} - ( m_{1,j'} m_{0,j'} ) \cdot \delta_{ij}
166 \right) \\
167 A_{1i0j} & = & \mu^{(c)}_{01} \cdot \omega^{(c)}_{01} \cdot \left( 2 \cdot m_{1,i} \cdot m_{0,j}
168 - m_{0,i} \cdot m_{1,j} - ( m_{1,j'} m_{0,j'} ) \cdot \delta_{ij} \right) \\
169 A_{1i1j} & = & \mu_1 \cdot \omega^{(1)}_{1i} \cdot \delta_{ij} + \mu^{(c)}_{01} \cdot \omega^{(c)}_{01} \cdot
170 \left( (m_{0,j'}m_{0,j'} ) \cdot \delta_{ij} - m_{0,i} \cdot m_{0,j} ) \right)
171 \end{array}
172 \end{equation}
173
174

  ViewVC Help
Powered by ViewVC 1.1.26