/[escript]/trunk/doc/inversion/CostFunctions.tex
ViewVC logotype

Annotation of /trunk/doc/inversion/CostFunctions.tex

Parent Directory Parent Directory | Revision Log Revision Log


Revision 6681 - (hide annotations)
Mon Jun 11 05:40:04 2018 UTC (14 months, 1 week ago) by jfenwick
File MIME type: application/x-tex
File size: 20906 byte(s)
Deal with overfull boxes

Scaled some of the compound figures to 95%

1 gross 4138 \chapter{Cost Function}\label{chapter:ref:inversion cost function}
2 caltinay 4139 The general form of the cost function minimized in the inversion is given in the form (see also Chapter~\ref{chapter:ref:Drivers})
3 gross 4133 \begin{equation}\label{REF:EQU:DRIVE:10}
4     J(m) = J^{reg}(m) + \sum_{f} \mu^{data}_{f} \cdot J^{f}(p^f)
5     \end{equation}
6     where $m$ represents the level set function, $J^{reg}$ is the regularization term, see Chapter~\ref{Chp:ref:regularization},
7 jfenwick 4438 and $J^{f}$ are a set of cost functions for forward models, (see Chapter~\ref{Chp:ref:forward models}) depending on
8 gross 4133 physical parameters $p^f$. The physical parameters $p^f$ are known functions
9     of the level set function $m$ which is the unknown to be calculated by the optimization process.
10     $\mu^{data}_{f}$ are trade-off factors. It is pointed out that the regularization term includes additional trade-off factors
11     The \class{InversionCostFunction} is class to define cost functions of an inversion. It is pointed out that
12     the \class{InversionCostFunction} class implements the \class{CostFunction} template class, see Chapter~\ref{chapter:ref:Minimization}.
13 gross 4131
14 gross 4133 In the simplest case there is a single forward model using a single physical parameter which is
15     derived form single-values level set function. The following script snippet shows the creation of the
16 jfenwick 6681 cost function for the case of a gravity inversion:
17 gross 4133 \begin{verbatim}
18     p=DensityMapping(...)
19     f=GravityModel(...)
20     J=InversionCostFunction(Regularization(...), \
21     mappings=p, \
22     forward_models=f)
23     \end{verbatim}
24     The argument \verb|...| refers to an appropriate argument list.
25 gross 4131
26 gross 4133 If two forward models are coming into play using two different physical parameters
27     the \member{mappings} and \member{forward_models} are defined as lists in the following form:
28     \begin{verbatim}
29     p_rho=DensityMapping(...)
30     p_k=SusceptibilityMapping(...)
31     f_mag=MagneticModel(...)
32     f_grav=GravityModel(...)
33    
34     J=InversionCostFunction(Regularization(...), \
35     mappings=[p_rho, p_k], \
36     forward_models=[(f_mag, 1), (f_grav,0)])
37     \end{verbatim}
38     Here we define a joint inversion of gravity and magnetic data. \member{forward_models} is given as a list of
39     a tuple of a forward model and an index which referring to parameter in the \member{mappings} list to be used as an input.
40 caltinay 4285 The magnetic forward model \member{f_mag} is using the second parameter (=\member{p_k}) in \member{mappings} list.
41 gross 4133 In this case the physical parameters are defined by a single-valued level set function. It is also possible
42     to link physical parameters to components of a level set function:
43     \begin{verbatim}
44     p_rho=DensityMapping(...)
45     p_k=SusceptibilityMapping(...)
46     f_mag=MagneticModel(...)
47     f_grav=GravityModel(...)
48    
49     J=InversionCostFunction(Regularization(numLevelSets=2,...), \
50     mappings=[(p_rho,0), (p_k,1)], \
51     forward_models=[[(f_mag, 1), (f_grav,0)])
52     \end{verbatim}
53     The \member{mappings} argument is now a list of pairs where the first pair entry specifies the parameter mapping and
54     the second pair entry specifies the index of the component of the level set function to be used to evaluate the parameter.
55     In this case the level set function has two components, where the density mapping uses the first component of the level set function
56     while the susceptibility mapping uses the second component.
57    
58 gross 4142 \section{\class{InversionCostFunction} API}\label{chapter:ref:inversion cost function:api}
59 gross 4133
60 caltinay 4273 The \class{InversionCostFunction} implements a \class{CostFunction} class used
61     to run optimization solvers, see Section~\ref{chapter:ref:Minimization: costfunction class}.
62     Its API is defined as follows:
63 gross 4142
64 gross 4133 \begin{classdesc}{InversionCostFunction}{regularization, mappings, forward_models}
65     Constructor for the inversion cost function. \member{regularization} sets the regularization to be used, see Chapter~\ref{Chp:ref:regularization}.
66     \member{mappings} is a list of pairs where each pair comprises of a
67     physical parameter mapping (see Chapter~\ref{Chp:ref:mapping}) and an index which refers to the component of level set function
68 caltinay 4273 defined by the \member{regularization} to be used to calculate the corresponding physical parameter.
69     If the level set function has a single component the index can be omitted.
70     If in addition there is a single physical parameter the mapping can be given instead of a list.
71     \member{forward_models} is a list of pairs where the first pair component is a
72     forward model (see Chapter~\ref{Chp:ref:forward models}) and the second pair
73     component refers to the physical parameter in the \member{mappings} list
74     providing the physical parameter for the model.
75     If a single physical parameter is present the index can be omitted.
76     If in addition a single forward model is used this forward model can be
77     assigned to \member{forward_models} in replacement of a list.
78 gross 4376 The \member{regularization} and all \member{forward_models} must use the same
79     \class{ReferenceSystem}, see Section~\ref{sec:ref:reference systems}.
80 gross 4119 \end{classdesc}
81 gross 4133
82     \begin{methoddesc}[InversionCostFunction]{getDomain}{}
83 gross 4142 returns the \escript domain of the inversion, see~\cite{ESCRIPT}.
84 gross 4133 \end{methoddesc}
85    
86     \begin{methoddesc}[InversionCostFunction]{getNumTradeOffFactors}{}
87 caltinay 4273 returns the total number of trade-off factors.
88     The count includes the trade-off factors $\mu^{data}_{f}$ for the forward
89     models and (hidden) trade-off factors in the regularization term,
90     see Definition~\ref{REF:EQU:DRIVE:10}.
91 gross 4133 \end{methoddesc}
92    
93 gross 4142 \begin{methoddesc}[InversionCostFunction]{getForwardModel}{\optional{idx=\None}}
94 caltinay 4273 returns the forward model with index \member{idx}.
95     If the cost function contains one model only argument \member{idx} can be omitted.
96 gross 4133 \end{methoddesc}
97    
98     \begin{methoddesc}[InversionCostFunction]{getRegularization}{}
99 gross 4142 returns the regularization component of the cost function, see \class{regularization} in Chapter~\ref{Chp:ref:regularization}.
100 gross 4133 \end{methoddesc}
101    
102 gross 4142 \begin{methoddesc}[InversionCostFunction]{setTradeOffFactorsModels}{\optional{mu=\None}}
103 caltinay 4273 sets the trade-off factors $\mu^{data}_{f}$ for the forward model components.
104     If a single model is present \member{mu} must be a floating point number.
105     Otherwise \member{mu} must be a list of floating point numbers.
106     It is assumed that all numbers are positive.
107     The default value for all trade-off factors is one.
108 gross 4133 \end{methoddesc}
109 gross 4142
110     \begin{methoddesc}[InversionCostFunction]{getTradeOffFactorsModels}{}
111 caltinay 4273 returns the values of the trade-off factors $\mu^{data}_{f}$ for the forward model components.
112 gross 4142 \end{methoddesc}
113    
114     \begin{methoddesc}[InversionCostFunction]{setTradeOffFactorsRegularization}{\optional{mu=\None}, \optional{mu_c=\None}}
115 caltinay 4273 sets the trade-off factors for the regularization component of the cost function.
116     \member{mu} defines the trade-off factors for the level-set variation part and
117     \member{mu_c} sets the trade-off factors for the cross-gradient variation part.
118     This method is a shortcut for calling \member{setTradeOffFactorsForVariation}
119     and \member{setTradeOffFactorsForCrossGradient} for the underlying the
120     regularization.
121     Please see \class{Regularization} in Chapter~\ref{Chp:ref:regularization} for
122     more details on the arguments \member{mu} and \member{mu_c}.
123 gross 4142 \end{methoddesc}
124 gross 4133
125 gross 4142 \begin{methoddesc}[InversionCostFunction]{setTradeOffFactors}{\optional{mu=\None}}
126 caltinay 4273 sets the trade-off factors for the forward model and regularization terms.
127     \member{mu} is a list of positive floats. The length of the list is the total
128     number of trade-off factors given by the method \method{getNumTradeOffFactors}.
129     The first part of \member{mu} defines the trade-off factors $\mu^{data}_{f}$
130     for the forward model components while the remaining entries define the
131     trade-off factors for the regularization components of the cost function.
132     By default all values are set to one.
133 gross 4133 \end{methoddesc}
134    
135 gross 4142 \begin{methoddesc}[InversionCostFunction]{getProperties}{m}
136     returns the physical properties from a given level set function \member{m}
137 caltinay 4273 using the mappings of the cost function. The physical properties are
138 gross 4142 returned in the order in which they are given in the \member{mappings} argument
139     in the class constructor.
140 gross 4133 \end{methoddesc}
141    
142     \begin{methoddesc}[InversionCostFunction]{createLevelSetFunction}{*props}
143 caltinay 4273 returns the level set function corresponding to set of given physical properties.
144 gross 4142 This method is the inverse of the \method{getProperties} method.
145 caltinay 4273 The arguments \member{props} define a tuple of values for the physical
146     properties where the order needs to correspond to the order in which the
147     physical property mappings are given in the \member{mappings} argument in the
148     class constructor. If a value for a physical property is given as \None the
149     corresponding component of the returned level set function is set to zero.
150     If no physical properties are given all components of the level set function
151     are set to zero.
152 gross 4133 \end{methoddesc}
153    
154 gross 4142 \begin{methoddesc}[InversionCostFunction]{getNorm}{m}
155     returns the norm of a level set function \member{m} as a floating point number.
156 gross 4133 \end{methoddesc}
157    
158     \begin{methoddesc}[InversionCostFunction]{getArguments}{m}
159 caltinay 4273 returns pre-computed values for the evaluation of the cost function and its
160     gradient for a given value \member{m} of the level set function.
161     In essence the method collects pre-computed values for the underlying
162     regularization and forward models\footnote{Using pre-computed values can
163     significantly speed up the optimization process when the value of the cost
164     function and its gradient are needed for the same level set function.}.
165 gross 4133 \end{methoddesc}
166    
167 caltinay 4273 \begin{methoddesc}[InversionCostFunction]{getValue}{m\optional{, *args}}
168 gross 4142 returns the value of the cost function for a given level set function \member{m}
169 caltinay 4273 and corresponding pre-computed values \member{args}.
170     If the pre-computed values are not supplied \member{getArguments} is called.
171 gross 4133 \end{methoddesc}
172    
173 caltinay 4273 \begin{methoddesc}[InversionCostFunction]{getGradient}{m\optional{, *args}}
174     returns the gradient of the cost function at level set function \member{m}
175     using the corresponding pre-computed values \member{args}.
176     If the pre-computed values are not supplied \member{getArguments} is called.
177     The gradient is represented as a tuple $(Y,X)$ where in essence $Y$ represents
178     the derivative of the cost function kernel with respect to the level set
179     function and $X$ represents the derivative of the cost function kernel with
180     respect to the gradient of the level set function, see
181     Section~\ref{chapter:ref:inversion cost function:gradient} for more details.
182 gross 4133 \end{methoddesc}
183 gross 4142
184     \begin{methoddesc}[InversionCostFunction]{getDualProduct}{m, g}
185 caltinay 4273 returns the dual product of a level set function \member{m} with a gradient
186     \member{g}, see Section~\ref{chapter:ref:inversion cost function:gradient} for more details.
187     This method uses the dual product of the regularization.
188 gross 4142 \end{methoddesc}
189 gross 4133
190 gross 4142 \begin{methoddesc}[InversionCostFunction]{getInverseHessianApproximation}{m, g \optional{, *args}}
191 caltinay 4273 returns an approximative evaluation of the inverse of the Hessian operator of
192     the cost function for a given gradient \member{g} at a given level set function
193     \member{m} using the corresponding pre-computed values \member{args}.
194     If no pre-computed values are present \member{getArguments} is called.
195     In the current implementation contributions to the Hessian operator from the
196     forward models are ignored and only contributions from the regularization and
197     cross-gradient term are used.
198 gross 4133 \end{methoddesc}
199    
200 caltinay 4279
201 gross 4142 \section{Gradient calculation}\label{chapter:ref:inversion cost function:gradient}
202     In this section we briefly discuss the calculation of the gradient and the Hessian operator.
203     If $\nabla$ denotes the gradient operator (with respect to the level set function $m$)
204     the gradient of $J$ is given as
205     \begin{equation}\label{REF:EQU:DRIVE:10b}
206     \nabla J(m) = \nabla J^{reg}(m) + \sum_{f} \mu^{data}_{f} \cdot \nabla J^{f}(p^f) \; .
207     \end{equation}
208     We first focus on the calculation of $\nabla J^{reg}$. In fact the
209     regularization cost function $J^{reg}$ is given through a cost function
210     kernel\index{cost function!kernel} $K^{reg}$ in the form
211     \begin{equation}\label{REF:EQU:INTRO 2a}
212     J^{reg}(m) = \int_{\Omega} K^{reg} \; dx
213     \end{equation}
214     where $K^{reg}$ is a given function of the
215     level set function $m_k$ and its spatial derivative $m_{k,i}$. If $n$ is an increment to the level set function
216     then the directional derivative of $J^{ref}$ in the direction of $n$ is given as
217 gross 4147 \begin{equation}\label{REF:EQU:INTRO 2aa}
218 gross 4142 <n, \nabla J^{reg}(m)> = \int_{\Omega} \frac{ \partial K^{reg}}{\partial m_k} n_k + \frac{ \partial K^{reg}}{\partial m_{k,i}} n_{k,i} \; dx
219     \end{equation}
220     where $<.,.>$ denotes the dual product, see Chapter~\ref{chapter:ref:Minimization}. Consequently, the gradient $\nabla J^{reg}$
221     can be represented by a pair of values $Y$ and $X$
222     \begin{equation}\label{ref:EQU:CS:101}
223     \begin{array}{rcl}
224     Y_k & = & \displaystyle{\frac{\partial K^{reg}}{\partial m_k}} \\
225     X_{ki} & = & \displaystyle{\frac{\partial K^{reg}}{\partial m_{k,i}}}
226     \end{array}
227     \end{equation}
228     while the dual product $<.,.>$ of a level set increment $n$ and a gradient increment $g=(Y,X)$ is given as
229 gross 4147 \begin{equation}\label{REF:EQU:INTRO 2aaa}
230 gross 4142 <n,g> = \int_{\Omega} Y_k n_k + X_{ki} n_{k,i} \; dx
231     \end{equation}
232     We also need to provide (an approximation of) the value $p$ of the inverse of the Hessian operator $\nabla \nabla J$
233     for a given gradient increment $g=(Y,X)$. This means we need to (approximatively) solve the variational problem
234     \begin{equation}\label{REF:EQU:INTRO 2b}
235     <n,\nabla \nabla J p > = \int_{\Omega} Y_k n_k + X_{ki} n_{k,i} \; dx
236     \end{equation}
237     for all increments $n$ of the level set function. If we ignore contributions
238     from the forward models the left hand side takes the form
239     \begin{equation}\label{REF:EQU:INTRO 2c}
240     <n,\nabla \nabla J^{reg} p > = \int_{\Omega}
241     \displaystyle{\frac{\partial Y_k}{\partial m_l}} p_l n_k +
242     \displaystyle{\frac{\partial Y_k}{\partial m_{l,j}}} p_{l,j} n_k +
243     \displaystyle{\frac{\partial X_{ki}}{\partial m_l}} p_l n_{k,i} +
244     \displaystyle{\frac{\partial X_{ki}}{\partial m_{l,j}}} p_{l,j} n_{k,i}
245     \; dx
246     \end{equation} We follow the concept as outlined in section~\ref{chapter:ref:inversion cost function:gradient}.
247     Notice that equation~\ref{REF:EQU:INTRO 2b} defines a system of linear PDEs
248     which is solved using \escript \class{LinearPDE} class. In the \escript notation we need to provide
249     \begin{equation}\label{ref:EQU:REG:600}
250     \begin{array}{rcl}
251     A_{kilj} & = & \displaystyle{\frac{\partial X_{ki}}{\partial m_{l,j}}} \\
252     B_{kil} & = & \displaystyle{\frac{\partial X_{ki}}{\partial m_l}} \\
253     C_{klj} & = & \displaystyle{\frac{\partial Y_k}{\partial m_{l,j}}} \\
254     D_{kl} & = & \displaystyle{\frac{\partial Y_k}{\partial m_l}} \\
255     \end{array}
256     \end{equation}
257     The calculation of the gradient of the forward model component is more complicated:
258 jfenwick 4438 the data defect $J^{f}$ for forward model $f$ is expressed using a cost function kernel $K^{f}$
259 gross 4147 \begin{equation}\label{REF:EQU:INTRO 2bb}
260 gross 4142 J^{f}(p^f) = \int_{\Omega} K^{f} \; dx
261     \end{equation}
262     In this case the cost function kernel $K^{f}$ is a function of the
263     physical parameter $p^f$, which again is a function of the level-set function,
264     and the state variable $u^f_{k}$ and its gradient $u^f_{k,i}$. For the sake of a simpler
265     presentation the upper index $f$ is dropped.
266    
267     The gradient $\nabla_{p} J$ of the $J$ with respect to
268     the physical property $p$ is given as
269     \begin{equation}\label{REF:EQU:costfunction 100b}
270     <q, \nabla_{p} J(p)> = \int_{\Omega}
271     \displaystyle{\frac{\partial K }{\partial u_k } } \displaystyle{\frac{\partial u_k }{\partial q } } +
272     \displaystyle{\frac{\partial K }{\partial u_{k,i} } } \left( \displaystyle{\frac{\partial u_k }{\partial q } } \right)_{,i}+
273     \displaystyle{\frac{\partial K }{\partial p } } q \; dx
274     \end{equation}
275     for any $q$ as an increment to the physical parameter $p$. If the change
276     of the state variable
277     $u_f$ for physical parameter $p$ in the direction of $q$ is denoted as
278     \begin{equation}\label{REF:EQU:costfunction 100c}
279     d_k =\displaystyle{\frac{\partial u_k }{\partial q } }
280     \end{equation}
281     equation~\ref{REF:EQU:costfunction 100b} can be written as
282     \begin{equation}\label{REF:EQU:costfunction 100d}
283     <q, \nabla_{p} J(p)> = \int_{\Omega}
284     \displaystyle{\frac{\partial K }{\partial u_k } } d_k +
285     \displaystyle{\frac{\partial K }{\partial u_{k,i} } } d_{k,i}+
286     \displaystyle{\frac{\partial K }{\partial p } } q \; dx
287     \end{equation}
288     The state variable are the solution of PDE which in variational from is given
289     \begin{equation}\label{REF:EQU:costfunction 100}
290     \int_{\Omega} F_k \cdot r_k + G_{li} \cdot r_{k,i} \; dx = 0
291     \end{equation}
292     for all increments $r$ to the stat $u$. The functions $F$ and $G$ are given and describe the physical
293     model. They depend of the state variable $u_{k}$ and its gradient $u_{k,i}$ and the physical parameter $p$. The change
294     $d_k$ of the state
295     $u_f$ for physical parameter $p$ in the direction of $q$ is given from the equation
296 gross 4147 \begin{equation}\label{REF:EQU:costfunction 100bb}
297 gross 4142 \int_{\Omega}
298     \displaystyle{\frac{\partial F_k }{\partial u_l } } d_l r_k +
299     \displaystyle{\frac{\partial F_k }{\partial u_{l,j}} } d_{l,j} r_k +
300     \displaystyle{\frac{\partial F_k }{\partial p} }q r_k +
301     \displaystyle{\frac{\partial G_{ki}}{\partial u_l} } d_l r_{k,i} +
302     \displaystyle{\frac{\partial G_{ki}}{\partial u_{l,j}} } d_{l,j} r_{k,i}+
303     \displaystyle{\frac{\partial G_{ki}}{\partial p} } q r_{k,i}
304     \; dx = 0
305     \end{equation}
306     to be fulfilled for all functions $r$. Now let $d^*_k$ be the solution of the
307     variational equation
308 gross 4147 \begin{equation}\label{REF:EQU:costfunction 100dd}
309 gross 4142 \int_{\Omega}
310     \displaystyle{\frac{\partial F_k }{\partial u_l } } h_l d^*_k +
311     \displaystyle{\frac{\partial F_k }{\partial u_{l,j}} } h_{l,j} d^*_k +
312     \displaystyle{\frac{\partial G_{ki}}{\partial u_l} } h_l d^*_{k,i} +
313     \displaystyle{\frac{\partial G_{ki}}{\partial u_{l,j}} } h_{l,j} d^*_{k,i}
314     \; dx
315     = \int_{\Omega}
316     \displaystyle{\frac{\partial K }{\partial u_k } } h_k +
317     \displaystyle{\frac{\partial K }{\partial u_{k,i} } } h_{k,i} \; dx
318     \end{equation}
319     for all increments $h_k$ to the physical property $p$. This problem
320     is solved using \escript \class{LinearPDE} class. In the \escript notation we need to provide
321 gross 4147 \begin{equation}\label{ref:EQU:REG:600b}
322 gross 4142 \begin{array}{rcl}
323     A_{kilj} & = & \displaystyle{\frac{\partial G_{lj}}{\partial u_{k,i}} } \\
324     B_{kil} & = & \displaystyle{\frac{\partial F_l }{\partial u_{k,i}} } \\
325     C_{klj} & = & \displaystyle{\frac{\partial G_{lj}}{\partial u_k} } \\
326     D_{kl} & = & \displaystyle{\frac{\partial F_l }{\partial u_k } } \\
327     Y_{k} & = & \displaystyle{\frac{\partial K }{\partial u_k } } \\
328     X_{ki} & = & \displaystyle{\frac{\partial K }{\partial u_{k,i} } } \\
329     \end{array}
330     \end{equation}
331     Notice that these coefficient are transposed to the coefficients used to solve for the
332     state variables in equation~\ref{REF:EQU:costfunction 100}.
333 gross 4133
334 gross 4142 Setting $h_l=d_l$ in equation~\ref{REF:EQU:costfunction 100d} and
335     $r_k=d^*_k$ in equation~\ref{REF:EQU:costfunction 100b} one gets
336 gross 4147 \begin{equation}\label{ref:EQU:costfunction:601}
337 gross 4142 \int_{\Omega}
338     \displaystyle{\frac{\partial K }{\partial u_k } } d_k +
339     \displaystyle{\frac{\partial K }{\partial u_{k,i} } } d_{k,i}+
340     \displaystyle{\frac{\partial F_k }{\partial p} } q d^*_k +
341     \displaystyle{\frac{\partial G_{ki}}{\partial p} } q d^*_{k,i}
342     \; dx = 0
343     \end{equation}
344     which is inserted into equation~\ref{REF:EQU:costfunction 100d} to get
345     \begin{equation}\label{REF:EQU:costfunction 602}
346     <q, \nabla_{p} J(p)> = \int_{\Omega} \left(
347     \displaystyle{\frac{\partial K }{\partial p } } - \displaystyle{\frac{\partial F_k }{\partial p} } d^*_k
348     - \displaystyle{\frac{\partial G_{ki}}{\partial p} } d^*_{k,i} \right) q \; dx
349     \end{equation}
350     We need in fact the gradient of $J^f$ with respect to the level set function which is given as
351     \begin{equation}\label{REF:EQU:costfunction 603}
352     <n, \nabla J^f> = \int_{\Omega} \left(
353     \displaystyle{\frac{\partial K^f}{\partial p^f} } - \displaystyle{\frac{\partial F^f_k }{\partial p^f} } d^{f*}_k
354     - \displaystyle{\frac{\partial G^f_{ki}}{\partial p^f} } d^{f*}_{k,i} \right)
355     \cdot \displaystyle{\frac{\partial p^f }{\partial m_l} } n_l \; dx
356     \end{equation}
357     for any increment $n$ to the level set function. So in summary we get
358 gross 4147 \begin{equation}\label{ref:EQU:CS:101b}
359 gross 4142 \begin{array}{rcl}
360     Y_k & = & \displaystyle{\frac{\partial K^{reg}}{\partial m_k}} +
361     \sum_{f} \mu^{data}_{f} \left(
362     \displaystyle{\frac{\partial K^f}{\partial p^f} } - \displaystyle{\frac{\partial F^f_l }{\partial p^f} } d^{f*}_l
363     - \displaystyle{\frac{\partial G^f_{li}}{\partial p^f} } d^{f*}_{l,i} \right)
364     \cdot \displaystyle{\frac{\partial p^f }{\partial m_k} }
365 caltinay 4139
366 gross 4142 \\
367     X_{ki} & = & \displaystyle{\frac{\partial K^{reg}}{\partial m_{k,i}}}
368     \end{array}
369     \end{equation}
370     to represent $\nabla J$ as the tuple $(Y,X)$. Contributions of the forward model to the
371     Hessian operator are ignored.
372    

  ViewVC Help
Powered by ViewVC 1.1.26