1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
|
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html401/loose.dtd">
<html>
<!-- Copyright (C) 2004, 2005, 2006, 2007 Alain Lahellec
Copyright (C) 2004, 2005, 2006, 2007 Patrice Dumas
Copyright (C) 2004, Ste'phane Hallegatte
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.1 or
any later version published by the Free Software Foundation; with no
Invariant Sections, with no Front-Cover text and with no Back-Cover Text.
A copy of the license is included in the section entitled "GNU Free
Documentation License."
-->
<!-- Created on a sunny day by texi2html
texi2html was written by:
Lionel Cons <Lionel.Cons@cern.ch> (original author)
Karl Berry <karl@freefriends.org>
Olaf Bachmann <obachman@mathematik.uni-kl.de>
and many others.
Maintained by: Many creative people.
Send bugs and suggestions to <texi2html-bug@nongnu.org>
-->
<head>
<title>Miniker 102 manual: 4.2 Adjoint model and optimisation with Miniker</title>
<meta name="description" content="Miniker 102 manual: 4.2 Adjoint model and optimisation with Miniker">
<meta name="keywords" content="Miniker 102 manual: 4.2 Adjoint model and optimisation with Miniker">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="texi2html">
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<style type="text/css">
<!--
@import "mini_ker_tex4ht_math.css";
@import "mini_ker_tex4ht_tex.css";
a.summary-letter {text-decoration: none}
blockquote.smallquotation {font-size: smaller}
pre.display {font-family: serif}
pre.format {font-family: serif}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
pre.smalldisplay {font-family: serif; font-size: smaller}
pre.smallexample {font-size: smaller}
pre.smallformat {font-family: serif; font-size: smaller}
pre.smalllisp {font-size: smaller}
span.roman {font-family:serif; font-weight:normal;}
span.sansserif {font-family:sans-serif; font-weight:normal;}
ul.toc {list-style: none}
-->
</style>
</head>
<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
<a name="Adjoint-model-and-optimisation"></a>
<ul class="toc"><li><a href="mini_ker.html#Top">Miniker 102 manual</a> </li>
<li><ul class="toc"><li><a href="Dynamic-system-analysis.html#Dynamic-system-analysis">4. Dynamic analysis of systems in Miniker</a> </li>
</ul></li>
</ul>
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="Sensitivities.html#Advance-matrix-sensitivity" title="Previous section in reading order"> < </a>]</td>
<td valign="middle" align="left">[<a href="#Overview-of-optimisation-with-Miniker" title="Next section in reading order"> > </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="mini_ker.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="Concepts-index.html#Concepts-index" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="mini_ker_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<a name="Adjoint-model-and-optimisation-with-Miniker"></a>
<h2 class="section"> 4.2 Adjoint model and optimisation with Miniker </h2>
<ul class="toc">
<li> <a href="#Overview-of-optimisation-with-Miniker">4.2.1 Overview of optimisation with Miniker</a> </li>
<li> <a href="#Control-laws">4.2.2 Control laws</a> </li>
<li> <a href="#Cost-function-coding-and-adjoint-modeling">4.2.3 Cost function coding and adjoint modeling</a> </li>
<li> <a href="#Sensitivity-of-cost-function-to-parameters">4.2.4 Sensitivity of cost function to parameters</a> </li>
</ul>
<p>In the following a possible use of Miniker for optimisation is discussed.
More precisely the use of adjoint and control laws in Miniker are presented.
Optimisation isn’t the only application of these tools, but it is the most
common one. In that case the adjoint may be used to determine the gradient of a
functional to perturbations in the control laws, and an optimisation process
can use this
information to search for the optimum.
Another application of the adjoint is to compute the sensitivity of a
cost function to parameters (the ones declared in the <code>free_parameters:</code>’ list.
Note that the cost function can be sensitive to probe’s variables, even if these are
uncoupled with standard variables in the forward calculations; this is the case
when minimizing a quadratic distance function between probes (from the model)
and the corresponding measurements.
</p>
<p>The code is close transcription of the mathematical calculus described
in<br> <a href="http://www.lmd.jussieu.fr/ZOOM/doc/Adjoint.pdf">http://www.lmd.jussieu.fr/ZOOM/doc/Adjoint.pdf</a> . It essentialy reverse time and
transpose the four Jacobian matrices: states and transfers are saved in array dimensionned
with <code>maxstep</code> Fortran parameter.
</p>
<hr size="2">
<a name="Overview-of-optimisation-with-Miniker"></a>
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="#Adjoint-model-and-optimisation" title="Previous section in reading order"> < </a>]</td>
<td valign="middle" align="left">[<a href="#Control-laws" title="Next section in reading order"> > </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="mini_ker.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="Concepts-index.html#Concepts-index" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="mini_ker_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<a name="Overview-of-optimisation-with-Miniker-1"></a>
<h3 class="subsection"> 4.2.1 Overview of optimisation with Miniker </h3>
<a name="index-adjoint"></a>
<a name="index-optimisation"></a>
<p>In the proposed method, Miniker is run twice, one time forward and then
backward to determine the trajectory and the adjoint model. After that the
control laws are modified by a program external to Miniker. The same steps
are repeated until convergence. More pecisely,
</p>
<dl compact="compact">
<dt> <strong>forward</strong></dt>
<dd><p>The command law h(t) is given (by an explicit law or taken from a file).
The trajectory is computed in a classical way, with the additionnal computation
of the functional to be optimised, J, prescribed with specific
<code>f_set</code> macros. The states, transfers and control laws are stored.
</p></dd>
<dt> <strong>backward</strong></dt>
<dd><p>The adjoint variable is computed from the last time T backward. The
time increment is re-read as it could have changed during the forward
simulation. The system is solved by using the same technics as in the forward
simulation, but with a negative time step.
</p></dd>
<dt> <strong>external phase</strong></dt>
<dd><p>Now the command should be corrected. This step isn’t covered here, but, for
example, minuit the optimisation tool from the CERN could be used.
In order to ease such a use of Miniker, the principal program has to be
compiled as a subroutine to be driven by an external program
(see <a href="Calling-the-model-code.html#Calling-the-model-code">Calling the model code</a>).
</p></dd>
</dl>
<p>The functionnal J to be optimised is defined as
</p>
<p>
<center class="math-display" >
<img
src="mini_ker_tex4ht_tex15x.png" alt=" ∫ T
J = ψ [η(T),φ(T),h(T)]+ l[η(τ),φ (τ ),h(τ)]dτ
0 " class="math-display" ></center>
</p>
<a name="index-final-cost"></a>
<a name="index-integrand-cost"></a>
<p>Where ψ is the final cost function, l is the integrand
cost function and h represents the control laws variations.
</p>
<p>The general use of the adjoint model of a system is the determination of the
gradient of this J functional to be optimised, with respect to perturbations
of the original conditions of the reference trajectory, that is, along its
GTLS<a name="DOCF6" href="mini_ker_fot.html#FOOT6">(6)</a>.
</p>
<hr size="2">
<a name="Control-laws"></a>
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="#Overview-of-optimisation-with-Miniker" title="Previous section in reading order"> < </a>]</td>
<td valign="middle" align="left">[<a href="#Cost-function-coding-and-adjoint-modeling" title="Next section in reading order"> > </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="mini_ker.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="Concepts-index.html#Concepts-index" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="mini_ker_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<a name="Control-laws-1"></a>
<h3 class="subsection"> 4.2.2 Control laws </h3>
<a name="index-zcommand"></a>
<a name="index-command-law"></a>
<p>Each control law is associated with one cell or transfer equation, meaning that a command
associated with an equation does not appear in any other equation.
It is still possible
to add commands acting anywhere by defining a transfer equal to that command.
</p>
<p>The control laws associated with states are in the <code>ux_com(.)</code> array,
control laws associated with transfers are in the <code>uy_com(.)</code> array.
The control laws may be prescribed even when there is no adjoint computed,
nor any optimisation, and they are used during simulation, in which case they will
act as external sources. To enable
the use of commands, the logical flag <code>Zcommand</code> should be <code>.true.</code>.
</p>
<a name="index-uxcom_002edata"></a>
<a name="index-uycom_002edata"></a>
<p>The command can be given either as:
</p><ol>
<li> a table of numerical
values in the files ‘<tt>uxcom.data</tt>’ and ‘<tt>uycom.data</tt>’.
</li><li> a function
<a name="index-zlaw"></a>
<a name="index-zcmd_005flaw"></a>
<a name="index-zcmd_005flaw_002einc"></a>
of the problem variables. To turn that feature on the logical flag
<code>Zlaw</code> should be set to <code>.true.</code> in ‘<tt>zinit</tt>’. The sequence
‘<tt>zcmd_law</tt>’ should hold
the code filling the <code>ux_com(.)</code> and <code>uy_com(.)</code> arrays, as the code
from that sequence is used whenever the control laws are needed.
In that case the files ‘<tt>uxcom.data</tt>’ and ‘<tt>uycom.data</tt>’ will
be filled by the command values generated by the function along the trajectory.
</li></ol>
<p>For example in the Lotka-Volterra model, the parameter <code>apar</code> could
be a control variable.
In that case, <code>apar</code> would be defined as the variable <code>ux_com(1)</code>,
and either entered as a law
in the sequence ‘<tt>zcmd_law</tt>’ , either written in the file ‘<tt>uxcom.data</tt>’
step by step. In that case, there must be a perfect corresponodence between time
of the commands and time of the run.
</p>
<hr size="2">
<a name="Cost-function-coding-and-adjoint-modeling"></a>
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="#Control-laws" title="Previous section in reading order"> < </a>]</td>
<td valign="middle" align="left">[<a href="#Sensitivity-of-cost-function-to-parameters" title="Next section in reading order"> > </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="mini_ker.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="Concepts-index.html#Concepts-index" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="mini_ker_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<a name="Cost-function-coding-and-adjoint-modeling-1"></a>
<h3 class="subsection"> 4.2.3 Cost function coding and adjoint modeling </h3>
<a name="index-zback"></a>
<a name="index-cout_005fPsi"></a>
<a name="index-cout_005fl"></a>
<p>First of all the flag <code>zback</code> should be set to <code>.true.</code> in order to
allow adjoint model computation:
</p>
<table><tr><td> </td><td><pre class="example">Zback=.true.;
</pre></td></tr></table>
<p>The two functions <code>cout_Psi</code> corresponding with the final cost and
<code>cout_l</code> corresponding with the integrand cost are set up with the
<code>f_set</code> macros.
</p>
<dl>
<dt><a name="index-f_005fset-2"></a><u>Macro:</u> <b>f_set</b><i> cout_Psi = f(eta(.),ff(.),ux_com(.),uy_com(.))</i></dt>
<dd><p>This macro defines the final cost function.
<code>f</code> is a fortran
expression which may be function of cell state variables,
‘<samp>eta(1)</samp>’…‘<samp>eta(np)</samp>’, transfers
‘<samp>ff(1)</samp>’…‘<samp>ff(mp)</samp>’,
state control laws
‘<samp>ux_com(1)</samp>’…‘<samp>ux_com(np)</samp>’, and transfer control laws
‘<samp>uy_com(1)</samp>’…‘<samp>uy_com(mp)</samp>’.
</p></dd></dl>
<dl>
<dt><a name="index-f_005fset-3"></a><u>Macro:</u> <b>f_set</b><i> cout_l = f(eta(.),ff(.),ux_com(.),uy_com(.))</i></dt>
<dd><p>This macro defines the integrand cost function.
<code>f</code> is a fortran
expression which may be function of cell state variables,
‘<samp>eta(1)</samp>’…‘<samp>eta(np)</samp>’, transfers
‘<samp>ff(1)</samp>’…‘<samp>ff(mp)</samp>’,
state control laws
‘<samp>ux_com(1)</samp>’…‘<samp>ux_com(np)</samp>’, and transfer control laws
‘<samp>uy_com(1)</samp>’…‘<samp>uy_com(mp)</samp>’.
</p></dd></dl>
<p>For example, the following code sets a cost function for the masselottes
model:
</p>
<table><tr><td> </td><td><pre class="example">! Initialisation
F_set cout_Psi = eta_move(inode:1);
!and f_set cout_l integrand in the functionnal
F_set cout_l = 0.;
</pre></td></tr></table>
<p>In that example the functional is reduced to the final value
of the first state component.
Here, the adjoint vector will correspond to the final sensitivity
(at t = 0) of
that component (here the first masselotte position) to a perturbation in
all initial conditions<a name="DOCF7" href="mini_ker_fot.html#FOOT7">(7)</a>.
</p>
<p>The following variables are set during the backward phase, and output
in the associated files:
</p>
<table>
<thead><tr><th>var</th><th>file</th><th>explanation</th></tr></thead>
<tr><td><code>v_adj(.)</code></td><td>‘<tt>vadj.data</tt>’</td><td>adjoint to <code>eta(.)</code></td></tr>
<tr><td><code>w_adj(.)</code></td><td>‘<tt>wadj.data</tt>’</td><td>adjoint to <code>ff(.)</code></td></tr>
<tr><td><code>wadj(mp+.)</code></td><td>‘<tt>gradmuj.data</tt>’</td><td>adjoint to <code>ff(mp+.)</code></td></tr>
<tr><td><code>graduej(.)</code></td><td>‘<tt>gradxj.data</tt>’</td><td>adjoint to <code>ux_com(.)</code></td></tr>
<tr><td><code>gradufj(.)</code></td><td>‘<tt>gradyj.data</tt>’</td><td>adjoint to <code>uy_com(.)</code></td></tr>
<tr><td><code>hamilton</code></td><td>‘<tt>hamilton.data</tt>’</td><td>time increment, hamiltonian, cost function increment</td></tr>
</table>
<hr size="2">
<a name="Sensitivity-of-cost-function-to-parameters"></a>
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="#Cost-function-coding-and-adjoint-modeling" title="Previous section in reading order"> < </a>]</td>
<td valign="middle" align="left">[<a href="Kalman-filter.html#Kalman-filter" title="Next section in reading order"> > </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="mini_ker.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="Concepts-index.html#Concepts-index" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="mini_ker_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<a name="Sensitivity-of-cost-function-to-parameters-1"></a>
<h3 class="subsection"> 4.2.4 Sensitivity of cost function to parameters </h3>
<a name="index-gradpj_002edata"></a>
<p>The sensitivity of the cost function to all the parameters given as
arguments of <code>Free_parameters</code> is computed. For the
predator model the sensitivity of a cost function consisting in
the integral of the predator population with respect with
<code>apar</code> an <code>cpar</code> is obtained with a number of parameters
set to 2 in ‘<tt>dimetaphi</tt>’:
</p>
<table><tr><td> </td><td><pre class="example">parameter (lp=2);
</pre></td></tr></table>
<p>And the cost function and <code>Free_parameters</code> list in ‘<tt>zinit</tt>’:
</p>
<table><tr><td> </td><td><pre class="example">f_set cout_Psi = eta(2);
f_set cout_l = eta(2);
Free_parameters: apar,cpar;
</pre></td></tr></table>
<p><code>apar</code> and <code>cpar</code> also have to be given a value.
The result is output in ‘<tt>gradpj.data</tt>’.
</p>
<hr size="2">
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="mini_ker.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="Concepts-index.html#Concepts-index" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="mini_ker_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<p>
<font size="-1">
This document was generated by <em>a tester</em> on <em>a sunny day</em> using <a href="http://www.nongnu.org/texi2html/"><em>texi2html</em></a>.
</font>
<br>
</p>
</body>
</html>
|
