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10<TITLE>Monte Carlo Methods</TITLE>
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53
54<H1><A NAME="SECTION02200000000000000000"></A> <A NAME="secmessel"></A>
55<BR>
56Monte Carlo Methods
57</H1>
58
59<P>
60The Geant4 toolkit uses a combination of the composition and
61rejection Monte Carlo methods. Only the basic formalism of these methods is
62outlined here. For a complete account of the Monte Carlo methods, the
63interested user is referred to the publications of Butcher and Messel,
64Messel and Crawford, or Ford and Nelson [#!m.butch!#,#!m.messel!#,#!m.egs4!#].
65
66<P>
67Suppose we wish to sample <IMG
68 WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
69 SRC="img1.gif"
70 ALT="$ x$"> in the interval <!-- MATH
71 $[x_1,\ x_2]$
72 -->
73<IMG
74 WIDTH="66" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
75 SRC="img113.gif"
76 ALT="$ [x_1, x_2]$"> from the
77distribution <IMG
78 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
79 SRC="img114.gif"
80 ALT="$ f(x)$"> and the <I>normalised</I> probability density function can
81be written as :
82<P></P>
83<DIV ALIGN="CENTER"><!-- MATH
84 \begin{equation}
85f(x) = \sum_{i=1}^{n} N_{i} f_{i} (x) g_{i} (x)
86\end{equation}
87 -->
88<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
89<TR VALIGN="MIDDLE">
90<TD NOWRAP ALIGN="CENTER"><IMG
91 WIDTH="197" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
92 SRC="img115.gif"
93 ALT="$\displaystyle f(x) = \sum_{i=1}^{n} N_{i} f_{i} (x) g_{i} (x)$"></TD>
94<TD NOWRAP WIDTH="10" ALIGN="RIGHT">
95(2.1)</TD></TR>
96</TABLE></DIV>
97<BR CLEAR="ALL"><P></P>
98where <IMG
99 WIDTH="60" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
100 SRC="img116.gif"
101 ALT="$ N_i&gt;0$">, <IMG
102 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
103 SRC="img117.gif"
104 ALT="$ f_i(x)$"> are <I>normalised</I> density functions on <!-- MATH
105 $[x_1,\ x_2]$
106 -->
107<IMG
108 WIDTH="66" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
109 SRC="img113.gif"
110 ALT="$ [x_1, x_2]$">
111, and <!-- MATH
112 $0 \leq g_i (x) \leq 1$
113 -->
114<IMG
115 WIDTH="114" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
116 SRC="img118.gif"
117 ALT="$ 0 \leq g_i (x) \leq 1$">.
118
119<P>
120According to this method, <IMG
121 WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
122 SRC="img1.gif"
123 ALT="$ x$"> can sampled in the following way:
124
125<OL>
126<LI>select a random integer <!-- MATH
127 $i \in \{1,2,\cdots n\}$
128 -->
129<IMG
130 WIDTH="126" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
131 SRC="img119.gif"
132 ALT="$ i \in \{1,2,\cdots n\}$">
133with probability proportional to <IMG
134 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
135 SRC="img120.gif"
136 ALT="$ N_i $">
137</LI>
138<LI>select a value <IMG
139 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
140 SRC="img121.gif"
141 ALT="$ x_0$"> from the distribution <IMG
142 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
143 SRC="img117.gif"
144 ALT="$ f_i(x)$">
145</LI>
146<LI>calculate <IMG
147 WIDTH="52" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
148 SRC="img122.gif"
149 ALT="$ g_i (x_0)$"> and accept <IMG
150 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
151 SRC="img123.gif"
152 ALT="$ x = x_0$"> with probability <IMG
153 WIDTH="52" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
154 SRC="img122.gif"
155 ALT="$ g_i (x_0)$">;
156</LI>
157<LI>if <IMG
158 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
159 SRC="img121.gif"
160 ALT="$ x_0$"> is rejected restart from step 1.
161</LI>
162</OL>
163It can be shown that this scheme is correct and the mean
164number of tries to accept a value is <!-- MATH
165 $\sum_{i} N_i$
166 -->
167<IMG
168 WIDTH="54" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
169 SRC="img124.gif"
170 ALT="$ \sum_{i} N_i $">.
171
172<P>
173In practice, a good method of sampling from the distribution <IMG
174 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
175 SRC="img114.gif"
176 ALT="$ f(x)$"> has the
177following properties:
178
179<UL>
180<LI>all the subdistributions <IMG
181 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
182 SRC="img117.gif"
183 ALT="$ f_i(x)$"> can be sampled easily;
184</LI>
185<LI>the rejection functions <IMG
186 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
187 SRC="img125.gif"
188 ALT="$ g_i(x)$"> can be evaluated easily/quickly;
189</LI>
190<LI>the mean number of tries is not too large.
191</LI>
192</UL>
193Thus the different possible decompositions of the distribution
194<IMG
195 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
196 SRC="img114.gif"
197 ALT="$ f(x)$"> are not equivalent from the practical point of view (e.g. they
198can be very different in computational speed) and it can be useful
199to optimise the decomposition.
200
201<P>
202A remark of practical importance : if our distribution is not
203normalised
204<!-- MATH
205 \begin{displaymath}
206\int_{x_1}^{x_2} f(x)dx = C > 0
207\end{displaymath}
208 -->
209<P></P>
210<DIV ALIGN="CENTER">
211<IMG
212 WIDTH="173" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
213 SRC="img126.gif"
214 ALT="$\displaystyle \int_{x_1}^{x_2} f(x)dx = C &gt; 0$">
215</DIV><P></P>
216the method can be used in the same
217manner; the mean number of tries in this
218case is <!-- MATH
219 $\sum_ {i} N_i/C$
220 -->
221<IMG
222 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
223 SRC="img127.gif"
224 ALT="$ \sum_ {i} N_i/C$">.
225
226<P>
227<BR><HR>
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