forked from johannesgerer/jburkardt-cpp
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathpce_ode_hermite.html
219 lines (187 loc) · 5.72 KB
/
pce_ode_hermite.html
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
<html>
<head>
<title>
PCE_ODE_HERMITE - Hermite Polynomial Chaos Expansion for a Scalar ODE
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
PCE_ODE_HERMITE <br> Hermite Polynomial Chaos Expansion for a Scalar ODE
</h1>
<hr>
<p>
<b>PCE_ODE_HERMITE</b>
is a C++ library which
defines and solves a time-dependent scalar exponential decay ODE
with uncertain decay coefficient, using a polynomial chaos expansion,
in terms of Hermite polynomials.
</p>
<p>
The deterministic equation is
<pre>
du/dt = - alpha * u,
u(0) = u0
</pre>
In the stochastic version, it is assumed that the decay coefficient
ALPHA is a Gaussian random variable with mean value ALPHA_MU and variance
ALPHA_SIGMA^2.
</p>
<p>
The exact expected value of the stochastic equation is known to be
<pre>
u(t) = u0 * exp ( t^2/2)
</pre>
This should be matched by the first component of the polynomial chaos
expansion.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>PCE_ODE_HERMITE</b> is available in
<a href = "../../c_src/pce_ode_hermite/pce_ode_hermite.html">a C version</a> and
<a href = "../../cpp_src/pce_ode_hermite/pce_ode_hermite.html">a C++ version</a> and
<a href = "../../f77_src/pce_ode_hermite/pce_ode_hermite.html">a FORTRAN77 version</a> and
<a href = "../../f_src/pce_ode_hermite/pce_ode_hermite.html">a FORTRAN90 version</a> and
<a href = "../../m_src/pce_ode_hermite/pce_ode_hermite.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/black_scholes/black_scholes.html">
BLACK_SCHOLES</a>,
a C++ library which
implements some simple approaches to
the Black-Scholes option valuation theory,
by Desmond Higham.
</p>
<p>
<a href = "../../cpp_src/hermite_polynomial/hermite_polynomial.html">
HERMITE_POLYNOMIAL</a>,
a C++ library which
evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial,
the Hermite function, and related functions.
</p>
<p>
<a href = "../../cpp_src/pce_burgers/pce_burgers.html">
PCE_BURGERS</a>,
a C++ program which
defines and solves a version of the time-dependent viscous Burgers equation,
with uncertain viscosity, using a polynomial chaos expansion in terms
of Hermite polynomials,
by Gianluca Iaccarino.
</p>
<p>
<a href = "../../cpp_src/sde/sde.html">
SDE</a>,
a C++ library which
illustrates the properties of stochastic differential equations (SDE's), and
common algorithms for their analysis,
by Desmond Higham;
</p>
<p>
<a href = "../../cpp_src/stochastic_rk/stochastic_rk.html">
STOCHASTIC_RK</a>,
a C++ library which
applies a Runge Kutta (RK) scheme to a stochastic differential equation.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Roger Ghanem, Pol Spanos,<br>
Stochastic Finite Elements: A Spectral Approach,<br>
Revised Edition,<br>
Dover, 2003,<br>
ISBN: 0486428184,<br>
LC: TA347.F5.G56.
</li>
<li>
Dongbin Xiu,<br>
Numerical Methods for Stochastic Computations: A Spectral Method Approach,<br>
Princeton, 2010,<br>
ISBN13: 978-0-691-14212-8,<br>
LC: QA274.23.X58.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "pce_ode_hermite.cpp">pce_ode_hermite.cpp</a>, the source code.
</li>
<li>
<a href = "pce_ode_hermite.hpp">pce_ode_hermite.hpp</a>, the source code.
</li>
<li>
<a href = "pce_ode_hermite.sh">pce_ode_hermite.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "pce_ode_hermite_prb.cpp">pce_ode_hermite_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "pce_ode_hermite_prb.sh">pce_ode_hermite_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "pce_ode_hermite_prb_output.txt">pce_ode_hermite_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>HE_DOUBLE_PRODUCT_INTEGRAL:</b> integral of He(i,x)*He(j,x)*e^(-x^2/2).
</li>
<li>
<b>HE_TRIPLE_PRODUCT_INTEGRAL:</b> integral of He(i,x)*He(j,x)*He(k,x)*e^(-x^2/2).
</li>
<li>
<b>PCE_ODE_HERMITE</b> applies the polynomial chaos expansion to a scalar ODE.
</li>
<li>
<b>R8_FACTORIAL</b> computes the factorial of N.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last modified on 18 March 2012.
</i>
<!-- John Burkardt -->
</body>
</html>