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hermite_example.cpp
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6// Stokhos Package
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43
44// hermite_example
45//
46// usage:
47// hermite_example
48//
49// output:
50// prints the Hermite Polynomial Chaos Expansion of the simple function
51//
52// v = 1/(log(u)^2+1)
53//
54// where u = 1 + 0.4*H_1(x) + 0.06*H_2(x) + 0.002*H_3(x), x is a zero-mean
55// and unit-variance Gaussian random variable, and H_i(x) is the i-th
56// Hermite polynomial.
57
58#include "Stokhos.hpp"
59#include "Teuchos_GlobalMPISession.hpp"
60
61int main(int argc, char **argv)
62{
63 using Teuchos::Array;
64 using Teuchos::RCP;
65 using Teuchos::rcp;
66
67 // If applicable, set up MPI.
68 Teuchos::GlobalMPISession mpiSession (&argc, &argv);
69
70 try {
71
72 // Basis of dimension 3, order 5
73 const int d = 3;
74 const int p = 5;
75 Array< RCP<const Stokhos::OneDOrthogPolyBasis<int,double> > > bases(d);
76 for (int i=0; i<d; i++) {
77 bases[i] = rcp(new Stokhos::HermiteBasis<int,double>(p-i));
78 }
79 RCP<const Stokhos::CompletePolynomialBasis<int,double> > basis =
81
82 // Quadrature method
83 RCP<const Stokhos::Quadrature<int,double> > quad =
85
86 // Triple product tensor
87 RCP<Stokhos::Sparse3Tensor<int,double> > Cijk =
88 basis->computeTripleProductTensor();
89
90 // Expansion method
92
93 // Polynomial expansions
94 Stokhos::OrthogPolyApprox<int,double> u(basis), v(basis), w(basis);
95 u.term(0,0) = 1.0;
96 for (int i=0; i<d; i++) {
97 if (bases[i]->order() >= 1)
98 u.term(i,1) = 0.4 / d;
99 if (bases[i]->order() >= 2)
100 u.term(i,2) = 0.06 / d;
101 if (bases[i]->order() >= 3)
102 u.term(i,3) = 0.002 / d;
103 }
104
105 // Compute expansion
106 expn.log(v,u);
107 expn.times(w,v,v);
108 expn.plusEqual(w,1.0);
109 expn.divide(v,1.0,w);
110
111 // Print u and v
112 std::cout << "v = 1.0 / (log(u)^2 + 1):" << std::endl;
113 std::cout << "\tu = ";
114 u.print(std::cout);
115 std::cout << "\tv = ";
116 v.print(std::cout);
117
118 // Compute moments
119 double mean = v.mean();
120 double std_dev = v.standard_deviation();
121
122 // Evaluate PCE and function at a point = 0.25 in each dimension
123 Array<double> pt(d);
124 for (int i=0; i<d; i++)
125 pt[i] = 0.25;
126 double up = u.evaluate(pt);
127 double vp = 1.0/(std::log(up)*std::log(up)+1.0);
128 double vp2 = v.evaluate(pt);
129
130 // Print results
131 std::cout << "\tv mean = " << mean << std::endl;
132 std::cout << "\tv std. dev. = " << std_dev << std::endl;
133 std::cout << "\tv(0.25) (true) = " << vp << std::endl;
134 std::cout << "\tv(0.25) (pce) = " << vp2 << std::endl;
135
136 // Check the answer
137 if (std::abs(vp - vp2) < 1e-2)
138 std::cout << "\nExample Passed!" << std::endl;
139
140 Teuchos::TimeMonitor::summarize(std::cout);
141 }
142 catch (std::exception& e) {
143 std::cout << e.what() << std::endl;
144 }
145}
Multivariate orthogonal polynomial basis generated from a total-order complete-polynomial tensor prod...
Hermite polynomial basis.
Class to store coefficients of a projection onto an orthogonal polynomial basis.
Orthogonal polynomial expansions based on numerical quadrature.
Defines quadrature for a tensor product basis by tensor products of 1-D quadrature rules.
int main(int argc, char **argv)