int main(int argc, char **argv)
{
try {
const unsigned int d = 2;
const unsigned int pmin = 2;
const unsigned int pmax = 10;
const unsigned int np = pmax-pmin+1;
bool use_pce_quad_points = false;
bool normalize = true;
bool sparse_grid = true;
bool project_integrals = false;
#ifndef HAVE_STOKHOS_DAKOTA
sparse_grid = false;
#endif
Teuchos::Array<double> mean(np), mean_st(np), std_dev(np), std_dev_st(np);
Teuchos::Array<double> pt(np), pt_st(np);
Teuchos::Array< Teuchos::RCP<const Stokhos::OneDOrthogPolyBasis<int,double> > > bases(d);
Teuchos::Array<double> eval_pt(d, 0.5);
double pt_true;
// Loop over orders
unsigned int n = 0;
for (unsigned int p=pmin; p<=pmax; p++) {
std::cout << "p = " << p << std::endl;
// Create product basis
for (unsigned int i=0; i<d; i++)
bases[i] = Teuchos::rcp(new basis_type(p));
Teuchos::RCP<const Stokhos::CompletePolynomialBasis<int,double> > basis =
Teuchos::rcp(new Stokhos::CompletePolynomialBasis<int,double>(bases));
// Create approximation
Stokhos::OrthogPolyApprox<int,double> x(basis), u(basis), v(basis),
w(basis), w2(basis);
for (unsigned int i=0; i<d; i++) {
x.term(i, 1) = 1.0;
}
double x_pt = x.evaluate(eval_pt);
pt_true = std::exp(std::sin(x_pt));
// Quadrature
Teuchos::RCP<const Stokhos::Quadrature<int,double> > quad;
#ifdef HAVE_STOKHOS_DAKOTA
if (sparse_grid)
quad =
Teuchos::rcp(new Stokhos::SparseGridQuadrature<int,double>(basis, p));
#endif
if (!sparse_grid)
quad =
Teuchos::rcp(new Stokhos::TensorProductQuadrature<int,double>(basis));
// Triple product tensor
Teuchos::RCP<Stokhos::Sparse3Tensor<int,double> > Cijk =
basis->computeTripleProductTensor(basis->size());
// Quadrature expansion
Stokhos::QuadOrthogPolyExpansion<int,double> quad_exp(basis, Cijk, quad);
// Compute PCE via quadrature expansion
quad_exp.sin(u,x);
//quad_exp.times(u,u,10.0);
quad_exp.exp(w,u);
// Compute Stieltjes basis
Teuchos::Array< Teuchos::RCP<const Stokhos::OneDOrthogPolyBasis<int,double> > > st_bases(1);
Teuchos::RCP<const Stokhos::LanczosProjPCEBasis<int,double> > st_1d_basis
= Teuchos::rcp(new Stokhos::LanczosProjPCEBasis<int,double>(
p, Teuchos::rcp(&u,false), Cijk, normalize));
st_bases[0] = st_1d_basis;
Teuchos::RCP<const Stokhos::CompletePolynomialBasis<int,double> >
st_basis =
Teuchos::rcp(new Stokhos::CompletePolynomialBasis<int,double>(st_bases));
//std::cout << *st_basis << std::endl;
Stokhos::OrthogPolyApprox<int,double> u_st(st_basis), w_st(st_basis);
u_st.term(0, 0) = st_1d_basis->getNewCoeffs(0);
u_st.term(0, 1) = st_1d_basis->getNewCoeffs(1);
// Triple product tensor
Teuchos::RCP<Stokhos::Sparse3Tensor<int,double> > st_Cijk =
st_basis->computeTripleProductTensor(st_basis->size());
// Tensor product quadrature
Teuchos::RCP<const Stokhos::Quadrature<int,double> > st_quad;
if (!use_pce_quad_points) {
#ifdef HAVE_STOKHOS_DAKOTA
if (sparse_grid)
st_quad = Teuchos::rcp(new Stokhos::SparseGridQuadrature<int,double>(st_basis, p));
#endif
if (!sparse_grid)
st_quad = Teuchos::rcp(new Stokhos::TensorProductQuadrature<int,double>(st_basis));
}
else {
Teuchos::Array<double> st_points_0;
Teuchos::Array<double> st_weights_0;
Teuchos::Array< Teuchos::Array<double> > st_values_0;
st_bases[0]->getQuadPoints(p+1, st_points_0, st_weights_0, st_values_0);
Teuchos::Array<double> st_points_1;
Teuchos::Array<double> st_weights_1;
Teuchos::Array< Teuchos::Array<double> > st_values_1;
st_bases[1]->getQuadPoints(p+1, st_points_1, st_weights_1, st_values_1);
Teuchos::RCP< Teuchos::Array< Teuchos::Array<double> > > st_points =
Teuchos::rcp(new Teuchos::Array< Teuchos::Array<double> >(st_points_0.size()));
for (int i=0; i<st_points_0.size(); i++) {
(*st_points)[i].resize(2);
(*st_points)[i][0] = st_points_0[i];
(*st_points)[i][1] = st_points_1[i];
}
Teuchos::RCP< Teuchos::Array<double> > st_weights =
Teuchos::rcp(new Teuchos::Array<double>(st_weights_0));
Teuchos::RCP< const Stokhos::OrthogPolyBasis<int,double> > st_b =
st_basis;
st_quad =
Teuchos::rcp(new Stokhos::UserDefinedQuadrature<int,double>(st_b,
st_points,
st_weights));
}
// Quadrature expansion
Stokhos::QuadOrthogPolyExpansion<int,double> st_quad_exp(st_basis,
st_Cijk,
st_quad);
// Compute w_st = u_st*v_st in Stieltjes basis
st_quad_exp.exp(w_st, u_st);
// Project w_st back to original basis
pce_quad_func st_func(w_st, *st_basis);
quad_exp.unary_op(st_func, w2, u);
// std::cout.precision(12);
// std::cout << w;
// std::cout << w2;
// std::cout << w_st;
mean[n] = w.mean();
mean_st[n] = w2.mean();
std_dev[n] = w.standard_deviation();
std_dev_st[n] = w2.standard_deviation();
pt[n] = w.evaluate(eval_pt);
pt_st[n] = w2.evaluate(eval_pt);
n++;
}
n = 0;
int wi=10;
std::cout << "Statistical error:" << std::endl;
std::cout << "p "
<< std::setw(wi) << "mean" << " "
<< std::setw(wi) << "mean_st" << " "
<< std::setw(wi) << "std_dev" << " "
<< std::setw(wi) << "std_dev_st" << " "
<< std::setw(wi) << "point" << " "
<< std::setw(wi) << "point_st" << std::endl;
for (unsigned int p=pmin; p<pmax; p++) {
std::cout.precision(3);
std::cout.setf(std::ios::scientific);
std::cout << p << " "
<< std::setw(wi) << rel_err(mean[n], mean[np-1]) << " "
<< std::setw(wi) << rel_err(mean_st[n], mean[np-1]) << " "
<< std::setw(wi) << rel_err(std_dev[n], std_dev[np-1]) << " "
<< std::setw(wi) << rel_err(std_dev_st[n], std_dev[np-1])
<< " "
<< std::setw(wi) << rel_err(pt[n], pt_true) << " "
<< std::setw(wi) << rel_err(pt_st[n], pt_true)
<< std::endl;
n++;
}
}
catch (std::exception& e) {
std::cout << e.what() << std::endl;
}
}
int main(int argc, char **argv)
{
try {
const unsigned int d = 2;
const unsigned int p = 5;
// Create product basis
Teuchos::Array< Teuchos::RCP<const Stokhos::OneDOrthogPolyBasis<int,double> > > bases(d);
for (unsigned int i=0; i<d; i++)
bases[i] = Teuchos::rcp(new basis_type(p));
Teuchos::RCP<const Stokhos::CompletePolynomialBasis<int,double> > basis =
Teuchos::rcp(new Stokhos::CompletePolynomialBasis<int,double>(bases));
// Create approximation
Stokhos::OrthogPolyApprox<int,double> x(basis), u(basis), v(basis),
w(basis), w2(basis), w3(basis);
for (unsigned int i=0; i<d; i++) {
x.term(i, 1) = 1.0;
}
// Tensor product quadrature
Teuchos::RCP<const Stokhos::Quadrature<int,double> > quad =
Teuchos::rcp(new Stokhos::TensorProductQuadrature<int,double>(basis));
// Triple product tensor
Teuchos::RCP<Stokhos::Sparse3Tensor<int,double> > Cijk =
basis->computeTripleProductTensor();
// Quadrature expansion
Stokhos::QuadOrthogPolyExpansion<int,double> quad_exp(basis, Cijk, quad);
// Compute PCE via quadrature expansion
quad_exp.sin(u,x);
quad_exp.exp(v,x);
quad_exp.times(w,v,u);
// Compute tensor product Stieltjes basis for u and v
Teuchos::Array< Teuchos::RCP<const Stokhos::OneDOrthogPolyBasis<int,double> > > st_bases(2);
st_bases[0] =
Teuchos::rcp(new Stokhos::StieltjesPCEBasis<int,double>(
p, Teuchos::rcp(&u,false), quad, true));
st_bases[1] =
Teuchos::rcp(new Stokhos::StieltjesPCEBasis<int,double>(
p, Teuchos::rcp(&v,false), quad, true));
Teuchos::RCP<const Stokhos::CompletePolynomialBasis<int,double> > st_basis =
Teuchos::rcp(new Stokhos::CompletePolynomialBasis<int,double>(st_bases));
Stokhos::OrthogPolyApprox<int,double> u_st(st_basis), v_st(st_basis),
w_st(st_basis);
u_st.term(0, 0) = u.mean();
u_st.term(0, 1) = 1.0;
v_st.term(0, 0) = v.mean();
v_st.term(1, 1) = 1.0;
// Use Gram-Schmidt to orthogonalize Stieltjes basis of u and v
Teuchos::Array<double> st_points_0;
Teuchos::Array<double> st_weights_0;
Teuchos::Array< Teuchos::Array<double> > st_values_0;
st_bases[0]->getQuadPoints(p+1, st_points_0, st_weights_0, st_values_0);
Teuchos::Array<double> st_points_1;
Teuchos::Array<double> st_weights_1;
Teuchos::Array< Teuchos::Array<double> > st_values_1;
st_bases[1]->getQuadPoints(p+1, st_points_1, st_weights_1, st_values_1);
Teuchos::Array< Teuchos::Array<double> > st_points(st_points_0.size());
for (int i=0; i<st_points_0.size(); i++) {
st_points[i].resize(2);
st_points[i][0] = st_points_0[i];
st_points[i][1] = st_points_1[i];
}
Teuchos::Array<double> st_weights = st_weights_0;
Teuchos::RCP< Stokhos::GramSchmidtBasis<int,double> > gs_basis =
Teuchos::rcp(new Stokhos::GramSchmidtBasis<int,double>(st_basis,
st_points,
st_weights,
1e-15));
// Create quadrature for Gram-Schmidt basis using quad points and
// and weights from original basis mapped to Stieljtes basis
Teuchos::RCP< const Teuchos::Array< Teuchos::Array<double> > > points =
Teuchos::rcp(&st_points,false);
Teuchos::RCP< const Teuchos::Array<double> > weights =
Teuchos::rcp(&st_weights,false);
Teuchos::RCP<const Stokhos::Quadrature<int,double> > gs_quad =
Teuchos::rcp(new Stokhos::UserDefinedQuadrature<int,double>(gs_basis,
points,
weights));
// Triple product tensor
Teuchos::RCP<Stokhos::Sparse3Tensor<int,double> > gs_Cijk =
gs_basis->computeTripleProductTensor();
// Gram-Schmidt quadrature expansion
Stokhos::QuadOrthogPolyExpansion<int,double> gs_quad_exp(gs_basis,
gs_Cijk,
gs_quad);
Stokhos::OrthogPolyApprox<int,double> u_gs(gs_basis), v_gs(gs_basis),
w_gs(gs_basis);
// Map expansion in Stieltjes basis to Gram-Schmidt basis
gs_basis->transformCoeffs(u_st.coeff(), u_gs.coeff());
gs_basis->transformCoeffs(v_st.coeff(), v_gs.coeff());
// Compute w_gs = u_gs*v_gs in Gram-Schmidt basis
gs_quad_exp.times(w_gs, u_gs, v_gs);
// Project w_gs back to original basis
pce_quad_func gs_func(w_gs, *gs_basis);
quad_exp.binary_op(gs_func, w2, u, v);
// Triple product tensor
Teuchos::RCP<Stokhos::Sparse3Tensor<int,double> > st_Cijk =
st_basis->computeTripleProductTensor();
// Stieltjes quadrature expansion
Teuchos::RCP<const Stokhos::Quadrature<int,double> > st_quad =
Teuchos::rcp(new Stokhos::UserDefinedQuadrature<int,double>(st_basis,
points,
weights));
Stokhos::QuadOrthogPolyExpansion<int,double> st_quad_exp(st_basis,
st_Cijk,
st_quad);
// Compute w_st = u_st*v_st in Stieltjes basis
st_quad_exp.times(w_st, u_st, v_st);
// Project w_st back to original basis
pce_quad_func st_func(w_st, *st_basis);
quad_exp.binary_op(st_func, w3, u, v);
std::cout.precision(12);
std::cout << "w = " << std::endl << w;
std::cout << "w2 = " << std::endl << w2;
std::cout << "w3 = " << std::endl << w3;
std::cout << "w_gs = " << std::endl << w_gs;
std::cout << "w_st = " << std::endl << w_st;
std::cout.setf(std::ios::scientific);
std::cout << "w.mean() = " << w.mean() << std::endl
<< "w2.mean() = " << w2.mean() << std::endl
<< "w3.mean() = " << w3.mean() << std::endl
<< "w_gs.mean() = " << w_gs.mean() << std::endl
<< "w_st.mean() = " << w_st.mean() << std::endl
<< "w.std_dev() = " << w.standard_deviation() << std::endl
<< "w2.std_dev() = " << w2.standard_deviation() << std::endl
<< "w3.std_dev() = " << w3.standard_deviation() << std::endl
<< "w_gs.std_dev() = " << w_gs.standard_deviation() << std::endl
<< "w_st.std_dev() = " << w_st.standard_deviation() << std::endl;
}
catch (std::exception& e) {
std::cout << e.what() << std::endl;
}
}
Stieltjes_PCE_Setup(bool use_pce_quad_points_) :
func(), use_pce_quad_points(use_pce_quad_points_)
{
rtol = 1e-8;
atol = 1e-12;
const OrdinalType d = 3;
const OrdinalType p = 5;
// Create product basis
Teuchos::Array< Teuchos::RCP<const Stokhos::OneDOrthogPolyBasis<OrdinalType,ValueType> > > bases(d);
for (OrdinalType i=0; i<d; i++)
bases[i] =
Teuchos::rcp(new Stokhos::LegendreBasis<OrdinalType,ValueType>(p));
basis =
Teuchos::rcp(new Stokhos::CompletePolynomialBasis<OrdinalType,ValueType>(bases));
// Create approximation
sz = basis->size();
Stokhos::OrthogPolyApprox<OrdinalType,ValueType> x(basis);
for (OrdinalType i=0; i<d; i++)
x.term(i, 1) = 1.0;
// Tensor product quadrature
Teuchos::RCP<const Stokhos::Quadrature<OrdinalType,ValueType> > quad =
Teuchos::rcp(new Stokhos::TensorProductQuadrature<OrdinalType,ValueType>(basis, 4*p));
// Triple product tensor
Teuchos::RCP<Stokhos::Sparse3Tensor<int,double> > Cijk =
basis->computeTripleProductTensor(basis->size());
// Quadrature expansion
exp = Teuchos::rcp(new Stokhos::QuadOrthogPolyExpansion<OrdinalType,ValueType>(basis, Cijk, quad));
// Compute PCE via quadrature expansion
u.reset(basis);
v.reset(basis);
func.eval(*exp, x, u);
exp->times(v,u,u);
// Compute Stieltjes basis
st_1d_basis =
Teuchos::rcp(new Stokhos::StieltjesPCEBasis<OrdinalType,ValueType>(
p, Teuchos::rcp(&u,false), quad, use_pce_quad_points));
Teuchos::Array< Teuchos::RCP<const Stokhos::OneDOrthogPolyBasis<OrdinalType,ValueType> > > st_bases(1);
st_bases[0] = st_1d_basis;
st_basis =
Teuchos::rcp(new Stokhos::CompletePolynomialBasis<OrdinalType,ValueType>(st_bases, 1e-15));
st_sz = st_basis->size();
u_st.reset(st_basis);
v_st.reset(st_basis);
u_st[0] = u.mean();
u_st[1] = 1.0;
// Tensor product quadrature
st_quad =
Teuchos::rcp(new Stokhos::TensorProductQuadrature<OrdinalType,ValueType>(st_basis));
// Triple product tensor
Teuchos::RCP<Stokhos::Sparse3Tensor<int,double> > st_Cijk =
st_basis->computeTripleProductTensor(st_basis->size());
// Quadrature expansion
Stokhos::QuadOrthogPolyExpansion<OrdinalType,ValueType> st_exp(st_basis,
st_Cijk,
st_quad);
st_exp.times(v_st, u_st, u_st);
}
int main(int argc, char **argv)
{
try {
const int d = 5;
const int pmin = 1;
const int pmax = 10;
const int np = pmax-pmin+1;
const double a = 1.5;
bool use_pce_quad_points = false;
bool normalize = false;
bool project_integrals = false;
bool lanczos = true;
bool sparse_grid = true;
#ifndef HAVE_STOKHOS_DAKOTA
sparse_grid = false;
#endif
Teuchos::Array<double> mean(np), mean_st(np), std_dev(np), std_dev_st(np);
Teuchos::Array<double> pt(np), pt_st(np);
Teuchos::Array< Teuchos::RCP<const Stokhos::OneDOrthogPolyBasis<int,double> > > bases(d);
Teuchos::Array<double> eval_pt(d, 0.56789);
double pt_true;
// Loop over orders
int n = 0;
for (int p=pmin; p<=pmax; p++) {
std::cout << "p = " << p << std::endl;
// Create product basis
for (int i=0; i<d; i++)
bases[i] = Teuchos::rcp(new basis_type(p));
Teuchos::RCP<const Stokhos::CompletePolynomialBasis<int,double> > basis =
Teuchos::rcp(new Stokhos::CompletePolynomialBasis<int,double>(bases));
// Create approximation
Stokhos::OrthogPolyApprox<int,double> s(basis), t1(basis), t2(basis);
Teuchos::Array< Stokhos::OrthogPolyApprox<int,double>* > xi(d);
for (int i=0; i<d; i++) {
xi[i] = new Stokhos::OrthogPolyApprox<int,double>(basis);
xi[i]->term(i, 1) = 1.0;
}
// Quadrature
Teuchos::RCP<const Stokhos::Quadrature<int,double> > quad;
#ifdef HAVE_STOKHOS_DAKOTA
if (sparse_grid)
quad =
Teuchos::rcp(new Stokhos::SparseGridQuadrature<int,double>(basis, p));
#endif
if (!sparse_grid)
quad =
Teuchos::rcp(new Stokhos::TensorProductQuadrature<int,double>(basis));
// Triple product tensor
Teuchos::RCP<Stokhos::Sparse3Tensor<int,double> > Cijk =
basis->computeTripleProductTensor();
// Quadrature expansion
Stokhos::QuadOrthogPolyExpansion<int,double> quad_exp(basis, Cijk, quad);
// Compute PCE via quadrature expansion
s_quad_func<d> s_func(a);
const Stokhos::OrthogPolyApprox<int,double> **xip =
new const Stokhos::OrthogPolyApprox<int,double>*[d];
for (int i=0; i<d; i++)
xip[i] = xi[i];
quad_exp.nary_op(s_func,s,xip);
quad_exp.divide(t1,1.0,s);
delete [] xip;
// compute true point
Teuchos::Array<double> xx(d);
for (int i=0; i<d; i++)
xx[i] = xi[i]->evaluate(eval_pt);
pt_true = s_func(&(xx[0]));
pt_true = 1.0/pt_true;
// Compute Stieltjes basis
Teuchos::Array< Teuchos::RCP<const Stokhos::OneDOrthogPolyBasis<int,double> > > st_bases(1);
Teuchos::RCP< Stokhos::LanczosProjPCEBasis<int,double> > stp_basis_s;
Teuchos::RCP< Stokhos::LanczosPCEBasis<int,double> > st_basis_s;
if (lanczos) {
if (project_integrals) {
stp_basis_s =
Teuchos::rcp(new Stokhos::LanczosProjPCEBasis<int,double>(
p, Teuchos::rcp(&s,false), Cijk, normalize, true));
st_bases[0] = stp_basis_s;
}
else {
st_basis_s =
Teuchos::rcp(new Stokhos::LanczosPCEBasis<int,double>(
p, Teuchos::rcp(&s,false), quad, normalize, true));
st_bases[0] = st_basis_s;
}
}
else {
st_bases[0] =
Teuchos::rcp(new Stokhos::StieltjesPCEBasis<int,double>(
p, Teuchos::rcp(&s,false), quad, use_pce_quad_points,
normalize, project_integrals, Cijk));
}
Teuchos::RCP<const Stokhos::CompletePolynomialBasis<int,double> >
st_basis =
Teuchos::rcp(new Stokhos::CompletePolynomialBasis<int,double>(st_bases));
//std::cout << *st_basis << std::endl;
Stokhos::OrthogPolyApprox<int,double> s_st(st_basis), t_st(st_basis);
if (lanczos) {
if (project_integrals) {
s_st.term(0, 0) = stp_basis_s->getNewCoeffs(0);
s_st.term(0, 1) = stp_basis_s->getNewCoeffs(1);
}
else {
s_st.term(0, 0) = st_basis_s->getNewCoeffs(0);
s_st.term(0, 1) = st_basis_s->getNewCoeffs(1);
}
}
else {
s_st.term(0, 0) = s.mean();
s_st.term(0, 1) = 1.0;
}
// Triple product tensor
Teuchos::RCP<Stokhos::Sparse3Tensor<int,double> > st_Cijk =
st_basis->computeTripleProductTensor();
// Tensor product quadrature
Teuchos::RCP<const Stokhos::Quadrature<int,double> > st_quad;
#ifdef HAVE_STOKHOS_DAKOTA
if (sparse_grid)
st_quad = Teuchos::rcp(new Stokhos::SparseGridQuadrature<int,double>(st_basis, p));
#endif
if (!sparse_grid)
st_quad = Teuchos::rcp(new Stokhos::TensorProductQuadrature<int,double>(st_basis));
// Quadrature expansion
Stokhos::QuadOrthogPolyExpansion<int,double> st_quad_exp(st_basis,
st_Cijk,
st_quad);
// Compute t_st = 1/s_st in Stieltjes basis
st_quad_exp.divide(t_st, 1.0, s_st);
// Project t_st back to original basis
pce_quad_func st_func(t_st, *st_basis);
quad_exp.unary_op(st_func, t2, s);
// std::cout.precision(12);
// std::cout << w;
// std::cout << w2;
// std::cout << w_st;
mean[n] = t1.mean();
mean_st[n] = t2.mean();
std_dev[n] = t1.standard_deviation();
std_dev_st[n] = t2.standard_deviation();
pt[n] = t1.evaluate(eval_pt);
pt_st[n] = t2.evaluate(eval_pt);
n++;
for (int i=0; i<d; i++)
delete xi[i];
}
n = 0;
int wi=10;
std::cout << "Statistical error:" << std::endl;
std::cout << "p "
<< std::setw(wi) << "mean" << " "
<< std::setw(wi) << "mean_st" << " "
<< std::setw(wi) << "std_dev" << " "
<< std::setw(wi) << "std_dev_st" << " "
<< std::setw(wi) << "point" << " "
<< std::setw(wi) << "point_st" << std::endl;
for (int p=pmin; p<pmax; p++) {
std::cout.precision(3);
std::cout.setf(std::ios::scientific);
std::cout << p << " "
<< std::setw(wi) << rel_err(mean[n], mean[np-1]) << " "
<< std::setw(wi) << rel_err(mean_st[n], mean[np-1]) << " "
<< std::setw(wi) << rel_err(std_dev[n], std_dev[np-1]) << " "
<< std::setw(wi) << rel_err(std_dev_st[n], std_dev[np-1])
<< " "
<< std::setw(wi) << rel_err(pt[n], pt_true) << " "
<< std::setw(wi) << rel_err(pt_st[n], pt_true)
<< std::endl;
n++;
}
}
catch (std::exception& e) {
std::cout << e.what() << std::endl;
}
}
