exp_moment_example.cpp

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#include <iostream>
#include <iomanip>

#include "Stokhos.hpp"

typedef Stokhos::LegendreBasis<int,double> basis_type;

// This example uses PC expansions for computing moments of
//
// u = exp(x1 + ... + xd)
//
// where x1, ..., xd are uniform random variables on [-1,1].  The methods
// are compared to computing the "true" value via very high-order quadrature.
// Because of the structure of the exponential, the moments can easily
// be computed in one dimension.

int main(int argc, char **argv)
{
  try {

    // Compute "true" 1-D mean, std. dev using quadrature
    const unsigned int true_quad_order = 200;
    basis_type tmp_basis(1);
    Teuchos::Array<double> true_quad_points, true_quad_weights;
    Teuchos::Array< Teuchos::Array<double> > true_quad_values;
    tmp_basis.getQuadPoints(true_quad_order, true_quad_points, 
          true_quad_weights, true_quad_values);
    double mean_1d = 0.0;
    double sd_1d = 0.0;
    for (unsigned int qp=0; qp<true_quad_points.size(); qp++) {
      double t = std::exp(true_quad_points[qp]);
      mean_1d += t*true_quad_weights[qp];
      sd_1d += t*t*true_quad_weights[qp];
    }

    const unsigned int dmin = 1;
    const unsigned int dmax = 4;
    const unsigned int pmin = 1;
    const unsigned int pmax = 5;

    // Loop over dimensions
    for (unsigned int d=dmin; d<=dmax; d++) {

      // compute "true" values
      double true_mean = std::pow(mean_1d,static_cast<double>(d));
      double true_sd = std::pow(sd_1d,static_cast<double>(d)) - 
  true_mean*true_mean;
      true_sd = std::sqrt(true_sd);
      std::cout.precision(12);
      std::cout << "true mean = " << true_mean << "\t true std. dev. = "
                << true_sd << std::endl;

      Teuchos::Array< Teuchos::RCP<const Stokhos::OneDOrthogPolyBasis<int,double> > > bases(d); 

      // Loop over orders
      for (unsigned int p=pmin; p<=pmax; p++) {

  // 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
  int sz = basis->size();
  Stokhos::OrthogPolyApprox<int,double> x(basis), u(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.exp(u,x);
  double mean = u.mean(); 
  double sd = u.standard_deviation();
        
  std::cout.precision(4);
  std::cout.setf(std::ios::scientific);
  std::cout << "d = " << d << " p = " << p
      << " sz = " << sz
      << "\tmean err = " 
      << std::fabs(true_mean-mean) << "\tstd. dev. err = "
      << std::fabs(true_sd-sd) << std::endl;
      }
      
    }
  }
  catch (std::exception& e) {
    std::cout << e.what() << std::endl;
  }
}