Quadrature in C++

The following example shows how to integrate in SG++, using both direct integration of a sparse grid function and the use of Monte Carlo integration.

As in the tutorial.cpp (Start Here) example, we deal with the function

\[ f\colon [0, 1]^2 \to \mathbb{R},\quad f(x_0, x_1) := 16 (x_0 - 1) x_0 (x_1 - 1) x_1 \]

which we first interpolate. We then integrate the interpolant, then the function itself using 100000 Monte Carlo points, and we then compute the L2-error.

The function, which sgpp::base::OperationQuadratureMC takes, has three parameters. First, the dimensionality (int), then a double* with the coordinates of the grid point \(\in[0,1]^d\), and finally a void* with clientdata for the function, see sgpp::base::FUNC.

// include all SG++ base headers
#include <sgpp_base.hpp>

#include <iostream>

using sgpp::base::OperationHierarchisation;

// function to interpolate
double f(int dim, double* x, void* clientdata) {
  double res = 1.0;

  for (int i = 0; i < dim; i++) {
    res *= 4.0 * x[i] * (1.0 - x[i]);
  }

  return res;
}

int main() {

Create a two-dimensional piecewise bi-linear grid of level 3

  int dim = 2;
  std::unique_ptr<sgpp::base::Grid> grid(sgpp::base::Grid::createLinearGrid(dim));
  sgpp::base::GridStorage& gridStorage = grid->getStorage();
  std::cout << "dimensionality:        " << gridStorage.getDimension() << std::endl;

  // create regular grid, level 3
  int level = 3;
  grid->getGenerator().regular(level);
  std::cout << "number of grid points: " << gridStorage.getSize() << std::endl;

Calculate the surplus vector alpha for the interpolant of \( f(x)\). Since the function can be evaluated at any point. Hence. we simply evaluate it at the coordinates of the grid points to obtain the nodal values. Then we use hierarchization to obtain the surplus value.

  sgpp::base::DataVector alpha(gridStorage.getSize());
  double p[2];

  for (size_t i = 0; i < gridStorage.getSize(); i++) {
    sgpp::base::GridPoint& gp = gridStorage.getPoint(i);
    p[0] = gp.getStandardCoordinate(0);
    p[1] = gp.getStandardCoordinate(1);
    alpha[i] = f(2, p, NULL);
  }

  std::unique_ptr<OperationHierarchisation>(sgpp::op_factory::createOperationHierarchisation(*grid))
      ->doHierarchisation(alpha);

Now we compute and compare the quadrature using four different methods available in SG++.

  // direct quadrature
  std::unique_ptr<sgpp::base::OperationQuadrature> opQ(
      sgpp::op_factory::createOperationQuadrature(*grid));
  double res = opQ->doQuadrature(alpha);
  std::cout << "exact integral value:  " << res << std::endl;

  // Monte Carlo quadrature using 100000 paths
  sgpp::base::OperationQuadratureMC opMC(*grid, 100000);
  res = opMC.doQuadrature(alpha);
  std::cout << "Monte Carlo value:     " << res << std::endl;
  res = opMC.doQuadrature(alpha);
  std::cout << "Monte Carlo value:     " << res << std::endl;

  // Monte Carlo quadrature of a function
  res = opMC.doQuadratureFunc(f, NULL);
  std::cout << "MC value:              " << res << std::endl;

  // Monte Carlo quadrature of error
  res = opMC.doQuadratureL2Error(f, NULL, alpha);
  std::cout << "MC L2-error:           " << res << std::endl;
}  // end of main

This results in an output similar to:

dimensionality:        2
number of grid points: 17
exact integral value:  0.421875
Monte Carlo value:     0.421298
Monte Carlo value:     0.421971
MC value:              0.444917
MC L2-error:           0.0242639