SG++-Doxygen-Documentation

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>
// 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.

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))

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

std::cout << "exact integral value: " << res << std::endl;
// Monte Carlo quadrature using 100000 paths
std::cout << "Monte Carlo value: " << res << std::endl;
std::cout << "Monte Carlo value: " << res << std::endl;
// Monte Carlo quadrature of a function
std::cout << "MC value: " << res << std::endl;
// Monte Carlo quadrature of error
dimensionality:        2