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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 (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.

For instructions on how to run the example, please see Installation and Usage.

The function, which sgpp::base::OperationQuadratureMC takes, has one parameter, a sequence (C++ provides a tuple) with the coordinates of the grid point \(\in [0,1]^d\).

# import pysgpp library
import pysgpp
# the standard parabola (arbitrary-dimensional)
def f(x):
res = 1.0
for i in range(len(x)):
res *= 4.0*x[i]*(1.0-x[i])
return res
# create a two-dimensional piecewise bi-linear grid
dim = 2
grid = pysgpp.Grid.createLinearGrid(dim)
gridStorage = grid.getStorage()
print "dimensionality: {}".format(dim)
# create regular grid, level 3
level = 3
gridGen = grid.getGenerator()
print "number of grid points: {}".format(gridStorage.getSize())
# create coefficient vector
alpha = pysgpp.DataVector(gridStorage.getSize())
for i in xrange(gridStorage.getSize()):
gp = gridStorage.getPoint(i)
alpha[i] = f((gp.getStandardCoordinate(0), gp.getStandardCoordinate(1)))
# direct quadrature
opQ = pysgpp.createOperationQuadrature(grid)
res = opQ.doQuadrature(alpha)
print "exact integral value: {}".format(res)
# Monte Carlo quadrature using 100000 paths
opMC = pysgpp.OperationQuadratureMC(grid, 100000)
res = opMC.doQuadrature(alpha)
print "Monte Carlo value: {:.6f}".format(res)
res = opMC.doQuadrature(alpha)
print "Monte Carlo value: {:.6f}".format(res)
# Monte Carlo quadrature of a function
res = opMC.doQuadratureFunc(f)
print "MC value (f): {:.6f}".format(res)
# Monte Carlo quadrature of error
res = opMC.doQuadratureL2Error(f, alpha)
print "MC L2-error (f-u) {:.7f}".format(res)

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