SG++
PCE with Combigrids

This simple example shows how to create a Polynomial Chaos Expansion from an adaptively refined combigrid.

First we have to define a model to approximate.

Then we can create a refined combigrid

// create polynomial basis
auto basisFunction = std::make_shared<sgpp::combigrid::OrthogonalPolynomialBasis1D>(config);
for (size_t q = 6; q < 7; ++q) {
// create sprarse grid interpolation operation
auto tensor_op =
basisFunction, ishigamiModel.numDims, func);
stopwatch.start();

and construct a PCE representation to easily calculate statistical features of our model.

// create polynomial chaos surrogate from sparse grid
config.loadFromCombigridOperation(tensor_op, false);
config.basisFunction = basisFunction;
stopwatch.log();
stopwatch.start();
// compute mean, variance and sobol indices
double mean = pce->mean();
double variance = pce->variance();
sgpp::base::DataVector sobol_indices;
sgpp::base::DataVector total_sobol_indices;
pce->getComponentSobolIndices(sobol_indices);
pce->getTotalSobolIndices(total_sobol_indices);
// print results
std::cout << "Time: "
<< stopwatch.elapsedSeconds() / static_cast<double>(tensor_op->numGridPoints())
<< std::endl;
std::cout << "---------------------------------------------------------" << std::endl;
std::cout << "#gp = " << tensor_op->getLevelManager()->numGridPoints() << std::endl;
std::cout << "E(u) = " << mean << std::endl;
std::cout << "Var(u) = " << variance << std::endl;
std::cout << "Sobol indices = " << sobol_indices.toString() << std::endl;
std::cout << "Sum Sobol indices = " << sobol_indices.sum() << std::endl;
std::cout << "Total Sobol indices = " << total_sobol_indices.toString() << std::endl;
std::cout << "---------------------------------------------------------" << std::endl;
}

Output:

Time: 3.74635s.
Time: 3.73112s.
Time: 4.96569e-05
---------------------------------------------------------
#gp = 1825
E(u) = 3.5
Var(u) = 13.8446
Sobol indices = [3.13905191147811180041e-01, 4.42411144790040733454e-01,
9.56029390935037928152e-31, 5.58403133152096916677e-32, 2.43683664062148142015e-01,
6.16110611418130297137e-32, 8.27081513075557474542e-32]
Sum Sobol indices = 1
Total Sobol indices = [5.57588855209959377568e-01, 4.42411144790040733454e-01,
2.43683664062148142015e-01]