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Refinement Example

Here we demonstrate how to refine a grid.

As a refinement indicator, we take the surpluses of the grid points directly. We start with a regular sparse grid of level 3 with linear basis functions and refine five times. In each refinement step, we refine the grid point with the highest absolute surplus.

function to interpolate. This is a two-dimensional parabola.

double f(double x0, double x1) {
return 16.0 * (x0 - 1) * x0 * (x1 - 1) * x1;
int main() {

create a two-dimensional piecewise bilinear grid

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

create regular sparse grid, level 3

size_t level = 3;
std::cout << "number of initial grid points: " << gridStorage.getSize() << std::endl;

create coefficient vector with size corresponding to the grid size. Initially, all the values are set to zero.

DataVector alpha(gridStorage.getSize());
std::cout << "length of alpha vector: " << alpha.getSize() << std::endl;

Obtain function values and refine adaptively 5 times

for (int step = 0; step < 5; step++) {
// set function values in alpha
for (size_t i = 0; i < gridStorage.getSize(); i++) {
GridPoint& gp = gridStorage.getPoint(i);
alpha[i] = f(gp.getStandardCoordinate(0), gp.getStandardCoordinate(1));

Each time, we have to hierarchize the grid again, because in the previous interation, new grid points have been added.

Refine a single grid point each time. The SurplusRefinementFunctor chooses the grid point with the highest absolute surplus. Refining the point means, that all children of this point (if not already present) are added to the grid. Also all missing parents are added (recursively).

SurplusRefinementFunctor functor(alpha, 1);
std::cout << "refinement step " << step + 1 << ", new grid size: " << alpha.getSize()
<< std::endl;

Extend alpha vector (new entries uninitialized). Note that right now, the surplus vector has the correct size again, but the values of the new points are set to zero. The correct surplus values will be inserted in the next iteration of the refinement loop.