 SG++-Doxygen-Documentation
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.

The following example interpolates the (non-symmetric) function f : [0,1]x[0,1] -> R, f(x,y) := 16 (x-1) * x * (y-1) * y

The number of grid points is printed in each iteration. After refinement, the surplusses have to be set for all new grid points, i.e., the alpha-Vector has to be extended.

Function to interpolate. This is a two-dimensional parabola. - nonsymmetric(!).

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

Create a two-dimensional piecewise bi-linear 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;
grid->getGenerator().regular(level);
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());
alpha.setAll(0.0);
std::cout << "length of alpha vector: " << alpha.getSize() << std::endl;

Create a vector for storing (possibly expensive) function evaluations at each gridpoint.

DataVector funEvals(gridStorage.getSize());
for (size_t i = 0; i < gridStorage.getSize(); i++) {
GridPoint& gp = gridStorage.getPoint(i);
funEvals[i] = f(gp.getStandardCoordinate(0), gp.getStandardCoordinate(1));
}

create a vector for storing newly added points by their sequence id.

Refine adaptively 5 times.

for (int step = 0; step < 5; step++) {

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). All new points are appended to the addedPoints vector.

SurplusRefinementFunctor functor(alpha, 1);

Extend alpha and funEval vector (new entries uninitialized). Note that right now, the size of both vectors matches number of gridpoints again, but the values of the new points are set to zero.

alpha.resize(gridStorage.getSize());
funEvals.resize(gridStorage.getSize());

Evaluate the function f at the newly created gridpoints and set the corresponding entries in the funEval vector with these values.

for (size_t i = 0; i < addedPoints.size(); i++) {
size_t seq = addedPoints[i];
GridPoint& gp = gridStorage.getPoint(seq);
funEvals[seq] = f(gp.getStandardCoordinate(0), gp.getStandardCoordinate(1));
}

Reset the alpha vector to function evals to prepare for hierarchisation.

alpha.copyFrom(funEvals);

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

Clear the addedPoints vector for the next iteration.

std::cout << "refinement step " << step + 1 << ", new grid size: " << alpha.getSize()
<< std::endl;
}
for (size_t i = 0; i < alpha.getSize(); i++) {
std::cout << alpha[i] << std::endl;
}
}

This results in the following output:

dimensionality:                   2
number of initial grid points:    17
length of alpha vector:           17
refinement step 1, new grid size: 21
refinement step 2, new grid size: 24
refinement step 3, new grid size: 27
refinement step 4, new grid size: 29
refinement step 5, new grid size: 33

There are clearly more efficient approaches than to hierarchize the whole grid each time. But this works even where no efficient alternatives are available and suffices for demonstration purposes.

This use of the SurplusRefinementFunctor takes as arguments the coefficient vector (it doesn't have to be the coefficient vector, it could be something modified!) and the number of grid points to refine (if available). It bases its refinement decision on the absolute values of the vector's entries, choosing the largest ones. Other refinement functors are available or can be implemented.