SG++
Spatially-Dimension-Adaptive Refinement in C++

We compute the sparse grid interpolant of the function \( f(x) = \sin(\pi x).\) We perform spatially-dimension-adaptive refinement of the sparse grid model, which means we refine a particular grid point (locality) only in some dimensions (dimensionality).

For details on spatially-dimension-adaptive refinement see

V. Khakhutskyy and M. Hegland: Spatially-Dimension-Adaptive Sparse Grids for Online Learning.
In D. Pflüger and J. Garcke (ed.), Sparse Grids and Applications - Stuttgart 2014, Volume 109 of
LNCSE, p. 133–162. Springer International Publishing, March 2016.

The example can be found in the file predictiveRefinement.cpp.

We define the function \( f(x) = \sin(\pi x).\) to interpolate. / double f(double x0, double x1) { return sin(x0 * M_PI); }

Spatially-dimension-adaptive refinement uses squared prediction error on a dataset to compute refinement indicators. Hence, here we define a function to compute these squared errors.

DataVector& calculateError(const DataMatrix& dataSet, Grid& grid, const DataVector& alpha,
DataVector& error) {
std::cout << "calculating error" << std::endl;
// traverse dataSet
DataVector vec(2);
std::unique_ptr<OperationEval> opEval(sgpp::op_factory::createOperationEval(grid));
for (unsigned int i = 0; i < dataSet.getNrows(); i++) {
dataSet.getRow(i, vec);
error[i] = pow(f(dataSet.get(i, 0), dataSet.get(i, 1)) - opEval->eval(alpha, vec), 2);
}

Start with the main function

int main() {

create a two-dimensional piecewise bilinear grid

size_t dim = 2;
std::unique_ptr<Grid> grid(Grid::createModLinearGrid(dim));
GridStorage& gridStorage = grid->getStorage();
std::cout << "dimensionality: " << gridStorage.getDimension() << std::endl;
// create regular grid, level 3
size_t level = 1;
grid->getGenerator().regular(level);
std::cout << "number of initial grid points: " << gridStorage.getSize() << std::endl;

To create a dataset we use points on a regular 2d grid with a step size of 1 / rows and 1 / cols.

int rows = 100;
int cols = 100;
DataMatrix dataSet(rows * cols, dim);
DataVector vals(rows * cols);
for (int i = 0; i < rows; i++) {
for (int j = 0; j < cols; j++) {
// xcoord
dataSet.set(i * cols + j, 0, i * 1.0 / rows);
// ycoord
dataSet.set(i * cols + j, 1, j * 1.0 / cols);
vals[i * cols + j] = f(i * 1.0 / rows, j * 1.0 / cols);
}

We refine adaptively 20 times. In every step we recompute the vector of surpluses alpha, the vector with squared errors on the dataset errorVector, and then call the refinement routines.

// create coefficient vector
DataVector alpha(gridStorage.getSize());
alpha.setAll(0.0);
std::cout << "length of alpha vector: " << alpha.getSize() << std::endl;
for (int step = 0; step < 20; step++) {

Step 1: calculate the surplus vector alpha. In data mining with do it by solving a regression problem as shown in example Classification Example. Here, 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.

// set function values in alpha
DataVector gridPointCoordinates(dim);
for (size_t i = 0; i < gridStorage.getSize(); i++) {
gridStorage.getPoint(i).getStandardCoordinates(gridPointCoordinates);
alpha[i] = f(gridPointCoordinates[0], gridPointCoordinates[1]);
}
// hierarchize

Step 2: calculate squared errors.

DataVector errorVector(dataSet.getNrows());
calculateError(dataSet, *grid, alpha, errorVector);

Step 3: call refinement routines. PredictiveRefinement implements the decorator pattern and extends the functionality of HashRefinement. PredictiveRefinement requires a special kind of refinement functor – PredictiveRefinementIndicator that can access the dataset and the error vector. The refinement itself if performed by calling .free_refine() same for normal refinement in HashRefinement.

// refinement stuff
HashRefinement refinement;
PredictiveRefinement decorator(&refinement);
// refine a single grid point each time
std::cout << "Error over all = " << errorVector.sum() << std::endl;
PredictiveRefinementIndicator indicator(*grid, dataSet, errorVector, 1);
decorator.free_refine(gridStorage, indicator);
std::cout << "Refinement step " << step + 1 << ", new grid size: " << gridStorage.getSize()
<< std::endl;
// extend alpha vector (new entries uninitialized)
alpha.resize(gridStorage.getSize());
}

The output of the program should look like this

dimensionality:                   2
number of initial grid points:    1
length of alpha vector:           1
calculating error
Error over all = 2268.65
Refinement step 1, new grid size: 3
calculating error
Error over all = 264.09
Refinement step 2, new grid size: 5
calculating error
Error over all = 125.378
Refinement step 3, new grid size: 7
calculating error
Error over all = 3.48359
Refinement step 4, new grid size: 9
calculating error
Error over all = 1.99757
Refinement step 5, new grid size: 11
calculating error
Error over all = 0.845349
Refinement step 6, new grid size: 13
calculating error
Error over all = 0.464096
Refinement step 7, new grid size: 15
calculating error
Error over all = 0.0828432
Refinement step 8, new grid size: 17
calculating error
Error over all = 0.0828432
Refinement step 9, new grid size: 19
calculating error
Error over all = 0.068976
Refinement step 10, new grid size: 21
calculating error
Error over all = 0.0551672
Refinement step 11, new grid size: 23
calculating error
Error over all = 0.0413584
Refinement step 12, new grid size: 25
calculating error
Error over all = 0.0330229
Refinement step 13, new grid size: 27
calculating error
Error over all = 0.0230578
Refinement step 14, new grid size: 29
calculating error
Error over all = 0.0130926
Refinement step 15, new grid size: 31
calculating error
Error over all = 0.00856834
Refinement step 16, new grid size: 33
calculating error
Error over all = 0.00404405
Refinement step 17, new grid size: 35
calculating error
Error over all = 0.00404405
Refinement step 18, new grid size: 37
calculating error
Error over all = 0.00404405
Refinement step 19, new grid size: 41
calculating error
Error over all = 0.00404405
Refinement step 20, new grid size: 45