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refinement.py

The following example interpolates the (non-symmetric) function

\[ f\colon [0, 1]^2 \to \mathbb{R},\quad f(x_0, x_1) := 16 (x_0 - 1) x_0 (x_1 - 1) x_1 \]

  • starting from a regular grid with level 2 and
  • refining 5 times one grid point each.

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.

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

# import pysgpp library
import pysgpp
# 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()
gridGen.regular(level)
print "number of initial grid points: {}".format(gridStorage.getSize())
# definition of function to interpolate - nonsymmetric(!)
f = lambda x0, x1: 16.0 * (x0-1)*x0 * (x1-1)*x1*x1
# create coefficient vector
alpha = pysgpp.DataVector(gridStorage.getSize())
print "length of alpha vector: {}".format(alpha.getSize())
# now refine adaptively 5 times
for refnum in range(5):
# set function values in alpha
for i in xrange(gridStorage.getSize()):
gp = gridStorage.getPoint(i)
pysgpp.createOperationHierarchisation(grid).doHierarchisation(alpha)
# refine a single grid point each time
gridGen.refine(pysgpp.SurplusRefinementFunctor(alpha, 1))
print "refinement step {}, new grid size: {}".format(refnum+1, gridStorage.getSize())
# extend alpha vector (new entries uninitialized)
alpha.resize(gridStorage.getSize())

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 set the function values for all grid points and 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.