Hierarchical sparse grid tutorial

This tutorial shows how to use a sparse Stochastic Collocation (SC) sampler in EasyVVUQ. We will assume you are familiar with the basic building blocks of an EasyVVUQ Campaign. If not, see the basic tutorial here.

The complete code for this example can be found here. This file demonstates the sparse grid using a analytic test function, for which we compute exact reference statistics.

Sparse grids

Let us briefly describe the concept behind sparse grids. In a standard EasyVVUQ Campaign, an SC sampler object might be created via:

my_sampler = uq.sampling.SCSampler(vary=vary, polynomial_order=3,

Here the specified polynomial_order, and the number of inputs in vary, determine the number of samples, which increases exponentially fast with an increasing amount of inputs. This is the so-called curse of dimensionality. Sparse grids do not circumvent the curse of dimensionality, although they can postpone its effect to higher dimensions. In the case of a standard EasyVVUQ Canpaign, by setting polynomial_order=3 we create a sampling plan through a single tensor product of one-dimensional quadrature rules with order 3 for every input. It is this tensor product construction that leads to the exponential rise in cost. Sparse grids on the other hand, do not create a single tensor product, but build the sampling plan from the ground up by using a linear combination of tensor products involving 1D quadrature rules of different orders. For two inputs, we might for instance consider using 1D quadrature rules of order [1, 1], [1, 2] and [2, 1], whereas a standard EasyVVUQ campaign with polynomial_order=2 uses just [2,2]. If the chosen quadrature rule generates 1 point for order 1 and 3 points for order 2, then [2, 2] (the 2nd order rule for both dimensions) will generate 3*3 = 9 points. For the sparse grid we have a linear combination of:

  • [1, 1]: a single point in the 2D domain (X, Y)
  • [1, 2]: a line of 3 points with constant X
  • [2, 1]: a line of 3 points with constant Y

In the case of sparse grids it is common to select a nested quadrature rule. This means that the quadrature rule of order p contains all points of the same rule of order p-1. When taking the linear combinations, a nested rule ensures that many points will conincide, which yields efficient sampling plans, especially in higher dimensions. If our 1D rule of order 1 and 2 generates the points [0.5] and [0, 0.5, 1] we obtain a sampling plan consisting of

  • [1, 1]: [0.5, 0.5]
  • [1, 2]: [0.5, 0.0], [0.5, 0.5], [0.5, 1.0]
  • [2, 1]: [0.0, 0.5], [0.5, 0.5], [1.0, 0.5],

which gives a total of 5 unique points, compared to the 9 points of [2, 2].

Create a sparse SC sampler

An example sparse SC sampler is given by:

my_sampler = uq.sampling.SCSampler(vary=vary, polynomial_order=poly_order,
                                   quadrature_rule="C", sparse=True,

Here "C" stands for the Clenshaw Curtis rule, which can be made nested by turning on the growth flag. You can also select other quadrature rules, e.g. the standard Gaussian option ("G"). Not all rules can be made nested though, see the Chaospy documentation for a list of all quadrature rules and their properties.

The rest of the Campaign proceed exactly as it would in the non-sparse case. The only exception is in the case of a nested rule, in which case the sampling plan can be isotropically refined. This is done with the following command:

#update the sparse grid to the next level

#draw the new samples

In the example the grid is refined once, after which the Sobol sensitivity indices are calculated and compared against the reference.