SparseGridsKit.jl

A simple implementation of sparse grid polynomial interpolation and corresponding interpolation.

To add the package use the following command:

] add SparseGridsKit

The construction is based upon the sparse operators introduced by Smolyak [1]. Key papers defining the approximation formulation include [2], [3] and [4]. This construction of the approximation is inspired by the Sparse Grids MATLAB Kit [5].

This documentation is example driven, with full documentation for each subset of functions given at the end of the relevant example. This is not a thorough mathematical description of sparse grid approximation, but a practical guide to using the software.

The package is still under development and by no means complete.

Statement of Need

This Julia implementation closely follows the mathematical description in the adaptive sparse grid approximation literature. This package aims to aid fast algorithm development and analysis, with an emphasis on applications in uncertainty quantification.

Contribution Guidelines

To report bugs, seek support or submit request features, please use the GitHub issue tracker. Please feel free to contribute and submit pull requests.

Example

Please see the documentation for extensive examples.

An example sparse grid construction is illustrated below.

using SparseGridsKit, Plots, LaTeXStrings
# Test create_sparsegrid
n,k =3,3
knots = [CCPoints(), CCPoints(), UniformPoints()]
rules = [Doubling(), Doubling(), Linear()]
mi_set = create_smolyak_miset(n,k)
sg = create_sparsegrid(mi_set, knots=knots, rule=rules)
nsteps = 100
@gif for i in range(0, stop = 2π, length = nsteps)
        plot(sg)
        plot!(
                title="Sparse Grid n,k="*string(n)*","*string(k),
                xlabel=L"y_1",
                ylabel=L"y_2",
                zlabel=L"y_3",
                camera = (20 * (1 + cos(i)),10 * (1 + cos(i)))
                )
end

Paired with function evaluations f_on_Z = [f(z) for z in get_grid_points(sg)] at the sparse grid points Z = get_grid_points(sg), this defines a sparse grid approximation.