Evaluation

UnifiedSparseGrids separates

choosing evaluation points (tensor grids, scattered points), and
building an evaluation plan (reusable) that maps sparse grid coefficients to values.

Point sets

Evaluation points are represented by AbstractPointSet:

where xs is a tuple of 1D point vectors and X is a d×N matrix.

For example, a tensor grid of Chebyshev–Gauss–Lobatto points with endpoints removed (a common choice for Dirichlet problems) in $D$ dimensions:

using UnifiedSparseGrids

D = 3
r = 6
x = points(ChebyshevGaussLobattoNodes(; endpoints=false), r)
P = TensorProductPoints(ntuple(_ -> x, Val(D)))

Or a scattered point cloud:

using UnifiedSparseGrids

D = 2
X = randn(D, 10_000)
P = ScatteredPoints(X)

To evaluate a sparse grid coefficient vector u on a point set P, build an evaluation plan and apply it:

plan = plan_evaluate(grid, DirichletLegendreBasis(), P, Float64)
values = evaluate(plan, u)

The plan can be reused to evaluate many coefficient vectors on the same point set.

Here basis declares how to interpret the coefficient vector u (e.g. Legendre modal coefficients with Dirichlet boundary conditions).

For convenience, evaluate(grid, f) samples a function f on the sparse grid points (ordered consistently with traverse(grid)):

vals = evaluate(grid) do x
    exp(-sum(abs2, x))
end

This is a common starting point for experiments and for assembling right-hand side vectors.

See the tutorials for end-to-end usage, including sparse Galerkin solves and time-dependent splitting methods.