Grid Formats in Globtim

This document explains the two grid representations used in Globtim.

Overview

Globtim uses two different grid formats:

1. Array{SVector} format - Natural Julia arrays of static vectors (from StaticArrays.jl)
2. Matrix format - Traditional matrix representation required for linear algebra operations

Array{SVector} Format

This is the natural output from generate_grid:

grid = generate_grid(2, 10, basis=:chebyshev)
# Returns: 11×11 Array{SVector{2,Float64},2}

Characteristics:

• Each element is an SVector{N,Float64} representing a point
• N-dimensional array structure preserves grid topology
• Efficient for point-wise function evaluation
• Natural for Julia's multiple dispatch

Used for:

• Function evaluation at grid points
• Grid traversal and manipulation
• Discrete L²-norm computation

Example:

f = x -> sum(x.^2)
grid = generate_grid(2, 10)
values = map(f, reshape(grid, :))

Matrix Format

Required for Vandermonde matrix construction. Convert using grid_to_matrix:

matrix_grid = grid_to_matrix(grid)
# Returns: 121×2 Matrix{Float64}

Characteristics:

• Each row represents a point
• Each column represents a dimension
• Compatible with BLAS operations
• Required by lambda_vandermonde

Used for:

• Vandermonde matrix construction
• Linear algebra operations
• Polynomial basis evaluation

Example:

grid = generate_grid(2, 10, basis=:chebyshev)
matrix_grid = grid_to_matrix(grid)
Lambda = SupportGen(2, 5)
V = lambda_vandermonde(Lambda, matrix_grid, basis=:chebyshev)

The ApproxPoly type stores grids in matrix format:

pol = Constructor(TR, 10, basis=:chebyshev)
# pol.grid is already in matrix format
V = lambda_vandermonde(Lambda, pol.grid, basis=pol.basis)