Grid

Let $G = S_1 \times S_2 \times \cdots \times S_{D}$ be a $D$-grid or simply a grid, defined as the cartesian product of $D$ finite and strictly ordered sets :

$$G = \prod_{j \in [1,D]} S_j ~~,~~ S_j = \{s_1^j<s_2^j<~...<s^j_{N_j}\} \in \mathbb{S}$$

We denote :

• $D$ the number of finite and strictly ordered sets
• $N_j$ the cardinality of $S_j \in \mathbb{S}$
• $s_{i}^{j}$ the $j^{th}$ element of the $i^{th}$ of $S_j$
• $I_j = [s_1^j,s_{N_j}^j] \in \mathbb{R}$ the interval in which $s^j$ evolves.

While this grid structure lays out a convenient representation of a $D$-dimensional discrete parametric space, we introduce for each parameter the continuous interval in which it varies as we are also interested to interpolate the results for any state of inputs.

A $D$-grid can be seen as an hyperrectangle, also known as a $D$-orthotope. It's a particular kind of convex polytope of $\mathbb{R}^d$.

This is a generalization from the easy-to-understand $\mathbb{R}^1$, $\mathbb{R}^2$ and $\mathbb{R}^3$ cases where :

• a 1-grid is a segments divided into smaller segment called edge
• a 2-grid is a rectangle divided into smaller rectangles called face
• a 3-grid is a parallelepiped divided into smaller parallelepipeds called cell

In $\mathbb{R}^d$, a polytope would indicate a geometric object of dimension $D$ whereas a facet would account for an object of dimension $N$.

One can easily introduce the notion of a normalized grid, leading to represent it as a hypercube of length 1, which is a regular polytope.