Discretizations¶

Mathematical background¶

In mathematics, the term discretization stands for the transition from abstract, continuous, often infinite-dimensional objects to concrete, discrete, finite-dimensional counterparts. We define discretizations as tuples encompassing all necessary aspects involved in this transition. Let $\mathcal{X}$ be an arbitrary set, $\mathbb{F}^n$ be the set of $n$ -tuples where each component lies in $\mathbb{F}$ . We define two mappings

$\mathcal{R}_\mathcal{X}: \mathcal{X} \to \mathbb{F}^n, \mathcal{E}_\mathcal{X}: \mathbb{F}^n \to \mathcal{X},$

which we call sampling and interpolation, respectively. Then, the discretization of $\mathcal{X}$ with respect to $\mathbb{F}^n$ and the above operators is defined as the tuple

$\mathcal{D}(\mathcal{X}) = (\mathcal{X}, \mathbb{F}^n, \mathcal{R}_\mathcal{X}, \mathcal{E}_\mathcal{X}).$

The following abstract diagram visualizes a discretization:

TODO: write up in more detail

Example¶

Let $\mathcal{X} = C([0, 1])$ be the space of real-valued continuous functions on the interval $[0, 1]$ , and let $x_1 < \dots < x_n$ be ordered sampling points in $[0, 1]$ .

Restriction operator:

We define the grid collocation operator as

$\mathcal{C}: \mathcal{X} \to \mathbb{R}^n, \mathcal{C}(f) := \big(f(x_1), \dots, f(x_n)\big).$

The abstract object in this case is the input function $f$ , and the operator evaluates this function at the given points, resulting in a vector in $\mathbb{R}^n$ .