# 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 be an arbitrary set, be the set of -tuples where each component lies in . We define two mappings

which we call sampling and interpolation, respectively. Then, the discretization of with respect to and the above operators is defined as the tuple

The following abstract diagram visualizes a discretization:

TODO: write up in more detail

## Example¶

Let be the space of real-valued continuous functions on the interval , and let be ordered sampling points in .

Restriction operator:

We define the grid collocation operator as

The abstract object in this case is the input function , and the operator evaluates this function at the given points, resulting in a vector in .

This operator is implemented as PointCollocation.

Extension operator:

Let discrete values be given. Consider the linear interpolation of those values at a point :

where is the index such that .

Then we can define the linear interpolation operator as

where stands for the function .

Hence, this operator maps the finite array to the abstract interpolating function .

This interpolation scheme is implemented in the LinearInterpolation operator.