Interval propagation
# Interval propagation

In numerical mathematics, interval propagation or interval constraint propagation is the problem of contracting interval domains associated to variables of R without removing any value that is consistent with a set of constraints (i.e., equations or inequalities). It can be used to propagate uncertainties in the situation where errors are represented by intervals. Interval propagation considers an estimation problem as a constraint satisfaction problem.

## Atomic contractors

A contractor associated to an equation involving the variables $x_1,...,x_n$ is an operator which contracts the intervals $[x_1],...,$ $[x_n]$ (that are supposed to enclose the $x_i's)$ without removing any value for the variables that is consistent with the equation.

A contractor is said to be atomic if it is not built as a composition of other contractors. The main theory that is used to build atomic contractors are based on interval analysis.

Example. Consider for instance the equation

    $x_{1}+x_{2}=x_{3},$

which involves the three variables $x_1,x_2$ and $x_3.$

The associated contractor is given by the following statements

    $[x_{3}]:=[x_{3}]\cap ([x_{1}]+[x_{2}])$
    $[x_{1}]:=[x_{1}]\cap ([x_{3}]-[x_{2}])$
    $[x_{2}]:=[x_{2}]\cap ([x_{3}]-[x_{1}])$

For instance, if

    $x_{1}\in [-\infty ,5],$
    $x_{2}\in [-\infty ,4],$
    $x_{3}\in [6,\infty ]$

the contractor performs the following calculus

    $x_{3}=x_{1}+x_{2}\Rightarrow x_{3}\in [6,\infty ]\cap ([-\infty ,5]+[-\infty ,4])=[6,\infty ]\cap [-\infty ,9]=[6,9].$
    $x_{1}=x_{3}-x_{2}\Rightarrow x_{1}\in [-\infty ,5]\cap ([6,\infty ]-[-\infty ,4])=[-\infty ,5]\cap [2,\infty ]=[2,5].$
    $x_{2}=x_{3}-x_{1}\Rightarrow x_{2}\in [-\infty ,4]\cap ([6,\infty ]-[-\infty ,5])=[-\infty ,4]\cap [1,\infty ]=[1,4].$

For other constraints, a specific algorithm for implementing the atomic contractor should be written. An illustration is the atomic contractor associated to the equation

    $x_{2}=\sin(x_{1}),$

is provided by Figures 1 and 2.

## Decomposition

For more complex constraints, a decomposition into atomic constraints (i.e., constraints for which an atomic contractor exists) should be performed. Consider for instance the constraint

    $x+\sin(xy)\leq 0,$

could be decomposed into

    $a=xy$
    $b=\sin(a)$
    $c=x+b.$

The interval domains that should be associated to the new intermediate variables are

    $a\in [-\infty ,\infty ],$
    $b\in [-1,1],$
    $c\in [-\infty ,0].$

## Propagation

The principle of the interval propagation is to call all available atomic contractors until no more contraction could be observed.

As a result of the Knaster-Tarski theorem, the procedure always converges to intervals which enclose all feasible values for the variables. A formalization of the interval propagation can be made thanks to the contractor algebra. Interval propagation converges quickly to the result and can deal with problems involving several hundred of variables.

## Example

Consider the electronic circuit of Figure 3.

Assume that from different measurements, we know that

    $E\in [23V,26V]$
    $I\in [4A,8A]$
    $U_{1}\in [10V,11V]$
    $U_{2}\in [14V,17V]$
    $P\in [124W,130W]$
    $R_{1}\in [0\Omega ,\infty ]$
    $R_{2}\in [0\Omega ,\infty ].$

From the circuit, we have the following equations

    $P=EI$
    $U_{1}=R_{1}I$
    $U_{2}=R_{2}I$
    $E=U_{1}+U_{2}.$

After performing the interval propagation, we get

    $E\in [24V,26V]$
    $I\in [4.769A,5.417A]$
    $U_{1}\in [10V,11V]$
    $U_{2}\in [14V,16V]$
    $P\in [124W,130W]$
    $R_{1}\in [1.846\Omega ,2.307\Omega ]$
    $R_{2}\in [2.584\Omega ,3.355\Omega ].$
