# 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 is can be used to propagate uncertainties in the situation where errors are represented by intervals .[1] Interval propagation considers an estimation problem as a constraint satisfaction problem.

## Atomic contractors

A contractor associated to an equation involving the variables x1,...,xn is an operator which contracts the intervals [x1],..., [xn] (that are supposed to enclose the xi'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 x1,x2 and x3.

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

$x_3=x_1+x_2 \Rightarrow x_3 \in [6,\infty ] \cap ([-\infty,5]+[-\infty ,4]) =[6,\infty ] \cap [-\infty ,9]=[6,9].$
$x_1=x_3-x_2 \Rightarrow x_1 \in [-\infty ,5]\cap ([6,\infty]-[-\infty ,4]) =[-\infty ,5] \cap [2,\infty ]=[2,5].$
$x_2=x_3-x_1 \Rightarrow x_2 \in [-\infty ,4]\cap ([6,\infty]-[-\infty ,5]) = [-\infty ,4] \cap [1,\infty ]=[1,4].$
Figure 1: boxes before contraction
Figure 2: boxes after contraction

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

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

## Propagation

The principle of the interval propagation is to call all available atomic contractors until no more contraction could be observed. [2] 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. [3]

## Example

Consider the electronic circuit of Figure 3.

Figure 3: File:Electronic circuit to illustrate the interval propagation

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

$E \in [24V,26V]$
$I \in [4.769A,5.417A]$
$U_1 \in [10V,11V]$
$U_2 \in [14V,16V]$
$P \in [124W,130W]$
$R_{1} \in [1.846 \Omega,2.307 \Omega]$
$R_{2}\in [2.584 \Omega,3.355 \Omega].$

## References

1. ^ Jaulin, L.; Braems, I.; Walter, E. (2002). Interval methods for nonlinear identification and robust control (PDF). In Proceedings of the 41st IEEE Conference on Decision and Control (CDC).
2. ^ Cleary, J.L. (1987). Logical arithmetic. Future Computing Systems.
3. ^ Jaulin, L. (2006). Localization of an underwater robot using interval constraints propagation (PDF). In Proceedings of CP 2006.