Interval propagation

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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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

  1. ^ Jaulin, L.; Braems, I.; Walter, E. (2002). Interval methods for nonlinear identification and robust control. 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. In Proceedings of CP 2006.