Set inversion

In mathematics, set inversion is the problem of characterizing the preimage X of a set Y by a function f, i.e., X = f⁻¹(Y) = {x ∈ Rⁿ | f(x) ∈ Y}.

In most applications, f is a function from Rⁿ to R^p and the set Y is a box of R^p (i.e. a Cartesian product of p intervals of R).

When f is nonlinear the set inversion problem can be solved using interval analysis combined with a branch and bound algorithm.

The main idea consists in building a paving of R^p made with non-overlapping boxes. For each box [x], we perform the following tests:

1. if f([x]) ⊂ Y we conclude that [x] ⊂ X;
2. if f([x]) ∩ Y = ∅ we conclude that [x] ∩ X = ∅;
3. Otherwise, the box [x] the box is bisected except if its width is smaller than a given precision.

To check the two first tests, we need an interval extension (or an inclusion function) [f] for f. Classified boxes are stored into subpavings, i.e., union of non overlapping boxes.

Example The set X = f⁻¹([4,9]) where f(x₁, x₂) = x₁² + x₂² is represented on the figure. For instance, since [-2,1]²+[4,5]²=[0,4]+[16,25]=[16,29] does not intersect the interval [4,9], we conclude that the box [-2,1]×[4,5] is outside X. Since [-1,1]²+[2,√5]²=[0,1]+[4,5]=[4,6] is inside [4,9], we conclude that the whole box [-1,1]×[2,√5] is inside X.

A ring defined as a set inversion problem

Application Set inversion is mainly used for path planning, for nonlinear parameter set estimation , for localization or for the characterization of stability domains of linear dynamical systems. .