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 }. It can also be viewed as the problem of describing the solution set of the quantified constraint "Y(f(x))", where Y(y) is a constraint, e.g. an inequality, describing the set Y.
In most applications, f is a function from Rn to Rp and the set Y is a box of Rp (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 Rp made with non-overlapping boxes. For each box [x], we perform the following tests: if f([x]) ⊂ Y we conclude that [x] ⊂ X; if f([x]) ∩ Y = ∅ we conclude that [x] ∩ X = ∅; 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. The algorithm can be made more efficient by replacing the inclusion tests by contractors.
Example
The set X = f−1([4,9]) where f(x1, x2) = x + x 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,]2 = [0,1] + [4,5] = [4,6] is inside [4,9], we conclude that the whole box [−1,1] × [2,] is inside X.
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.
References