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}. 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, for example, an inequality, describing the set Y. 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, for example, 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 , we perform the following tests: To check the two first tests, we need an interval extension (or an inclusion function) 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. The set X = f−1() where f(x1, x2) = x21 + x22 is represented on the figure. For instance, since 2 + 2 = + = does not intersect the interval , we conclude that the box × is outside X. Since 2 + 2 = + = is inside , we conclude that the whole box × is inside X. 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..