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, 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: 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. The algorithm can be made more efficient by replacing the inclusion tests by contractors. (Wikipedia).
