Set inversion

Set inversion is the problem of characterizing the reciprocical image 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.

The main idea consists in building a paving of ℝ^p made with nonoverlapping boxes. For each box [x], we perform the following tests: (i) if f([x]) ⊂ Y we conclude that [x] ⊂ X; (ii) if f([x]) ∩ Y = ∅ we conclude that [x] ∩ X = ∅; (iii) Otherwise, [x] the box is bissected.