Subpaving

In mathematics, a subpaving is a set of nonoverlapping boxes of Rn. A subset X of Rn can be approximated by two subpavings X− and X+ such that X− ⊂ X ⊂ X+. The three figures on the right show an approximation of the set X = {(x1, x2) ∈ R2 | x12 + x22 + sin(x1 + x2) ∈ [4,9]} with different accuracies. The set X− corresponds to red boxes and the set X+ contains all red and yellow boxes.

Subpavings which bracket a set with a low resolution
Subpavings which bracket the same set with a middle resolution
Subpavings which bracket the set with a high resolution

Combined with interval-based methods, subpavings are used to approximate the solution set of non-linear problems such as set inversion problems. [1] Subpavings can also be used to prove that a set defined by nonlinear inequalities is path connected [2] , to provide topological properties of such sets [3] , to solve piano-mover's problems [4] or to implement set computation [5] .

References
  1. Jaulin, Luc; Walter, Eric (1993). "Set inversion via interval analysis for nonlinear bounded-error estimation" (PDF). Automatica. 29 (4): 1053–1064. doi:10.1016/0005-1098(93)90106-4.
  2. Delanoue, N.; Jaulin, L.; Cottenceau, B. (2005). "Using interval arithmetic to prove that a set is path-connected" (PDF). Theoretical Computer Science. 351 (1).
  3. Delanoue, N.; Jaulin, L.; Cottenceau, B. (2006). "Counting the Number of Connected Components of a Set and Its Application to Robotics" (PDF). Applied Parallel Computing, Lecture Notes in Computer Science. Lecture Notes in Computer Science. 3732 (1): 93–101. doi:10.1007/11558958_11. ISBN 978-3-540-29067-4.
  4. Jaulin, L. (2001). "Path planning using intervals and graphs" (PDF). Reliable Computing. 7 (1).
  5. Kieffer, M.; Jaulin, L.; Braems, I.; Walter, E. (2001). "Guaranteed set computation with subpavings" (PDF). In W. Kraemer and J. W. Gudenberg (Eds), Scientific Computing, Validated Numerics, Interval Methods, Kluwer Academic Publishers: 167–178. doi:10.1007/978-1-4757-6484-0_14. ISBN 978-1-4419-3376-8.
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