On a Theoretical Justification of the Choice of Epsilon-Inflation in PASCAL-XSC

Abstract

In many interval computation methods, if we cannot guarantee a solution within a given interval, it often makes sense to "inflate" this interval a little bit. There exist many different "inflation" methods. The authors of PASCAL-XSC, after empirically comparing the behavior of different inflation methods, decided to implement the formula [x-,x+]ε = [(1 + ε)x- - ε · x+, (1 + ε)x+ - ε · x-]. A natural question is: Is this choice really optimal (in some reasonable sense), or is it only an empirical approximation to the truly optimal choice?

In this paper, we show that this empirical choice can be theoretically justified. Namely, we will give two justifications:

First, the inflation method used in PASCAL-XSC is the only inflation that is invariant w.r.t. some reasonable symmetries; and

Second, that this inflation method is optimal in some reasonable sense.