The idea is simple. When you do some interval computations, you return an interval [a, b] such that "real (exact) interval value" of an expression [ar, br] is guaranteed to be included in [a, b]. Sometimes when we have good luck [a, b] = [ar, br]; sometimes [a, b] is very pessimistic, but refinement is a very time consuming problem.
Suppose we reverse the problem a little bit, and look for an "inner" interval value, i.e. such an interval value [ai, bi] of the same expression which is
The point is that [ai, bi] is as easy to compute as [a, b] (much easier than exact interval [ar, br]), and having both [a, b] and [ai, bi] it is sometimes an easy task to get a good refinement for [ar, br].
Have you ever heard something like this? I would appreciate any information or references.
You explain it very clearly. I had heard of it before but never really used it.
I can point you to authors, but not to specific papers.
Martin Berz (firstname.lastname@example.org ) may have used similar ideas, too.
Yes, these are the following papers:
S.P. Shary, Algebraic approach to the interval linear static identification, tolerance and control problems, or One more application of Kaucher arithmetic, Reliable Computing, Vol. 2 (1996), # 1, pp. 3--33.
S.P. Shary, Algebraic solutions to interval linear equations and their applications, in: Numerical Methods and Error Bounds, G. Alefeld and J. Herzberger, eds. Berlin, Akademie Verlag, 1996, pp.224--233.
S.P. Shary, A new approach to the analysis of static systems under interval uncertainty, in: Scientific Computing and Validated Numerics, G. Alefeld, A. Frommer, and B. Lang, eds., Berlin: Akademie Verlag, 1996, pp. 118--132.
In the last paper, I attempted to generalize my approach to a subclass of nonlinear algebraic systems as well.
For interval linear systems my software is "public domain".
I agree with all explanations. Inner interval operations are part of interval arithmetic. For instance, inner subtraction X = A-^-B is either the solution X of B + X = A, or MINUS the solution X of A - X = B, whichever has a solution). Another definition of inner subtraction is : Y = A-_i B is the solution of B + X = A or the solution of A - X = B. In this form it is a special case of the Minkowski subtraction (introduced by Hadwiger for convex bodies) or of so-called Hukuhara difference. Indeed, inner operations are important for finding tight ranges (inner but also outer). In fact many authors use them tacitly (instead of A-^-B one can say "the solution of B + X = A"). There exists a vast literature on related topics.
I can give more details if I know what is needed.
If you have a question related to validated computing, interval analysis, or related matters, I recommend