Interval Superposition Arithmetic
For researchers in numerical computation and verified methods, this provides a more efficient and accurate approach to bounding function images, though it is an incremental improvement over existing interval arithmetics.
This paper introduces interval superposition arithmetic, a new set-based method for enclosing the image set of multivariate factorable functions. It achieves linear runtime complexity in the number of variables and demonstrates superior accuracy over existing set arithmetics in numerical examples.
This paper presents a novel set-based computing method, called interval superposition arithmetic, for enclosing the image set of multivariate factorable functions on a given domain. In order to construct such enclosures, the proposed arithmetic operates over interval superposition models which are parameterized by a matrix with interval components. Every point in the domain of a factorable function is then associated with a sequence of components of this matrix and the superposition, i.e. Minkowski sum, of these elements encloses the image of the function at this point. Interval superposition arithmetic has a linear runtime complexity with respect to the number of variables. Besides presenting a detailed theoretical analysis of the accuracy and convergence properties of interval superposition arithmetic, the paper illustrates its advantages compared to existing set arithmetics via numerical examples.