Limitations

Interval Arithmetic has some major limitations that one must be aware of. You can find a detailed analysis of its shortcomings in many places on the web (see 1, 2). The following gives a brief summary of the most common pitfalls and how to potentially address them.

Dependency problem

Different expressions of the same function, e.g.,

$f(x)=x^2 - 4x = (x-2)^2 - 4 = x(x-4)$

may result in intervals of varying tightness. This is because the dependency that the two instantiations of x must have the same value is not accurately tracked.

By-hand function rewriting.

As a rule of thumb, the fewer instances of a variable in an expression, the tighter the resulting interval will end up being.

In this example, $(x-2)^2 - 4$ results in the tightest output interval.

Not all such dependency problems can be resolved using symbolic rewriting, however. If such scenarios lead to unsatisfactory results, a different arithmetic like Affine Arithmetic should probably be prefered. Sometimes, through clever algorithmic adjustments or constraint propagation the intervals can still lead to useful bounds. Also, if only the relative width between different intervals is of interest (e.g., in verified neural network optimization), pure interval arithmetic can still be surprisingly effective (see here).

Wrapping Effect

In iterative solvers the interval width can increase in each iteration resulting in a blow-up.

Relaxations of fixed-point iterations and implicit functions.

Special purpose methods have been designed over the years to address this. See, 1, 2.

Relaxations of ODE integrators.

Special purpose methods have been designed over the years to address this. See, 1, 2, 3.

Hardware limitations

The hardware might not support rounded intrinsic operations.

CUDA supporting hardware

CUDA luckily has rounded intrinsics for the basic operations (add, sub, mul, div, sqrt), including variants for rounding towards ±∞. However, operations like exp, sin, cos, etc. often incur a couple ulp error which leads to further rounding. For more details see: On rounding of intervals.