Numerical Reproducibility and Parallel Computations: Issues for Interval Algorithms

What is called "numerical reproducibility" is the problem of getting the same result when the scientific computation is run several times, either on the same machine or on different machines, with different types and numbers of processing units, execution environments, computational loads etc. This problem is especially stringent for HPC numerical simulations. In what follows, the focus is on parallel implementations of interval arithmetic using floating-point arithmetic. For interval computations, numerical reproducibility is of course an issue for testing and debugging purposes. However, as long as the computed result encloses the exact and unknown result, the inclusion property, which is the main property of interval arithmetic, is satisfied and getting bit for bit identical results may not be crucial. Still, implementation issues may invalidate the inclusion property. Several ways to preserve the inclusion property are presented, on the example of the product of matrices with interval coefficients.