Probabilistic interval-valued computation: toward a practical surrogate for statistics inside CAD tools

Interval methods offer a general, fine-grain strategy for modeling correlated range uncertainties in numerical algorithms. We present a new, improved interval algebra that extends the classical affine form to a more rigorous statistical foundation. Range uncertainties now take the form of confidence intervals. In place of pessimistic interval bounds, we minimize the probability of numerical "escape"; this can tighten interval bounds by 10times, while yielding 10-100times speedups over Monte Carlo. The formulation relies on three critical ideas: liberating the affine model from the assumption of symmetric intervals; a unifying optimization formulation; and a concrete probabilistic model. We refer to these as probabilistic intervals, for brevity. Our goal is to understand where we might use these as a surrogate for expensive, explicit statistical computations. Results from sparse matrices and graph delay algorithms demonstrate the utility of the approach, and the remaining challenges