Probabilistic Numerics and Uncertainty in Computations

TL;DR

This paper advocates for probabilistic numerical methods that quantify uncertainty in computational tasks, enhancing decision-making in science and industry by interpreting classical algorithms as probabilistic inference and developing new, adaptable algorithms.

Contribution

It introduces a probabilistic framework for numerical methods, offering new algorithms with improved performance and uncertainty quantification, applicable to complex scientific problems.

Findings

Probabilistic interpretation of classical numerical algorithms.

New algorithms that adapt to specific application needs.

Demonstrated benefits on real scientific problems.

Abstract

We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such uncertainties, arising from the loss of precision induced by numerical calculation with limited time or hardware, are important for much contemporary science and industry. Within applications such as climate science and astrophysics, the need to make decisions on the basis of computations with large and complex data has led to a renewed focus on the management of numerical uncertainty. We describe how several seminal classic numerical methods can be interpreted naturally as probabilistic inference. We then show that the probabilistic view suggests new algorithms that can flexibly be adapted to suit application specifics, while delivering improved empirical performance. We provide concrete illustrations of the benefits of probabilistic numeric algorithms on real scientific problems from astrometry and astronomical imaging, while highlighting open problems with these new algorithms. Finally, we describe how probabilistic numerical methods provide a coherent framework for identifying the uncertainty in calculations performed with a combination of numerical algorithms (e.g. both numerical optimisers and differential equation solvers), potentially allowing the diagnosis (and control) of error sources in computations.