ProbNum UNS

Probabilistic Numerics under Non-Stationarity

01/03/2025 to 28/02/2027

Probability theory provides a mathematical language for decision-making and uncertainty, while the aim of numerical analysis is to design and study approximations to mathematical problems such as integration, optimisation and solving differential equations and linear systems. In probabilistic numerics (PN) the problems of numerical analysis are interpreted as statistical inference tasks to which the full power of modern Bayesian statistics can be brought to bear. A great advantage of this paradigm is that a PN method outputs a full probability distribution over possible solutions of a numerical problem rather than a single point estimate only. This distribution represents epistemic uncertainty inherent to the approximation and can be meaningfully propagated in computational pipelines. While the history of PN can be traced to the 1970s [Lar72] and later, the field has undergone a remarkable resurgence within the past decade due to challenges posed by modern scientific computing and artificial intelligence (AI), in large part due to the efforts by one of the PIs [Hen22].
We will develop new PN methods for optimisation and integration that (a) exploit and adapt to non- stationarity, yet remain (b) computationally scalable. The methods will be accompanied with rigorous error and uncertainty quantification (UQ) guarantees that take model misspecification into account.