A Modern Retrospective on Probabilistic Numerics
This is an incremental retrospective that synthesizes existing knowledge for researchers in mathematical statistics and numerical analysis.
The paper reviews the historical development of probabilistic numerics as a research field, tracing its origins to the 1960s and assessing its current state and future challenges.
This article attempts to place the emergence of probabilistic numerics as a mathematical-statistical research field within its historical context and to explore how its gradual development can be related both to applications and to a modern formal treatment. We highlight in particular the parallel contributions of Sul'din and Larkin in the 1960s and how their pioneering early ideas have reached a degree of maturity in the intervening period, mediated by paradigms such as average-case analysis and information-based complexity. We provide a subjective assessment of the state of research in probabilistic numerics and highlight some difficulties to be addressed by future works.