The game theoretic approach to Uncertainty Quantification, reduced order modeling and numerical analysis

Abstract

We discuss the development of Uncertainty Quantification framework founded upon a combination of game/decision theory and information based complexity. We suggest that such a framework could be used not only to guide decisions in presence of epistemic uncertainties and complexity management capabilities constraints but also to automate the process of discovery in (1) model form uncertainty quantification and design (2) model reduction (3) the design of fast, robust and scalable numerical solvers. Although these applications appear dissimilar, they are all based on the efficient processing of incomplete information with limited computational resources: (1) model form UQ and design require the management and processing of epistemic uncertainties and limited data (2) model reduction requires the approximation of the full state of a complex system through operations performed on a few (coarse/reduced) variables (3) fast and robust computation requires computation with partial information. The core idea of the proposed framework is to reformulate the process of computing with partial information and limited resources as that of playing underlying hierarchies of adversarial information games characterizing the adversarial and nested processing of hierarchies of partial/missing information.

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