Lessons in uncertainty quantification for turbulent dynamical systems
Andrew J. Majda - Department of Mathematics and Center for Atmosphere and Ocean Science, Courant Institute of Mathematical Sciences, New York University, United States (email)
Abstract: The modus operandi of modern applied mathematics in developing very recent mathematical strategies for uncertainty quantification in partially observed high-dimensional turbulent dynamical systems is emphasized here. The approach involves the synergy of rigorous mathematical guidelines with a suite of physically relevant and progressively more complex test models which are mathematically tractable while possessing such important features as the two-way coupling between the resolved dynamics and the turbulent fluxes, intermittency and positive Lyapunov exponents, eddy diffusivity parameterization and turbulent spectra. A large number of new theoretical and computational phenomena which arise in the emerging statistical-stochastic framework for quantifying and mitigating model error in imperfect predictions, such as the existence of information barriers to model improvement, are developed and reviewed here with the intention to introduce mathematicians, applied mathematicians, and scientists to these remarkable emerging topics with increasing practical importance.
Keywords: Uncertainty quantification, information theory, information barriers, model error, prediction, stochastic PDE's, turbulent systems, fluctuation-dissipation theorems.
Received: January 2012; Revised: March 2012; Published: April 2012.
2011 Impact Factor.913