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.
Highlights
The ‘inevitable reality’ when it comes to predicting the dynamical behavior of turbulent, high-dimensional systems from nature is that the employed mathematical and numerical models need to properly account for propagation of uncertainty arising due to the limited understanding and partial observations of the true dynamics
There is much current activity in disparate areas of mathematics, statistics, engineering and computer science leading to ideas and techniques which are relevant for Uncertainty Quantification (UQ) in dynamical systems; examples range from Bayesian hierarchical space-time models (e.g., [121]) to variational approximations (e.g., [118, 53]), 2000 Mathematics Subject Classification
Intermittency and fat-tailed probability densities are abundant in the inertial and dissipation range of stochastic turbulence models (e.g., [19]) and we show that in such important dynamical regimes Polynomial Chaos Expansions (PCE) performs, at best, to the simple Gaussian moment closure technique utilized earlier in a different context for UQ within a framework of Empirical Information Theory [11]
Summary
The ‘inevitable reality’ when it comes to predicting the dynamical behavior of turbulent, high-dimensional systems from nature is that the employed mathematical and numerical models need to properly account for propagation of uncertainty arising due to the limited understanding and partial observations of the true dynamics. It is clear that UQ techniques for partially observed high-dimensional turbulent dynamical systems require a synergistic framework capable of quantifying the model error and systematic model improvement leading to reliable predictions for a subset of coarse-grained variables of the perfect system In this context basic questions and new issues arise such as the following:. The main goal of this research expository paper is to describe these recent and ongoing developments emphasizing the remarkable new mathematical and physical phenomena that emerge from the modern applied mathematics modus operandi applied to uncertainty quantification in partially observable high-dimensional dynamical systems The use of these ideas in applied mathematics and numerical analysis for quantifying model error, as well as in climate change science and various engineering applications for improving imperfect predictions, is only beginning but already the wealth of new important concepts and approaches warrants a detailed treatment.
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